Bedingt durch das breite Aufgabenspektrum, eignet sich dieses Lösungsbuch für eine Vielzahl von Studiengängen. Neben den Studierenden aus den Ingenieurstudiengängen, profitieren auch in besonderer Weise Mathematik- und Lehramtsstudierende von der Aufgabenvielfalt.},
address = {Berlin-Heidelberg},
author = {Merz, Wilhelm and Knabner, Peter},
doi = {10.1007/978-3-642-54529-0},
faupublication = {yes},
isbn = {978-3-642-54528-3},
peerreviewed = {unknown},
publisher = {Springer},
series = {Springer-Lehrbuch},
title = {{Endlich} gelöst! {Aufgaben} zur {Mathematik} für {Ingenieure} und {Naturwissenschaftler}},
url = {http://www.springer.com/us/book/9783642545283},
year = {2014}
}
@article{faucris.215928880,
abstract = {We consider a system of non-linear reaction diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order is an element of, and the equations inside the layer depend on the parameter is an element of and an additional parameter gamma is an element of[-1, 1), which describes the size of the diffusion in the layer. We derive effective models for the limit is an element of -> 0, when the layer reduces to an interface Sigma between the two bulk domains. The effective solution is continuous across Sigma for all gamma is an element of [-1, 1). For gamma is an element of(-1, 1), the jump in the normal flux is given by a non-linear ordinary differential equation on Sigma. In the critical case gamma = -1, a dynamic transmission condition of Wentzell-type arises at the interface Sigma.},
author = {Gahn, Markus and Neuss-Radu, Maria and Knabner, Peter},
doi = {10.3934/dcdss.2017039},
faupublication = {yes},
journal = {Discrete and Continuous Dynamical Systems},
keywords = {Thin heterogeneous layer;homogenization;weak and strong two-scale convergence;non-linear reaction-diffusion systems;effective transmission conditions},
pages = {773-797},
peerreviewed = {Yes},
title = {{DERIVATION} {OF} {EFFECTIVE} {TRANSMISSION} {CONDITIONS} {FOR} {DOMAINS} {SEPARATED} {BY} {A} {MEMBRANE} {FOR} {DIFFERENT} {SCALING} {OF} {MEMBRANE} {DIFFUSIVITY}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2017/2017_GahnNeussraduKn_DerivOfEffectTransmissConditForDomainSeparByMembraneForDiffScalOfMembrDiffusivity},
volume = {10},
year = {2017}
}
@inproceedings{faucris.121978604,
abstract = {Detailed modelling of reactive transport processes in the underground often requires the consideration of a wide range of reactive species. A prominent example is natural attenuation, that is the assessment and monitoring of microbially catalysed degradation processes of organic contaminants in the subsoil or aquifer with full geochemistry. Often the reactions exhibit a wide range of relaxation times, which advises to model those reactions being much faster than the time scale of the transport processes in a quasistationary manner, e.g. as (algebraicly described) equilibrium processes. Additionally not only mobile species (in solution) appear, but also immobile ones (attached to the porous skeleton). In summary, the resulting system is not semilinear and parabolic, but rather quasilinear and couples partial differential equations (pde), ordinary differential equations and algebraic equations. An often used approach is operator splitting, in which transport and reaction becomes (iteratively) decoupled. This procedure either introduces a further consistency error (in the non-interative version) which can only be controlled by the time stepping, or applies a fixed point type iteration of unclear convergence properties. We rather propose, after appropriate (mixed) finite element discretization, to deal with the full discrete nonlinear system (the Global Implicit Approach,in principal by a damped Newton's method). To make the problem still feasible we advise two means: The first is concerned with the continuous model and aims at a transformation of the dependent variables such that as many as possible are determined by decoupled linear pde's or by local algebraic relations, leading to a smaller coupled system. The problem lies here in the combined appearance of kinetics and equilibrium and mobile and immobile species. Alternatively to this exact a priori decoupling we use an a posteriori decoupling on the level of the linear system of equation in the Newton's method by sparsening, i.e. ignoring weak couplings in the Jacobian matrix. The resulting benefit in the solution of the linear system should supersede a possible deterioration in the convergence of the iterative method, being now only an approximate Newtons's method. Iterative operator splitting can be viewed as of this type with two half-steps, first ignoring e.g. the reactive couplings (the transport step) and then ignoring the transport couplings (the reaction step). The introduced larger flexibility of the method helps to find selective couplings where there is still a good (Newton-like) convergence in cases where operator splitting fails. The benefit of the two approaches, which in principle are also combinable, will be elucidated for several large problems from the hydrological literatur},
author = {Kräutle, Serge and Knabner, Peter and Hoffmann, Joachim},
booktitle = {International Conference on Computational Methods in Water Resources},
doi = {10.4122/1.1000000224},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Efficient} and {Accurate} {Simulation} of {Large} {General} {Reactive} {Multicomponent} {Transport} {Processes} in {Porous} {Media} by {Model}-{Preserving} {Decoupling} {Techniques}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2006/2006_KraeutleKnHoffmann_EfficAccurSimulOfLargeGenerReactiveMulticompTranspProcessInPMbyModelPreservDecoupTechn},
venue = {Kopenhagen},
volume = {XVI},
year = {2006}
}
@article{faucris.204341533,
abstract = {We consider a macroscopic (averaged) model of transport and reaction in the porous subsurface. The model consists of PDEs for the concentrations of the mobile (dissolved) species and of ODEs for the immobile (mineral) species. For the reactions, we assume the kinetic mass action law. The constant activity of the mineral species leads to set-valued rate functions or complementarity conditions coupled to the PDEs and ODEs. In this paper we first prove the equivalence of several formulations in a weak sense. Then we prove the existence and the uniqueness of a global solution for a multispecies multireaction setting with the method of a priori estimates. In addition to mineral precipitation-dissolution reactions, the model also allows for aquatic reactions, i.e., reactions among the mobile species. In both the mineral precipitation-dissolution rates and the aquatic reaction rates we consider polynomial nonlinearities of arbitrarily high order.},
author = {Hoffmann, Joachim and Kräutle, Serge and Knabner, Peter},
doi = {10.1137/16M1109266},
faupublication = {yes},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {existence of global solution;reactive transport in porous media;kinetic mineral precipitation-dissolution;complementarity problems;law of mass action},
month = {Jan},
pages = {4812-4837},
peerreviewed = {Yes},
title = {{Existence} and uniqueness of a global solution for reactive transport with mineral precipitation-dissolution and aquatic reactions in porous media},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2017/2017_HoffmannKraeutleKn_ExistenceAndUniquenessOfGlobalSolutionForReactiveTransWithMineralPreDissAndAquaReactInPM},
volume = {49},
year = {2017}
}
@masterthesis{faucris.122809764,
author = {Kräutle, Serge},
faupublication = {yes},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Approximationen} der {Navier}-{Stokes}-{Gleichungen} mit {Finiten} {Differenzen}},
year = {1996}
}
@article{faucris.214978728,
abstract = {We consider a family of variational regularization functionals for a generic inverse problem, where the data fidelity and regularization term are given by powers of a Hilbert norm and an absolutely one-homogeneous functional, respectively. We investigate the small and large time behavior of the associated solution paths and, in particular, prove finite extinction time for a large class of functionals. Depending on the powers, we also show that the solution paths are of bounded variation or even Lipschitz continuous. In addition, it will turn out that the models are" almost" mutually equivalent in terms of the minimizers they admit. Finally, we apply our results to define and compare two different non-linear spectral representations of data and show that only one of it is able to decompose a linear combination of non-linear eigenfunctions into the individual eigenfunctions. For that purpose, we will also briefly address piecewise affine solution paths.

phytoplankton, and zooplankton (NPZ) mode},
author = {Hajduk, Hennes and Hodges, Ben R and Aizinger, Vadym and Reuter, Balthasar},
doi = {10.1016/j.envsoft.2018.01.003},
faupublication = {yes},
journal = {ENVIRON MODELL SOFTW },
keywords = {Discontinuous Galerkin finite element method; Multi-component advection-reaction system; Transport scheme; NPZ model; Shallow water equations},
pages = {185-198},
peerreviewed = {Yes},
title = {{Locally} {Filtered} {Transport} for computational efficiency in multi-component advection-reaction models},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_HajdukHodgesAizingerReuter_LocallyFilteredTransport.pdf},
volume = {102},
year = {2018}
}
@article{faucris.109102004,
abstract = {

In this paper, a stochastic model for the turbulent fiber lay-down in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corresponding to large noise values, the transient and stationary joint probability distributions are determined using the method of multiple scales and the Chapman–Enskog method. Moreover, exponential convergence towards the stationary solution is proven for the reduced problem. For special choices of the industrial parameters, the stochastic limit process is an Ornstein–Uhlenbeck process. It is a good approximation of the fiber motion even for moderate noise values. Moreover, as shown by Monte-Carlo simulations, the limiting process can be used to assess the quality of nonwoven materials in the industrial application by determining distributions of functionals of the process.

}, author = {Bonilla, Luis L. and Götz, Thomas and Klar, Axel and Marheineke, Nicole and Wegener, Raimund}, doi = {10.1137/070692728}, faupublication = {no}, journal = {SIAM Journal on Applied Mathematics}, pages = {648-665}, peerreviewed = {Yes}, title = {{Hydrodynamic} limit for the {Fokker}-{Planck} equation describing fiber lay-down models}, volume = {68}, year = {2007} } @article{faucris.117790464, abstract = {The authors study concentration profiles of solutes undergoing equilibrium absorption in the vicinity of a water well. For the case of a contamination event, the limit problem of vanishing well radius, which is of self-similar nature, is analysed in detail. Existence, uniqueness, and qualitative properties of solutions of the corresponding ordinary differential equations are shown. Some numerical examples are also presented. © 1994 Oxford University Pres}, author = {Van Duijn, C. J. (Hans) and Knabner, Peter}, doi = {10.1093/imamat/52.2.177}, faupublication = {yes}, journal = {IMA Journal of Applied Mathematics}, pages = {177-200}, peerreviewed = {Yes}, title = {{Flow} and reactive transport in porous media induced by well injection: {Similarity} solution}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1994/1994_DuijnKn_FlwReactTrnspInPMInducedByWellInjectSimSol}, volume = {52}, year = {1994} } @book{faucris.120697324, abstract = {This book contains the proceedings of the 17th European Conference on Mathematics for Industry, ECMI2012, held in Lund, Sweden, July 2012, at which ECMI celebrated its 25th anniversary. It covers mathematics in a wide range of applications and methods, from circuit and electromagnetic devices, environment, fibers, flow, medicine, robotics and automotive industry, further applications to methods and education.

The book includes contributions from leading figures in business, science and academia that promote the application of mathematics to industry and emphasize industrial sectors that offer the most exciting opportunities. The contributions reinforce the role of mathematics as being a catalyst for innovation as well as an overarching resource for industry and business.

The book features an accessible presentation of real-world problems in industry and finance, provides insight and tools for engineers and scientists who will help them to solve similar problems and offers modeling and simulation techniques that will provide mathematicians with a source of fresh ideas and inspiration.},
doi = {10.1007/978-3-319-05365-3},
edition = {1},
editor = {Fontes, Magnus and Günther, Michael and Marheineke, Nicole},
faupublication = {yes},
isbn = {978-3-319-05365-3},
peerreviewed = {automatic},
publisher = {Springer},
series = {The European Consortium for Mathematics in Industry},
title = {{Progress} in {Industrial} {Mathematics} at {ECMI} 2012},
volume = {19},
year = {2014}
}
@inproceedings{faucris.208634344,
abstract = {We report on the parallelization of the library FEMLISP (http://www.femlisp.org), which is a Common Lisp framework for solving partial differential equations using the finite element method.

p <2) or infinite extinction time (*p*≥2). We give upper bounds for the finite extinction time and establish convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and nonlocal versions of PDEs like *p*-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting.

We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.

t−(a(x)u_{x})_{x}=0, x∈(0, l), t∈.(0, T], u(0, t)=ϕ(t), u_{x}(0,t)=0, 0≦t≦T, assuming only L^{∞}
for a. In the case of weak a priori bounds on u, we derive stability
estimates on u of Hölder type in the interior and of logarithmic type at
the boundary. Also the continuous dependence on a is considere},
author = {Knabner, Peter and Vessella, Sergio},
doi = {10.1007/BF01773944},
faupublication = {no},
journal = {Annali Di Matematica Pura Ed Applicata},
keywords = {Cauchy Problem, Parabolic Equation, Stability Estimate, Continuous Dependence, Logarithmic Type},
pages = {393-409},
peerreviewed = {Yes},
title = {{Stabilization} of {Ill}-{Posed} {Cauchy} {Problems} for {Parabolic} {Equations}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1987/1987_KnVessella_StabilizationOfIllPosedCauchyProblemsForParaEquation},
volume = {149},
year = {1987}
}
@incollection{faucris.118079764,
address = {Frankfurt a. M.},
author = {Knabner, Peter and Kräutle, Serge and Oßmann, Stephan and Prechtel, Alexander},
booktitle = {Statusseminar Forschungsverbund KORA - Kontrollierter natürlicher Rückhalt und Abbau von Schadstoffen bei der Sanierung kontaminierter Grundwässer und Böden},
editor = {Dechema},
faupublication = {yes},
isbn = {389746071X},
pages = {417-427},
peerreviewed = {unknown},
title = {{Modellierung} des reaktiven {Transports} von {Schadstoffen} in der (un-)gesättigten {Bodenzone} zur {Prognose} der natürlichen {Selbstreinigung} - {Komplexe} {Probleme} berechenbar machen},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2005/2005_KnKraeutleOssmannPrechtel_Proj73ModellDesReaktTranspVonSchadstInDer(un)gesättBodenzonePrognSelbstreinig},
year = {2005}
}
@article{faucris.204144050,
author = {Rupp, Andreas and Knabner, Peter},
doi = {10.1002/num.22150},
faupublication = {yes},
journal = {Numerical Methods For Partial Differential Equations},
keywords = {instationary Darcy problem; error analysis; stability analysis; local discontinuous Galerkin method},
pages = {1374-1394},
peerreviewed = {Yes},
title = {{Convergence} order estimates of the local discontinuous {Galerkin} method for instationary {Darcy} flow},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2017/2017_RuppKn_ConvergenceOrderEstimatesOfTheLocalDisGMForInstDarcyFlow},
volume = {33},
year = {2017}
}
@incollection{faucris.121996864,
author = {Cibis, Thomas and Marheineke, Nicole},
booktitle = {Progress in Industrial Mathematics at ECMI 2012},
faupublication = {yes},
pages = {237-243},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Model}-based medical decision support for glucose balance in {ICU} patients: {Optimization} and analysis},
year = {2014}
}
@article{faucris.204147247,
abstract = {Using the parallel/orthogonal space method, we calculate the planar two-loop three-point diagram and two rotated reduced planar two-loop three-point diagrams. Together with the crossed topology, these diagrams are the most complicated ones in the two-loop corrections necessary, for instance, for the decay of the Z(0) boson. Instead of calculating particular decay processes, we present a new algorithm which allows us to perform arbitrary next-to-next-to-leading order (NNLO) calculations for massive planar two-loop vertex functions in the general mass case. All integration steps up to the last two are performed analytically and will be implemented under XLOOPS as part of the Mainz XLOOPS-GiNaC project. The last two integrations are done numerically using methods like VEGAS and Divonne. Thresholds originating from Landau singularities are found and discussed in detail. In order to demonstrate the numerical stability of our methods we consider particular Feynman integrals which contribute to different physical processes. Our results can be generalized to the case of the crossed topology.},
author = {Knodel, Markus and et al.},
author_hint = {Groote S, Knodel MM},
doi = {10.1140/epjc/s2006-02478-9},
faupublication = {no},
journal = {European Physical Journal C},
pages = {157-178},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Evaluating} massive planar two-loop tensor vertex integrals},
volume = {46},
year = {2006}
}
@article{faucris.119190104,
author = {Grün, Günther and Eck, Christof and Fontelos, Marco Antonio and Klingbeil, Fabian and Vantzos, Orestis},
faupublication = {yes},
journal = {Proceedings in Applied Mathematics and Mechanics},
pages = {1151205-1151207},
peerreviewed = {Yes},
title = {{A} phase-field model for electrowetting and related phenomena},
volume = {7},
year = {2007}
}
@inproceedings{faucris.120508564,
author = {Knabner, Peter and Igler, B. A.},
booktitle = {ENUMATH 97, World Scientific, Singapore},
editor = {Bock H},
faupublication = {yes},
title = {{Identification} of {Nonlinearities} in {PDEs}: {Sorption} {Isotherms} in {Reactive} {Flow} {Through} {Porous} {Media}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1998/1998_IglerKn_IdentifOfNonlinearitInPDEsSorptIsothermInReactFlowThroughPM},
venue = {Singapore},
year = {1998}
}
@incollection{faucris.116680564,
author = {Lorenz, Maike and Marheineke, Nicole and Wegener, Raimund},
booktitle = {Progress in Industrial Mathematics at ECMI 2012},
faupublication = {yes},
pages = {91-98},
peerreviewed = {unknown},
publisher = {Springer},
title = {{On} viscoelastic fiber spinning: {Die} swell effect in 1d uniaxial {UCM} model},
year = {2014}
}
@article{faucris.117787164,
abstract = {We present adaptive mixed hybrid finite element discretizations of the Richards equation, a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium. The approach simultaneously constructs approximations of the flux and the pressure head in Raviart-Thomas spaces. The resulting nonlinear systems of equations are solved by a Newton method. For the linear problems of the Newton iteration a multigrid algorithm is used. We consider two different kinds of error indicators for space adaptive grid refinement: superconvergence and residual based indicators. They can be calculated easily by means of the available finite element approximations. This seems attractive for computations since no additional (sub-)problems have to be solved. Computational experiments conducted for realistic water table recharge problems illustrate the effectiveness and robustness of the approach. © 2004 Elsevier Ltd. All rights reserve},
author = {Bause, Markus and Knabner, Peter},
doi = {10.1016/j.advwatres.2004.03.005},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {A posteriori error indicator; Mixed finite element method; Nonlinear elliptic-parabolic problem; Raviart-Thomas spaces; Saturated-unsaturated flow},
pages = {565-581},
peerreviewed = {Yes},
title = {{Computation} of variably saturated subsurface flow by adaptive mixed hybrid finite element methods},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_BauseKn_ComputOfVariabSaturSubsurfFlowByAdaptMixHyridFinitElemtMeth},
volume = {27},
year = {2004}
}
@incollection{faucris.116687164,
author = {Arne, Walter and Marheineke, Nicole and Wegener, Raimund},
booktitle = {Progress in Industrial Mathematics at ECMI 2010},
editor = {M. Günther A. Bartel, M. Brunk, S. Schöps, M. Striebel},
faupublication = {yes},
pages = {349-356},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Asymptotic} models of different complexity for viscous jets and their applicability regimes},
year = {2012}
}
@article{faucris.205335697,
abstract = {Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner-and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighboring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is a connected interval. Preliminary results indicate the existence of phase separation, i.e., spatial sorting of the different species. A detailed analysis is given in one spatial dimension. The existence and shape of segregated stationary solutions is shown via the Krein-Rutman theorem. Moreover, for small repulsion/nonlinear diffusion, also uniqueness of these stationary states is proved.},
author = {Burger, Martin and et al.},
author_hint = {Burger M, Di Francesco M, Fagioli S, Stevens A},
doi = {10.1137/17M1125716},
faupublication = {no},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {phase separation;spatial sorting;nonlinear diffusion;long-range attraction;stationary states},
month = {Jan},
pages = {3210-3250},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{SORTING} {PHENOMENA} {IN} {A} {MATHEMATICAL} {MODEL} {FOR} {TWO} {MUTUALLY} {ATTRACTING}/{REPELLING} {SPECIES}},
url = {https://arxiv.org/pdf/1704.04179.pdf},
volume = {50},
year = {2018}
}
@article{faucris.205335160,
abstract = {Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research.In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.},
author = {Burger, Martin and et al.},
author_hint = {Benning M, Burger M},
doi = {10.1017/S0962492918000016},
faupublication = {no},
journal = {Acta Numerica},
month = {Jan},
pages = {1-111},
peerreviewed = {unknown},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Modern} regularization methods for inverse problems},
url = {https://arxiv.org/pdf/1801.09922.pdf},
volume = {27},
year = {2018}
}
@article{faucris.117788264,
abstract = {We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart-Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452-1478, 2004), Schneid et al. (Numer Math 98:353-370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart-Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples. © 2008 Springer-Verla},
author = {Radu, Adrian Florin and Pop, Iuliu Sorin and Knabner, Peter},
doi = {10.1007/s00211-008-0139-9},
faupublication = {yes},
journal = {Numerische Mathematik},
pages = {285-311},
peerreviewed = {Yes},
title = {{Error} estimates for a mixed finite element discretization of some degenerate parabolic equations},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2008/2008_RaduPopKn_ErrorEstimatesForAMixedFiniteElementParabEqua},
volume = {109},
year = {2008}
}
@article{faucris.106667044,
author = {Grün, Günther and Fontelos, Marco Antonio},
faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
pages = {777-788},
peerreviewed = {Yes},
title = {{On} uniqueness results for an elliptic-parabolic system of pde arising in dynamic electrowetting},
volume = {93},
year = {2013}
}
@article{faucris.117787824,
abstract = {Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles describee by Ito equations. We establish consistency conditions relating the concentration statistics to that of the Ito process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwate},
author = {Suciu, Nicolae and Schüler, Lennart and Radu, Adrian Florin and Attinger, Sabine and Vamo̧s, Cǎlin and Knabner, Peter},
doi = {10.1515/auom-2015-0055},
faupublication = {yes},
journal = {Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica},
keywords = {Mixing; PDF methods; Porous media; Random walk},
pages = {187-208},
peerreviewed = {unknown},
title = {{Consistency} issues in pdf methods},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_SuciuSchuelerRaduAttingerVamosKn_ConsistencyIssuesInPDFMethods},
volume = {23},
year = {2015}
}
@article{faucris.204938230,
abstract = {We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from mu CT (micro computed tomography) imaging of porous rock and approximate a (on mu m scale) smooth domain with a certain resolution Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels However, planar surfaces in any other directions and curved surfaces yield a jagged/topologically rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90 degrees, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90 degrees on jagged voxel surfaces is amplified. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method However, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model. (C) 2018 Elsevier Inc. All rights reserved.},
author = {Frank, Florian and Liu, Chen and Scanziani, Alessio and Alpak, Faruk Omer and Rivière, Béatrice},
doi = {10.1016/j.jcis.2018.02.075},
faupublication = {yes},
journal = {Journal of Colloid and Interface Science},
keywords = {Cahn-Hilliard equation;Contact angle;Jagged surface;Topologically rough surface;mu CT imaging;Porous media},
pages = {282–291},
peerreviewed = {Yes},
title = {{An} energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods},
volume = {523},
year = {2018}
}
@incollection{faucris.215927944,
abstract = {It has been shown that hematopoietic stem cells migrate in vitro
and in vivo following the gradient of a chemotactic factor produced by
stroma cells. In this contribution, a quantitative model for this
process is presented. Themodel consists of chemotaxis equations coupled
with an ordinary differential equation on the boundary of the domain and
subjected to nonlinear boundary conditions. The existence and
uniqueness of a local solution is proved and the model is simulated
numerically. It turns out that for adequate parameter ranges, the
qualitative behavior of the stem cells observed in the experiment is in
good agreement with the numerical results. Our investigations represent a
first step in the process of elucidating the mechanism underlying the
homing of hematopoietic stem cells quantitativel},
author = {Neuss-Radu, Maria},
booktitle = {Managing Complexity, Reducing Perplexity. Modeling Biological Systems},
doi = {10.1007/978-3-319-03759-2_11},
editor = {Marcello Delitala, Giulia Ajmone Marsan},
faupublication = {no},
isbn = {9783319037585},
keywords = {Hematopoietic stem cells; Chemotaxis equations; Nonlinear boundary conditions; Homing},
pages = {101-108},
peerreviewed = {Yes},
publisher = {Springer New York LLC},
series = {Springer Proceedings in Mathematics & Statistics},
title = {{Homing} to the niche: {A} mathematical model describing the chemotactic migration of hematopoietic stem cells},
volume = {67},
year = {2014}
}
@masterthesis{faucris.216987080,
abstract = {Novel approaches for medical methods use Magnetic Nanoparticles (MNP) for the treatment process. These nanoparticles with a diameter of about 100nm can also be used for
imaging purposes. In Magnetorelaxometry Imaging (MRXI), magnetic nanoparticles
are exposed to an external magnetic field induced by coils outside the region of interest. This
causes a torque on the magnetization leading to an alignment of the MNPs. The overlay of
the magnetic field induced by each MNP itself can be measured using Superconducting
Quantum Interference Devices (SQUIDs). The corresponding mathematical operator that models the MRXI experiment is severely ill-posed and the corresponding inverse
problem of determining the particle distribution is difficult to solve directly. In the recent
literature only basic regularization techniques like Tikhonov regularization and Truncated Singular Value Decomposition were used to derive a reasonable solution. Using
first-order regularization techniques like the Alternating Direction Method of Multipliers (ADMM), it is possible to include a-priori knowledge about the anticipated solution.
In this work we derive a sophisticated mathematical model of the MRXI operator and compare the impact of different regularization techniques on the solution, in particular in the
case of noisy data, for various simulated settings.

We implement a phase-field model for the direct pore-scale simulation of incompressible flow of two immiscible fluids. The model naturally lends itself to the transport of fluids with large density and viscosity ratios. In the phase-field approach, the fluid-phase interfaces are expressed in terms of thin transition regions, the so-called diffuse interfaces, for increased computational efficiency. The conservation law of mass for binary mixtures leads to the advective Cahn–Hilliard equation and the condition that the velocity field is divergence free. Momentum balance, on the other hand, leads to the Navier–Stokes equations for Newtonian fluids modified for two-phase flow and coupled to the advective Cahn–Hilliard equation. Unlike the volume of fluid (VoF) and level-set methods, which rely on regularization techniques to describe the phase interfaces, the phase-field method facilitates a thermodynamic treatment of the phase interfaces, rendering it more physically consistent for the direct simulations of two-phase pore-scale flow. A novel geometric wetting (wall) boundary condition is implemented as part of the phase-field method for the simulation of two-fluid flows with moving contact lines. The geometric boundary condition accurately replicates the prescribed equilibrium contact angle and is extended to account for dynamic (non-equilibrium) effects. The coupled advective Cahn–Hilliard and modified Navier–Stokes (phase-field) system is solved by using a robust and accurate semi-implicit finite volume method. An extension of the momentum balance equations is also implemented for Herschel–Bulkley (non-Newtonian) fluids. Non-equilibrium-induced two-phase flow problems and dynamic two-phase flows in simple two-dimensional (2-D) and three-dimensional (3-D) geometries are investigated to validate the model and its numerical implementation. Quantitative comparisons are made for cases with analytical solutions. Two-phase flow in an idealized 2-D pore-scale conduit is simulated to demonstrate the viability of the proposed direct numerical simulation approac},
author = {Alpak, Faruk Omer and Rivière, Béatrice and Frank, Florian},
doi = {10.1007/s10596-015-9551-2},
faupublication = {no},
journal = {Computational Geosciences},
keywords = {phase-field method; two-phase flow; pore-scale flow simulation; CFD; digital rock; finite volume method; thermodynamics; contact line motion; dynamic contact angle; wetting},
pages = {881–908},
peerreviewed = {Yes},
title = {{A} phase-field method for the direct simulation of two-phase flows in pore-scale media using a non-equilibrium wetting boundary condition},
url = {https://link.springer.com/article/10.1007/s10596-015-9551-2},
volume = {20},
year = {2016}
}
@article{faucris.116634144,
abstract = {The space mapping technique aims at an efficient optimization of complex technical problems. Its success relies on the quality of the coarse model that might be deduced from either physical, algebraic or grid-based hierarchies of models. In this paper we perform a feasibility study for transport processes coming from the fields of fluid dynamics, semiconductors and radiation. We present appropriate model hierarchies and discuss the numerical performance. Space mapping shows to give very convincing suboptimal solutions to the actual fine model optimization regarding the balance of computational effort (speed and memory) and accuracy. © 2012 - IOS Press and the authors. All rights reserved.},
author = {Marheineke, Nicole and Pinnau, René},
doi = {10.3233/JCM-2012-0404},
faupublication = {yes},
journal = {Journal of Computational Methods in Sciences and Engineering},
keywords = {Aggressive space mapping; flow control; optimal control; radiation optimization; semiconductor design},
pages = {63-74},
peerreviewed = {unknown},
title = {{Model} hierarchies in space-mapping optimization: {Feasibility} study for transport processes},
volume = {12},
year = {2012}
}
@article{faucris.205334628,
abstract = {In dynamic imaging, a key challenge is to reconstruct image sequences with high temporal resolution from strong undersampling projections due to a relatively slow data acquisition speed. In this paper, we propose a variational model using the infimal convolution of Bregman distance with respect to total variation to model edge dependence of sequential frames. The proposed model is solved via an alternating iterative scheme, for which each subproblem is convex and can be solved by existing algorithms. The proposed model is formulated under both Gaussian and Poisson noise assumption and the simulation on two sets of dynamic images shows the advantage of the proposed method compared to previous methods.},
author = {Burger, Martin and et al.},
author_hint = {Ding QQ, Burger M, Zhang XQ},
doi = {10.1088/1361-6420/aa9a94},
faupublication = {no},
journal = {Inverse Problems},
keywords = {dynamic SPECT;sparsity;infimal convolution;Bregman distance;edge alignment;low rank representation},
month = {Jan},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Dynamic} {SPECT} reconstruction with temporal edge correlation},
url = {https://arxiv.org/pdf/1707.00158.pdf},
volume = {34},
year = {2018}
}
@article{faucris.223042292,
abstract = {We consider a one-dimensional two-phase Stefan-like free boundary problem for the heat equation with nonlinear Neumann boundary conditions. Local existence has been shown in [8], furthermore it is clear that cases exist, where global existence fails. The question arises, under which conditions the solution exists globally. In this paper it is shown that especially the natural sign conditions are suf- ficient for global existen},
author = {Knabner, Peter},
faupublication = {no},
journal = {Journal of Mathematical Analysis and Applications},
pages = {543-559},
peerreviewed = {Yes},
title = {{Global} {Existence} in a {General} {Stefan}-like {Problem}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1986/1986_Kn_GlobalExistInAGenStefanLikeProblm},
year = {1986}
}
@article{faucris.223043568,
abstract = {A multidimensional parabolic free boundary problem of the implicit type arises in models in

the soil sciences. Existence, uniqueness and asymptotic behaviour of weak and classical solutions are

considered.

The physical process of solute transport is described by the convection-dispersion soluteux

equation. The substance undergoes adsorption at the soil matrix, ie there is a mobile phase

in solution with concentration u and an immobile adsorbed phase with concentration},
author = {van Duijn, CJ Hans and et al.},
author_hint = {Duijn CJ,},
doi = {10.1515/crll.1991.415.1},
faupublication = {no},
journal = {Journal für die reine und angewandte Mathematik},
pages = {1-50},
peerreviewed = {unknown},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Solute} {Transport} through {Porous} {Media} with {Slow} {Adsorption}},
volume = {1991},
year = {1991}
}
@book{faucris.121348524,
abstract = {Ziel der Linearen Algebra ist die Einübung in die Theorie und Anwendung linearer Strukturen.

Der heutigen Bedeutung der Linearen Algebra als grundlegendes Werkzeug und Sprache für fast alle Teile der Mathematik entsprechend wurden die Inhalte bewußt breit gefasst und vernetzt:

Aspekte der affinen Geometrie (Lehramt), unendlich-dimensionale Vektorräume, Spektralanalyse und lineare Differentialgleichungen (Physik), allgemeine K-Vektorräume sowie algebraische Strukturen (Algebra), die Anfänge der linearen und quadratischen Optimierung (Wirtschaftsmathematik) und die LR-Zerlegung, Pseudoinverse und Singulärwertzerlegung (Numerische Mathematik und Optimierung).

Die erarbeitete Theorie und Algorithmik wird durchgängig mit innermathematischen Themen wie auch mit realen Anwendungen verbunden. Eine klare optische Struktur der Inhalte ermöglicht es dem Leser, den Kerntext von weiterführenden Bemerkungen leicht zu unterscheiden und somit das Buch als Lern- , Arbeits- wie auch als Nachschlagewerk zu benutzen.},
address = {Berlin-Heidelberg},
author = {Knabner, Peter and Barth, Wolf},
doi = {10.1007/978-3-642-32186-3},
edition = {1},
faupublication = {yes},
isbn = {978-3-642-32185-6},
peerreviewed = {unknown},
publisher = {Springer},
series = {Springer-Lehrbuch},
title = {{Lineare} {Algebra}: {Grundlagen} und {Anwendungen}},
url = {http://www.springer.com/de/book/9783642321856#aboutBook},
year = {2013}
}
@article{faucris.106871644,
author = {Grün, Günther and Becker, Jürgen and Blossey, Ralf and Neto, Chiara and Jacobs, Karin and Seemann, Ralf},
faupublication = {no},
journal = {Journal of Physics: Condensed Matter},
pages = {3355-3366},
peerreviewed = {Yes},
title = {{Satellite} hole formation during dewetting; experiment and simulation},
volume = {15},
year = {2003}
}
@article{faucris.123846404,
abstract = {Biodegradable collagen matrices have become a promising alternative to traditional drug delivery systems. The relevant mechanisms in controlled drug release are the diffusion of water into the collagen matrix, the swelling of the matrix coming along with drug release, and enzymatic degradation of the matrix with additional simultaneous drug release. These phenomena have been extensively studied in the past experimentally, via numerical simulations as well as analytically. However, a rigorous derivation of the macroscopic model description, which includes the evolving microstructure due to the degradation process, is still lacking. Since matrix degradation leads to the release of physically entrapped active agent, a good understanding of these phenomena is very important. We present such a derivation using formal twoscale asymptotic expansion in a level set framework and complete our results with numerical simulations in comparison with experimental data. Biodegradable collagen matrices have become a~promising alternative to traditional drug delivery systems. The relevant mechanisms in controlled drug release are the diffusion of water into the collagen matrix, the swelling of the matrix coming along with drug release, and enzymatic degradation of the matrix with additional simultaneous drug release. These phenomena have been extensively studied in the past experimentally, via numerical simulations as well as analytically. However, a rigorous derivation of the macroscopic model description, which includes the evolving microstructure due to the degradation process, is still lacking. Since matrix degradation leads to the release of physically entrapped active agent, a~good understanding of these phenomena is very important. The authors present such a~derivation using formal twoscale asymptotic expansion in a~level set framework and complete their results with numerical simulations in comparison with experimental data. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei},
author = {Ray, Nadja and van Noorden, Tycho and Radu, Adrian Florin and Friess, Wolfgang and Knabner, Peter},
doi = {10.1002/zamm.201200196},
faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
keywords = {Drug release; Evolving microstructure; Homogenization; Mixed finite element method; Numerical simulation; Porous media},
pages = {811-822},
peerreviewed = {Yes},
title = {{Drug} release from collagen matrices including an evolving microstructure},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2013/2013_RayNoordenRaduFriessKn_DrugReleaseFromCollMatIncludAnEvolvingMicrostruct},
volume = {93},
year = {2013}
}
@article{faucris.119229924,
abstract = {Optimal design problems for semiconductor devices with relevant thermal effects can be formulated by help of the energy transport model. In this paper we perform a sensitivity analysis to derive the first-order necessary condition for the optimization. Exploiting the special structure of the KKT system we use a special variant of the classical Gummel iteration to provide a very fast optimization algorithm. Numerical results for a ballistic diode underline the feasibility of our approach. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.},
author = {Drago, C.R. and Marheineke, Nicole and Pinnau, René},
doi = {10.1002/zamm.201100171},
faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
keywords = {Energy transport model; Fast optimal design; Gaussian doping profiles.; Semiconductor design},
pages = {700-705},
peerreviewed = {Yes},
title = {{Semiconductor} device optimization in the presence of thermal effects},
volume = {93},
year = {2013}
}
@article{faucris.208169582,
abstract = {A numerical method is formulated for the solution of the advective Cahn–Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a mass constraint and dissipation of free energy. Porous media/pore-scale problems specifically entail images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of elementwise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via inexact Newton's method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample.

}, author = {Liu, Chen and Frank, Florian and Rivière, Béatrice}, doi = {10.1002/num.22362}, faupublication = {yes}, journal = {Numerical Methods For Partial Differential Equations}, keywords = {Cahn–Hilliard equation; error analysis; interior penalty discontinuous Galerkin methods; stability analysis; unique solvability}, month = {Jan}, peerreviewed = {Yes}, title = {{Numerical} error analysis for nonsymmetric interior penalty discontinuous {Galerkin} method of {Cahn}–{Hilliard} equation}, year = {2019} } @masterthesis{faucris.209590658, author = {Prechtel, Alexander}, faupublication = {no}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Die} {Schätzung} von {Variogrammen} durch {Betrachtung} ihrer {Integrale}}, year = {1997} } @article{faucris.215925920, abstract = {A Galerkin approach for a class of multiscale reaction–diffusion systems with nonlinear coupling between the microscopic and macroscopic variables is presented. This type of models are obtained e.g. by upscaling of processes in chemical engineering (particularly in catalysis), biochemistry, or geochemistry. Exploiting the special structure of the models, the functions spaces used for the approximation of the solution are chosen as tensor products of spaces on the macroscopic domain and on the standard cell associated to the microstructure. Uniform estimates for the finite dimensional approximations are proven. Based on these estimates, the convergence of the approximating sequence is shown. This approach can be used as a basis for the numerical computation of the solutio}, author = {Muntean, Adrian and Neuss-Radu, Maria}, doi = {10.1016/j.jmaa.2010.05.056}, faupublication = {no}, journal = {Journal of Mathematical Analysis and Applications}, keywords = {Gas-liquid reactions; Galerkin approximation; Multiscale reaction-diffusion systems; Henry's law; Structured porous media; Nonlinear coupling}, pages = {705-718}, peerreviewed = {Yes}, title = {{A} multiscale {Galerkin} approach for a class of nonlinear coupled reaction-diffusion systems in complex media}, volume = {371}, year = {2010} } @article{faucris.211506717, abstract = {The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the regularised solutions and allow us to obtain convergence rates in terms of Bregman distances-as usual in inverse problems, under an additional assumption on the exact solution called the source condition. These results are obtained for general absolutely one-homogeneous functionals. In the special case of TV-based regularisation we also study the structure of regularised solutions and prove convergence of their level sets to those of an exact solution. Finally, using the developed theory, we adapt the concept of debiasing to inverse problems with imperfect operators and propose an approach to pointwise error estimation in TV-based regularisation.}, author = {Burger, Martin and Korolev, Yury and Rasch, Julian}, doi = {10.1088/1361-6420/aaf6f5}, faupublication = {yes}, journal = {Inverse Problems}, note = {CRIS-Team WoS Importer:2019-02-21}, peerreviewed = {Yes}, title = {{Convergence} rates and structure of solutions of inverse problems with imperfect forward models}, volume = {35}, year = {2019} } @article{faucris.223589201, abstract = {Wavelet (Besov) priors are a promising way of reconstructing indirectly measured fields in a regularized manner. We demonstrate how wavelets can be used as a localized basis for reconstructing permeability fields with sharp interfaces from noisy pointwise pressure field measurements in the context of the elliptic inverse problem. For this we derive the adjoint method of minimizing the Besov-norm-regularized misfit functional (this corresponds to determining the maximum a posteriori point in the Bayesian point of view) in the Haar wavelet setting. As it turns out, choosing a wavelet–based prior allows for accelerated optimization compared to established trigonometrically–based prior}, author = {Wacker, Philipp Konstantin and Knabner, Peter}, doi = {10.1007/s11009-019-09736-2}, faupublication = {yes}, journal = {Methodology and Computing in Applied Probability}, keywords = {Bayesian inverse problems; Besov priors; Optimization; Elliptical inverse problem}, pages = {1-27}, peerreviewed = {Yes}, title = {{Wavelet}-{Based} {Priors} {Accelerate} {Maximum}-a-{Posteriori} {Optimization} in {Bayesian} {Inverse} {Problems}}, year = {2019} } @incollection{faucris.109512524, abstract = {The paper is concerned with a hybrid finite element - spectral Chebyshev parallel solver of the incompressible Navier-Stokes equations. A domain decomposition, well adapted to the computation of wake or jet type flow, is assumed. Subdomains with complex geometry are handled with finite elements and the other ones with the highly accurate spectral method. The iterative resolution of the multi-domain problem is carried out with the "Conjugate Gradient Boundary Iteration" method. Here we focus on its optimal preconditioning when Gauss-Lobatto type grids axe involved. The resulting algorithm is then applied to the flow past a cylinder and over a backward facing step at higher Reynolds numbers. For the latter case the numerical results display some transitional laminar-turbulent behaviour of the flow arising from unstable steady states. Numerical results are presented for both convective and absolute instability regions.}, address = {Berlin Heidelberg}, author = {Borchers, Wolfgang and Kräutle, Serge and Pasquetti, Richard and Peyret, Roger and Rautmann, Reimund}, booktitle = {Numerical Flow Simulation III}, doi = {10.1007/978-3-540-45693-3_1}, editor = {Ernst Heinrich Hirschel}, faupublication = {yes}, isbn = {978-3-540-44130-4}, month = {Jan}, pages = {3-17}, peerreviewed = {unknown}, publisher = {Vieweg}, series = {Notes on Numerical Fluid Mechanics}, title = {{Multi}-domain finite element - {Spectral} {Chebyshev} parallel {Navier}-{Stokes} solver for viscous flow problems}, volume = {82}, year = {2003} } @article{faucris.117797284, abstract = {The appropriate prediction of the fate of the contaminant is an essential step when evaluating the risk of severe groundwater pollutions - in particular in the context of natural attenuation. We numerically study the reactive transport of phenanthrene at the field scale in a multilayer soil profile based on experimental data. The effect of carrier facilitation by dissolved organic carbon is emphasized and incorporated in the model. Previously published simulations are restricted to the saturated zone and/or to homogeneous soil columns at the laboratory scale. A numerical flow and transport model is extended and applied to understand and quantify the relevant processes in the case of a strongly sorbing hydrophobic organic compound that is subject to carrier facilitation in the unsaturated zone. The contaminant migration is investigated on long- and short-term time scales and compared to predictions without carrier facilitation. The simulations demonstrate the importance of carrier facilitation and suggest strongly to take this aspect into account. By carrier facilitation breakthrough times at the groundwater level decreased from 500 to approximately 8 years and concentration peaks increased by two orders of magnitude in the long-term simulation assuming a temporary spill in an initially unpolluted soil with a non-sorbing carrier. © 2002 Elsevier Science B.V. All rights reserve}, author = {Prechtel, Alexander and Knabner, Peter and Schneid, Eckhard and Totsche, Kai Uwe}, doi = {10.1016/S0169-7722(01)00211-X}, faupublication = {yes}, journal = {Journal of Contaminant Hydrology}, keywords = {Carrier facilitation; Dissolved organic matter; Hydrophobic organic compounds; Numerical transport model; Unsaturated flow}, pages = {209-225}, peerreviewed = {Yes}, title = {{Simulation} of carrier-facilitated transport of phenanthrene in a layered soil profile}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_PrechtelKnSchneidTotsche_SimulationOfCarrFaciTrnasportOfPhenanInALayerSoilProfile}, volume = {56}, year = {2002} } @article{faucris.108983424, author = {Grün, Günther and Klingbeil, Fabian}, faupublication = {yes}, journal = {Journal of Computational Physics}, pages = {708-725}, peerreviewed = {Yes}, title = {{Two}-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse interface model}, volume = {257}, year = {2014} } @article{faucris.215945774, abstract = {The aim of this paper is to provide a solid mathematical discussion of the inverse problem in magnetorelaxometry imaging (MRXI), a currently developed technique for quantitative biomedical imaging using magnetic nanoparticles. We provide a detailed discussion of the mathematical modeling of the forward problems including possible ways to magnetically activate the particles and measure its magnetic response, leading to a severely ill-posed linear inverse problem. Moreover, we formulate an idealized version of the inverse problem for infinitesimal small activation coils, which allows for a more detailed analysis of uniqueness issues.We propose a variational regularization approach to compute stable approximations of the solution and discuss its discretization and numerical solution. Results on synthetic data are presented and improvements to methods used previously in practice arc demonstrated.}, author = {Föcke, Janic and Baumgarten, Daniel and Burger, Martin}, doi = {10.1088/1361-6420/aadbbf}, faupublication = {no}, journal = {Inverse Problems}, keywords = {inverse source problem;magnetorelaxometry imaging;magnetic nanoparticles;total variation regularization;uniqueness;magnetic remanence;ADMM}, peerreviewed = {Yes}, title = {{The} inverse problem of magnetorelaxometry imaging}, volume = {34}, year = {2018} } @article{faucris.122616824, abstract = {In this paper we show first-order convergence of a multi-point flux approximation control volume method (MPFA) on unstructured triangular grids. In this approach the flux approximation is derived directly in the physical space. In order to do this, we introduce a perturbed mixed finite element method that is equivalent to the MPFA scheme and prove the first-order convergence of this approach. Moreover, we carefully compare the computational performance properties of the MPFA method with those of a lowest order Raviart-Thomas and Brezzi-Douglas-Marini mixed finite element approximation. © 2010 Springer-Verla}, author = {Bause, Markus and Hoffmann, Joachim and Knabner, Peter}, doi = {10.1007/s00211-010-0290-y}, faupublication = {yes}, journal = {Numerische Mathematik}, pages = {1-29}, peerreviewed = {Yes}, title = {{First}-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2010/2010_BauseHoffmannKn_FirstorderConvergenceOfMultiPointFlux}, volume = {116}, year = {2010} } @article{faucris.123981484, abstract = {The spinning of slender viscous jets can be asymptotically described by one-dimensional models that consist of systems of partial and ordinary differential equations. Whereas well-established string models only possess solutions for certain choices of parameters and configurations, the more sophisticated rod model is not limited by restrictions. It can be considered as an ε-regularized string model, but containing the slenderness ratio ε in the equations complicates its numerical treatment. We develop numerical schemes for fixed or enlarging (time-dependent) domains, using a finite volume approach in space with mixed central, up- and down-winded differences and stiffly accurate Radau methods for the time integration. For the first time, results of instationary simulations for a fixed or growing jet in a rotational spinning process are presented for arbitrary parameter ranges.}, author = {Arne, Walter and Marheineke, Nicole and Meister, Andreas and Schiessl, Stefan and Wegener, Raimund}, doi = {10.1016/j.jcp.2015.03.042}, faupublication = {yes}, journal = {Journal of Computational Physics}, keywords = {Finite volume scheme; Partial differential algebraic equations; Quaternions; Rotational spinning process; Special cosserat theory; Viscous fiber}, pages = {20-37}, peerreviewed = {unknown}, title = {{Finite} volume approach for the instationary {Cosserat} rod model describing the spinning of viscous jets}, volume = {294}, year = {2015} } @article{faucris.122619464, abstract = {We present a mass conservative finite element approach of second order accuracy for the numerical approximation of reactive solute transport in porous media modeled by a coupled system of advection-diffusion-reaction equations. The lowest order Brezzi-Douglas-Marini (BDM ) mixed finite element method is used. A modification based on the hybrid form of the approach is suggested for the discretization of the advective term. It is demonstrated numerically that this leads to optimal second order convergence of the flux variable. The modification improves the convergence behavior of the classical BDM scheme, which is known to be suboptimal of first order accuracy only for advection-diffusion problems; cf. [8]. Moreover, the new scheme shows more robustness for high Péclet numbers than the classical approach. A comparison with the Raviart-Thomas element (RT ) of second order accuracy for the approximation of the flux variable is also presented. For the case of strongly advection-dominated problems we propose a full upwind scheme. Various numerical studies, including also a nonlinear test problem, are presented to illustrate the numerical performance properties of the considered numerical methods. © 2011 Elsevier Lt}, author = {Brunner, Fabian and Radu, Adrian Florin and Bause, Markus and Knabner, Peter}, doi = {10.1016/j.advwatres.2011.10.001}, faupublication = {yes}, journal = {Advances in Water Resources}, keywords = {Mixed finite element methods; Optimal order convergence; Reactive transport}, pages = {163-171}, peerreviewed = {Yes}, title = {{Optimal} order convergence of a modified {BDM} 1 mixed finite element scheme for reactive transport in porous media}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2012/2012_BrunnerRaduBauseKn_OptimalOrderConvergenceOfModifiedBDMMixedFiniteInPM}, volume = {35}, year = {2012} } @article{faucris.106893864, abstract = {In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media: find {u(x, t),v(x,t)} such that ∂u + ∂v- δu = f in ω × (0,T] u = 0 on ∂ω × (0,T] ∂v = k((℘(u) - v) in ω × (0, T] u(·,0) = g1(·) v(·,0) = g2(·) in ω ⊂ R, 1≤d≤3 for given data k ∈ R, f, g1, g2 and a monotonically increasing ℘ ∈ C(R) ∩ C(-∞,0] ∪ (0,∞) satisfying ℘(0) = 0, which is only locally Hölder continuous with exponent p ∈ (0,1) at the origin, e.g., ℘(s) ≡ [s] . This lack of Lipschitz continuity at the origin limits the regularity of the unique solution {u,v} and leads to difficulties in the finite element error analysi}, author = {Barrett, John W. and Knabner, Peter}, doi = {10.1137/S0036142993249024}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {Degenerate parabolic systems; Error analysis; Finite elements; Porous medium}, pages = {201-227}, peerreviewed = {Yes}, title = {{Finite} {Element} {Approximation} of the {Transport} of {Reactive} {Solutes} in {Porous} {Media}. {Part} 1: {Error} {Estimates} for {Nonequilibrium} {Adsorption} {Processes}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_BarrettKn_FinEleApprOfTranspOfReactSolutInPMPart1}, volume = {34}, year = {1997} } @article{faucris.122012264, author = {Grün, Günther and Giacomelli, Lorenzo}, faupublication = {yes}, journal = {Interfaces and Free Boundaries}, pages = {111-129}, peerreviewed = {Yes}, title = {{Lower} bounds on waiting time for degenerate parabolic equations and systems}, volume = {8}, year = {2006} } @incollection{faucris.109115424, author = {Marheineke, Nicole}, booktitle = {Progress in Industrial Mathematics at ECMI 2004}, editor = {A. Di Bucchianico, R. M. M. Mattheij, M. A. Peletier}, faupublication = {no}, pages = {366-370}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Modeling} of turbulence effects on fiber motion}, year = {2005} } @article{faucris.215926918, abstract = {A system of reaction-diffusion equations in a multi-component medium with nonlinear flux-conditions and additional reaction-diffusion equations on the interfaces is considered. The model is motivated by metabolic processes in living cells. Especially, we are interested in modeling the central carbon metabolism in plant cells, with particular emphasis on metabolite channeling.

The nonlinear reaction terms arising in the equations and boundary conditions are described by structural conditions, which are fulfilled by the kinetics of multi-species enzymatic reactions encountered in cellular metabolism.

Starting from a mathematical model at subcellular level, where cellular structures like organelles are resolved, we derive an effective approximations for the cellular processes, by letting the scale parameter given by the ratio between the size of organelles and that of the cell going to zero. To show convergence of the nonlinear terms, we use homogenization concepts developed in \cite{Gahn}, based on estimates for the shifting operator for Banach-space-valued functions.

u =

Es werden Methoden aus der Linearen Algebra, der Analysis und der Theorie der gewöhnlichen und partiellen Differentialgleichungen benutzt bzw. sorgfältig eingeführt. Anwendungsbeispiele aus den Bereichen elektrische Netzwerke, chemische Reaktionskinetik, Populationsdynamik, Strömungsdynamik, Elastizitätstheorie und Kristallwachstum werden ausführlich behandelt. Der Stoffumfang des Buches eignet sich für bis zu zwei vierstündige Vorlesungen für Studierende der Mathematik und der Ingenieur- oder Naturwissenschaften ab dem vierten Semester.}, address = {Berlin Heidelberg}, author = {Eck, Christof and Garcke, Harald and Knabner, Peter}, doi = {10.1007/978-3-642-18424-6}, edition = {2}, faupublication = {yes}, isbn = {978-3-642-18423-9}, peerreviewed = {unknown}, publisher = {Springer}, series = {Springer-Lehrbuch}, title = {{Mathematische} {Modellierung}}, url = {http://www.springer.com/de/book/9783642184239#aboutBook}, year = {2011} } @article{faucris.117955684, abstract = {

We present convergence results for a fully discrete scheme based on the mixed finite element (MFE) method and an one-step Euler implicit (EI) method for simulating reactive solute transport in saturated/unsaturated soil. The results considered the low regularity of the solution of the degenerate parabolic equation describing the water flow in porous media.

},
author = {Radu, Adrian Florin and Pop, Iuliu Sorin and Attinger, Sabine and Knabner, Peter},
faupublication = {yes},
journal = {Proceedings in Applied Mathematics and Mechanics},
pages = {1024705-1024706},
peerreviewed = {Yes},
title = {{Error} estimates for an {Euler} implicit-mixed finite element scheme for reactive transport in saturated/unsaturated soil},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2007/2007_RaduPopAttingerKn_ErrorEstimatesForAnEulerImplicitMIxedFiniteElementScheme},
volume = {7},
year = {2007}
}
@article{faucris.117796184,
abstract = {Standard error estimates in the literature for finite element approximations of nonstationary convection-diffusion problems depend either reciprocally on the diffusion parameter ε or on higher order norms of the solution. Therefore, the estimates generally become worthless in the convection-dominated case with 0 < ε ≪ 1. In this work we develop a rigorous ε-uniform convergence theory for finite element discretizations of convection-dominated diffusion problems in Eulerian and Lagrangian coordinates. Here, the constants that arise in the error estimates depend on norms of the data and not of the solution and remain bounded in the hyperbolic limit ε → 0. In particular in the Lagrangian case this requires modifications to standard finite element error analyses. In the Eulerian formulation ε-uniform convergence of order one half is proven whereas in the Lagrangian framework ε-uniform convergence of optimal order is established. The estimates are based on ε-uniform a priori estimates for the solution of the continuous problems which are derived firs},
author = {Bause, Markus and Knabner, Peter},
doi = {10.1007/s10092-004-0081-x},
faupublication = {yes},
journal = {Calcolo},
pages = {1-26},
peerreviewed = {Yes},
title = {{On} uniform convergence rates for {Eulerian} and {Lagrangian} finite element approximations of convection-dominated diffusion problems},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_BauseKn_OnUniformConvergenceRatesForEulerianLagrangianFinElemApproxOfConvDominDiffProbl},
volume = {41},
year = {2004}
}
@article{faucris.204146417,
abstract = {The morphology of presynaptic specializations can vary greatly ranging from classical single-release-site boutons in the central nervous system to boutons of various sizes harboring multiple vesicle release sites. Multi-release-site boutons can be found in several neural contexts, for example at the neuromuscular junction (NMJ) of body wall muscles of Drosophila larvae. These NMJs are built by two motor neurons forming two types of glutamatergic multi-release-site boutons with two typical diameters. However, it is unknown why these distinct nerve terminal configurations are used on the same postsynaptic muscle fiber. To systematically dissect the biophysical properties of these boutons we developed a full three-dimensional model of such boutons, their release sites and transmitter-harboring vesicles and analyzed the local vesicle dynamics of various configurations during stimulation. Here we show that the rate of transmission of a bouton is primarily limited by diffusion-based vesicle movements and that the probability of vesicle release and the size of a bouton affect bouton-performance in distinct temporal domains allowing for an optimal transmission of the neural signals at different time scales. A comparison of our in silico simulations with in vivo recordings of the natural motor pattern of both neurons revealed that the bouton properties resemble a well-tuned cooperation of the parameters release probability and bouton size, enabling a reliable transmission of the prevailing firing-pattern at diffusion-limited boutons. Our findings indicate that the prevailing firing-pattern of a neuron may determine the physiological and morphological parameters required for its synaptic terminals.},
author = {Knodel, Markus and et al.},
author_hint = {Knodel MM, Geiger R, Ge L, Bucher D, Grillo A, Wittum G, Schuster CM, Queisser G},
doi = {10.3389/fncom.2014.00101},
faupublication = {no},
journal = {Frontiers in Computational Neuroscience},
keywords = {neuromuscular junction;boutons;modeling and simulation;structure-function relationships;morphology;firing pattern;adaption},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Synaptic} bouton properties are tuned to best fit the prevailing firing pattern},
volume = {8},
year = {2014}
}
@incollection{faucris.123843324,
abstract = {In this paper we discuss some iterative approaches for solving the nonlinear algebraic systems encountered as fully discrete counterparts of some degenerate (fast diffusion) parabolic problems. After regularization, we combine a mixed finite element discretization with the Euler implicit scheme. For the resulting systems we discuss three iterative methods and give sufficient conditions for convergenc},
address = {Berlin, Heidelberg},
author = {Radu, Adrian Florin and Pop, Iuliu Sorin and Knabner, Peter},
booktitle = {Numerical Mathematics and Advanced Applications},
doi = {10.1007/978-3-540-34288-5_120},
editor = {Alfredo Bermúdez de Castro, Dolores Gómez, Peregrina Quintela, Pilar Salgado},
faupublication = {yes},
isbn = {978-3-540-34287-8},
pages = {1192-1200},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Newton} - {Type} {Methods} for the {Mixed} {Finite} {Element} {Discretization} of {Some} {Degenerate} {Parabolic} {Equations}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2006/2006_RaduPopKn_NewtonTypMethFrMixedFinitElemtDiscretizOfSomeDegeneraParabolEquat},
year = {2006}
}
@article{faucris.205336232,
abstract = {The aim of this paper is to discuss potential advances in PET kinetic models and direct reconstruction of kinetic parameters. As a prominent example we focus on a typical task in perfusion imaging and derive a system of transport-reaction-diffusion equations, which is able to include macroscopic flow properties in addition to the usual exchange between arteries, veins, and tissues. For this system we propose an inverse problem of estimating all relevant parameters from PET data. We interpret the parameter identification as a nonlinear inverse problem, for which we formulate and analyze variational regularization approaches. For the numerical solution we employ gradient-based methods and appropriate splitting methods, which are used to investigate some test cases.},
author = {Reips, Louise and Burger, Martin and Engbers, Ralf},
doi = {10.1515/jiip-2015-0016},
faupublication = {no},
journal = {Journal of Inverse and Ill-posed Problems},
keywords = {Parameter identification;dynamic PET;inverse problems;image processing;forward-backward splitting;reaction;diffusion;transport},
pages = {185-200},
peerreviewed = {Yes},
title = {{Towards} dynamic {PET} reconstruction under flow conditions: {Parameter} identification in a {PDE} model},
volume = {26},
year = {2018}
}
@article{faucris.124249004,
abstract = {Although multicomponent reactive transport modeling is gaining wider application in various geoscience fields, it continues to present significant mathematical and computational challenges. There is a need to solve and compare the solutions to complex benchmark problems, using a variety of codes, because such intercomparisons can reveal promising numerical solution approaches and increase confidence in the application of reactive transport codes. In this contribution, the results and performance of five current reactive transport codes are compared for the 1D and 2D subproblems of the so-called easy test case of the MoMaS benchmark (Carrayrou et al., Comput Geosci, 2009, this issue). This benchmark presents a simple fictitious reactive transport problem that highlights the main numerical difficulties encountered in real reactive transport problems. As a group, the codes include iterative and noniterative operator splitting and global implicit solution approaches. The 1D easy advective and 1D easy diffusive scenarios were solved using all codes, and, in general, there was a good agreement, with solution discrepancies limited to regions with rapid concentration changes. Computational demands were typically consistent with what was expected for the various solution approaches. The differences between solutions given by the three codes solving the 2D problem are more important. The very high computing effort required by the 2D problem illustrates the importance of parallel computations. The most important outcome of the benchmark exercise is that all codes are able to generate comparable results for problems of significant complexity and computational difficulty. © Springer Science+Business Media B.V. 201},
author = {Carrayrou, Jérôme and Hoffmann, Joachim and Knabner, Peter and Kräutle, Serge and de Dieuleveult, Caroline and Erhel, Jocelyne and Van der Lee, Jan and Lagneau, Vincent and Mayer, K. Ulrich and MacQuarrie, Kerry T. B.},
doi = {10.1007/s10596-010-9178-2},
faupublication = {yes},
journal = {Computational Geosciences},
keywords = {Benchmark; Code intercomparison; Differential and algebraic equations (DAE); Direct substitution approach (DSA); MoMaS; Numerical methods for reactive transport; Sequential iterative approach (SIA); Sequential noniterative approach (SNIA)},
pages = {483-502},
peerreviewed = {Yes},
title = {{Comparison} of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems-the {MoMaS} benchmark case},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2010/2010_CarrayrouHoffmannKnKraeutleDieuleveultErhel_ComparisonOfNumericalMethodsForSimulStrongNonlinearAndHetere},
volume = {14},
year = {2010}
}
@inproceedings{faucris.117889244,
author = {Knabner, Peter and Ray, Nadja and Radu, Adrian Florin},
booktitle = {Proceedings of the 11th International Conference on Mathematical Methods in Science and Engineering},
faupublication = {yes},
pages = {975-986},
peerreviewed = {unknown},
title = {{Drug} release from collagen matrices and transport phenomena in porous media including an evolving microstructure},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2011/2011_KnRayRadu_DrugReleaseFromCollagMatrTranspPhenomInPMIncludAnEvolvMicrostruct},
venue = {Alicante},
year = {2011}
}
@article{faucris.215225684,
abstract = {Segmentation is one of the fundamental tasks in computer vision applications. For natural images there exist a vast amount of sophisticated segmentation methods in the literature. However, these standard methods tend to fail in the presence of non-Gaussian image noise, e.g., in biomedical imaging or astronomy. In this paper we propose an adequate variational segmentation model for segmentation of images perturbed by arbitrary noise models without adding a-priori assumptions about the unknown physical noise model. For this, we discuss the prominent minimal surface problem and two different numerical minimization schemes to solve it. The first approach efficiently computes the set of all possible minimal surface solutions for the given data via convex optimization and reduces the segmentation problem to the estimation of a proper threshold. The second approach is based on level set methods and is especially suitable for the separation of inhomogeneous image regions. The advantage of this approach is both its simpleness and robustness: the noise in the data does not have to be modeled explicitly since the image intensities are separated using histogram-based thresholding techniques. The proposed model can be interpreted as a generalization of many traditional segmentation methods which implicitly assume a perturbation by additive Gaussian noise. The superiority of this approach over standard methods such as the popular Chan–Vese formulation is demonstrated on synthetic images as well as real application data from biomedical imaging.
We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

}, author = {Lindner, Felix and Marheineke, Nicole and Stroot, Holger and Vibe, Alexander and Wegener, Raimund}, doi = {10.1142/S0219493717500162}, faupublication = {yes}, journal = {Stochastics and Dynamics}, keywords = {Stochastic elastic beam dynamics; stochastic differential algebraic equation; manifold-valued stochastic differential equation; nonlinear constraint; global solution theory}, peerreviewed = {Yes}, title = {{Stochastic} dynamics for inextensible fibers in a spatially semi-discrete setting}, year = {2016} } @article{faucris.116658124, author = {Grothaus, Martin and Marheineke, Nicole}, doi = {10.1093/imanum/drv056}, faupublication = {yes}, journal = {IMA Journal of Numerical Analysis}, peerreviewed = {Yes}, title = {{On} a nonlinear partial differential algebraic system arising in technical textile industry: {Analysis} and numerics}, year = {2015} } @book{faucris.204149148, author = {Knodel, Markus and et al.}, author_hint = {Knodel M., Nägel A., Reiter S., Rupp M., Vogel A., Lampe M., Targett-Adams P., Herrmann E., Wittum G.}, doi = {10.1007/978-3-319-24633-8_41}, faupublication = {no}, isbn = {9783319246338}, pages = {641-657}, peerreviewed = {Yes}, publisher = {Springer International Publishing}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{On} estimation of a viral protein diffusion constant on the curved intracellular {ER} surface}, year = {2016} } @incollection{faucris.206530022, author = {Luckner, Thomas and Sommer, Thomas and Luckner, Ludwig and Bilek, Felix and Knabner, Peter and Prechtel, Alexander}, booktitle = {KORA - Synopse "Systemanalyse, Modellierung und Prognose der Wirkungen natürlicher Schadstoffminderungsprozesse - eine rezente Synopse"}, faupublication = {yes}, pages = {1-81}, peerreviewed = {unknown}, series = {Gemeinsame Mitteilungen des Dresdner Grundwasserforschungszentrum e.V. und seiner Partner}, title = {{Grundlagen} der {Systemanalyse}, {Modellierung} und {Prognose}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2008/2008_LucknerSommerLucknerBilekKnPrechtel_GrundlagenDerSystemanalyseModellierungUndPrognose}, volume = {5}, year = {2008} } @article{faucris.121944504, abstract = {In this paper, we derive and investigate an asymptotic model for the dynamics of curved viscous inertial Newtonian fibres subjected to surface tension, as they occur in rotational spinning processes. Accordingly, we extend the slender body theory of Panda, Marheineke & Wegener (Math. Meth. Appl. Sci., vol. 31, 2008, p. 1153) by including surface tension and deducing boundary conditions for the free end of the fibre. The asymptotic model accounts for the inner viscous transport and places no restrictions on either the motion or the shape of the fibre centreline. Depending on the capillary number, the boundary conditions yield an explicit description for the temporal evolution of the fibre end. We study numerically the behaviour of the fibre as a function of the effects of viscosity, gravity, rotation and surface tension. © 2009 Cambridge University Press.}, author = {Marheineke, Nicole and Wegener, Raimund}, doi = {10.1017/S0022112008005259}, faupublication = {no}, journal = {Journal of Fluid Mechanics}, pages = {345-369}, peerreviewed = {Yes}, title = {{Asymptotic} model for the dynamics of curved viscous fibres with surface tension}, volume = {622}, year = {2009} } @misc{faucris.216987647, abstract = {SiMRX is a MRX simulation toolbox written in MATLAB for simulation of realistic 2D and 3D Magnetorelaxometry (MRX) setups, including coils, sensors and activation patterns. MRX is a new modality that uses magnetic nanoparticles (MNP) as contrast agent and shows promising results in medical applications, e.g. cancer treatment. Its basic principles were outlined in [Baumgarten et al., 2008], further elaborated in [Liebl et al., 2014], transferred into a rigorous mathematical model and analyzed in [Föcke et al., 2018].

Das Buch basiert auf jahrzehntelanger Lehrerfahrung an der Universität Erlangen und wichtige Teile des Buches entstammen aus dort entstandenen Skripten.}, address = {Berlin-Heidelberg}, author = {Merz, Wilhelm and Knabner, Peter}, doi = {10.1007/978-3-642-29980-3}, edition = {1}, faupublication = {yes}, isbn = {978-3-642-29979-7}, peerreviewed = {unknown}, publisher = {Springer}, series = {Springer-Lehrbuch}, title = {{Mathematik} für {Ingenieure} und {Naturwissenschaftler}}, url = {http://www.springer.com/de/book/9783642299797}, year = {2013} } @article{faucris.117922024, abstract = {Transport processes in heterogeneous media such as ionized plasmas, natural porous media, and turbulent atmosphere are often modeled as diusion processes in random velocity elds. Using the Ito ̂ formalism, we decompose the second spatial moments of the concentration and the equivalent eective dispersion coecients in terms corresponding to various physical factors which in uence the transport. We explicitly dene \ergodic " dispersion coecients, independent of the initial conditions and completely determined by local disper-sion coecients and velocity correlations. Ergodic coecients govern an upscaled process which describes the transport at large tine-space scales. The non-ergodic behavior at nite times shown by numerical experiments for large initial plumes is explained by \memory terms " accounting for correlations between initial po-sitions and velocity uctuations on the trajectories of the solute molecule}, author = {Suciu, Nicolae and Vamo̧s, Cǎlin and Vereecken, Harry and Sabelfeld, Karl and Knabner, Peter}, faupublication = {yes}, journal = {Revue d'Analyse Numérique et de Théorie de l'Approximation}, keywords = {second spatial moment; solute molecule; upscaled process; memory term; ergodic dispersion coecients; nite time; heterogeneous medium; initial positions; ito formalism; ergodic coecients; non-ergodic behavior; large tine-space scale; velocity correlation; equivalent eective dispersion; coecients; initial condition; natural porous medium; large initial plume; various physical factor; local disper-sion coecients; numerical experiment; turbulent atmosphere; random velocity eld; velocity uctuations; ionized plasma}, pages = {221-238}, peerreviewed = {Yes}, title = {{Ito} equation model for dispersion of solutes in heterogeneous media}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2008/2008_SuciuVamosVereeckenSabelfeldKn_ItoEquationModelForDisOfSolutesInHeteroMedia}, volume = {37}, year = {2008} } @article{faucris.207466853, author = {Burger, Martin and et al.}, author_hint = {Burger M, Hausser F, Stöcker C, Voigt A}, doi = {10.1016/j.jcp.2006.11.026}, faupublication = {no}, journal = {Journal of Computational Physics}, note = {WWU::5575}, pages = {183-205}, peerreviewed = {No}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{A} level set approach to anisotropic flows with curvature regularization}, volume = {225}, year = {2007} } @article{faucris.109108604, author = {Grün, Günther and Guillen-Gonzalez, Francisco and Metzger, Stefan}, faupublication = {yes}, journal = {Communications in Computational Physics}, pages = {1473-1502}, peerreviewed = {Yes}, title = {{On} fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities}, volume = {19}, year = {2016} } @inproceedings{faucris.117790684, abstract = {We apply a novel upwind stabilization of a mixed hybrid finite element method of lowest order to advection–diffusion problemswith dominant advection and compare it with a finite element scheme stabilized by finite volume upwinding. Both schemes are locally mass conservative and employ an upwind-weighting formula in the discretization of the advective term. Numerical experiments indicate that the upwind-mixed method is competitive with the finite volume method. It prevents the appearance of spurious oscillations and produces nonnegative solutions for strongly advection-dominated problems, while the amount of artificial diffusion is lower than that of the finite volume method. This makes the method attractive for applications in which too much numerical diffusion is critical and may lead to false predictions; e.g., if highly nonlinear reactive processes take place only in thin interaction region}, author = {Brunner, Fabian and Frank, Florian and Knabner, Peter}, booktitle = {Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects}, date = {2014-06-15/2014-06-20}, doi = {10.1007/978-3-319-05684-5_16}, editor = {Fuhrmann Jürgen, Ohlberger Mario, Rohde Christian}, faupublication = {yes}, isbn = {9783319056838}, pages = {177-185}, peerreviewed = {unknown}, publisher = {Springer New York LLC}, title = {{FV} upwind stabilization of {FE} discretizations for advection-diffusion problems}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2014/2014_BrunnerFrankKn_FVUpwindStabOfFEForAdvancedDiffProblems}, venue = {Berlin}, volume = {77}, year = {2014} } @incollection{faucris.122724844, author = {Knabner, Peter and Frolkovic, Peter}, booktitle = {Computational Methods in Water Resources}, faupublication = {yes}, pages = {93–100}, peerreviewed = {unknown}, publisher = {Computational Mechanics Publication, Southampton}, title = {{Consistent} {Velocity} {Approximations} in {Finite} {Element} or {Volume} {Discretizations} of {Density} {Driven} {Flow}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1996/1996_KnFrolkovic_ConsVeloApprForFinVolOrElemtDiscrOfDensDrivFlowInPM}, volume = {1}, year = {1996} } @article{faucris.106933024, abstract = {

Based on the local discontinuous Galerkin method, two substantially different mixed formulations for the subjective surfaces problem are compared using a number of numerical tests of various types. The work also performs the energy stability analysis for both schemes.}, author = {Fried, Michael and Aizinger, Vadym and Bungert, Leon}, doi = {10.1007/s00791-018-0291-4}, faupublication = {yes}, journal = {Computing and Visualization in Science}, keywords = {Local discontinuous Galerkin method; Image segmentation; Subjective surfaces; Stability analysis; Mixed formulation; Divergence form; Edge detection}, peerreviewed = {Yes}, title = {{Comparison} of two local discontinuous {Galerkin} formulations for the subjective surfaces problem}, url = {https://link.springer.com/article/10.1007%2Fs00791-018-0291-4}, year = {2018} } @article{faucris.123851904, abstract = {Proposed in this study is a model for transport of solutes in a porous medium participating in a dissolution/precipitation reaction, in general not in equilibrium. For an unbounded spatial domain, travelling wave solutions exist, if and only if the charge distribution is constant and the situation is a dissolution situation. The travelling wave in fact exhibits a sharp dissolution front. The wave is given in a nearly explicit manner. Also for the limit cases of equilibrium or no dispersion, travelling waves are established under the same conditions, but with different qualitative properties. © 199}, author = {Knabner, Peter and Van Duijn, C. J. (Hans) and Hengst, S.}, doi = {10.1016/0309-1708(95)00005-4}, faupublication = {yes}, journal = {Advances in Water Resources}, keywords = {crystal dissolution; mathematical analysis; porous media; transport; travelling wave}, pages = {171-185}, peerreviewed = {Yes}, title = {{An} analysis of crystal dissolution fronts in flows through porous media. {Part} 1: {Compatible} boundary conditions}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1995/1995_KnDuijnHengst_AnalysOfCrystDissolFroInFlowsThPM Part1CompBoundCond}, volume = {18}, year = {1995} } @incollection{faucris.123855424, abstract = {In this paper we perform a numerical bifurcation analysis of a one-dimensional problem arising in premixed combustion in porous inert media. The analysis shows that multiple steady solutions may occur, that a minimal mass flow rate is required for steady combustion, and that solutions with complete combustion exist for a large range of mass flow rates. Moreover, we show that a jump discontinuity in porosity and cooling contributes to the stability of the combustion zone. The one-dimensional results are compared with computations for a corresponding two-dimensional problem. The qualitative agreement between the solutions is good along the cooling boundary, and satisfactory along the axis of symmetry. The one-dimensional problem can therefore be used as preparation for a more complex two-dimensional simulatio}, address = {Berlin, Heidelberg}, author = {De Neef, Michel J. and Knabner, Peter and Summ, Gerhard}, booktitle = {High Performance Scientific and Engineering Computing}, doi = {10.1007/978-3-642-60155-2_4}, editor = {Hans-Joachim Bungartz, Franz Durst, Christoph Zenger}, faupublication = {yes}, pages = {39-50}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes in Computational Science and Engineering}, title = {{Numerical} {Bifurcation} {Analysis} of {Premixed} {Combustion} in {Porous} {Inert} {Media}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1999/1999_NeefKnSumm_NumericalBifurcationAnalysisOfPremixedCombInPorInertMed}, volume = {8}, year = {1999} } @article{faucris.109104864, author = {Grün, Günther}, faupublication = {no}, journal = {SIAM Journal on Mathematical Analysis}, pages = {992-1006}, peerreviewed = {Yes}, title = {{Droplet} spreading under weak slippage: a basic result on finite speed of propagation}, volume = {34}, year = {2003} } @article{faucris.201552267, abstract = {

In this paper, we derive mathematical models for the dynamics of superparamagnetic nanoparticles under the influence of applied magnetic fields. Such models are needed in many applications, e.g. magnetic drug targeting in cancer therapy. They are the starting point for the development of stable numerical approaches and for the formulation of optimization problems, which are needed for the optimal design of magnetic field configurations. Furthermore, we show the existence and uniqueness of classical radially symmetrical solutions and illustrate their qualitative behaviour by numerical simulations using Matlab.

We derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. Describing the polymers as bead chains governed by general elastic spring potentials, including in particular Hookean and finitely extensible, nonlinear elastic (FENE) potentials, it couples a Fokker-Planck type equation describing distribution and orientation of the polymer chains with Cahn–Hilliard and Navier–Stokes type equations describing the balance of mass and momentum. Allowing for different solubility properties which are modelled by Henry type energy functionals, the presented model covers the case of one Newtonian fluid and one non-Newtonian fluid as well as the case of two non-Newtonian fluids. In the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids.

In the case of dumbbell models, we show existence of solutions and present numerical simulations in two space dimensions on oscillating polymeric droplets.

}, author = {Grün, Günther and Metzger, Stefan}, booktitle = {Advances in Mathematical Fluid Mechanics}, faupublication = {yes}, pages = {291-303}, peerreviewed = {Yes}, publisher = {Springer}, series = {Transport processes at fluidic interfaces}, title = {{Micro}-macro-models for two-phase flow of dilute polymeric solutions: macroscopic limit, analysis, numerics}, year = {2017} } @article{faucris.123963004, abstract = {In this paper we present a model for crystal dissolution in porous media and analyse travelling wave solutions of the ensuing equations for a one-dimensional flow situation. We demonstrate the structure of the waves and we prove existence and uniqueness. The travelling wave description is characterized by a rate parameter k and a diffusion/dispersion parameter D. We investigate the limit processes as k→∞and D→0 and we obtain expressions for the rate of convergence. We also present some numerical result}, author = {Van Duijn, C. J. (Hans) and Knabner, Peter}, faupublication = {yes}, journal = {European Journal of Applied Mathematics}, pages = {49-72}, peerreviewed = {Yes}, title = {{Travelling} wave behaviour of crystal dissolution in porous media flow}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_DuijnKn_TravWaveBehavOfCrystDissInPMflow}, volume = {8}, year = {1997} } @book{faucris.206530978, address = {Aachen}, author = {Prechtel, Alexander}, faupublication = {yes}, isbn = {3-8322-4295-3}, peerreviewed = {Yes}, publisher = {Shaker}, title = {{Modelling} and {Efficient} {Numerical} {Solution} of {Hydrogeochemical} {Multicomponent} {Transport} {Problems} by {Process}-{Preserving} {Decoupling} {Techniques}}, year = {2005} } @phdthesis{faucris.111348644, author = {Kräutle, Serge}, faupublication = {yes}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{A} {Navier}-{Stokes} solver based on {CGBI} and the method of characteristics}, url = {https://www1.am.uni-erlangen.de/research/diss/PhD_Kraeutle.pdf}, year = {2001} } @article{faucris.106888364, abstract = {A mixed finite element discretization is applied to Richards' equation, a nonlinear, possibly degenerate parabolic partial differential equation modeling water flow through porous medium. The equation is considered in its pressure formulation and includes both variably and fully saturated flow regime. Characteristic for such problems is the lack in regularity of the solution. To handle this we use a time-integrated scheme. We analyze the scheme and present error estimates showing its convergenc}, author = {Schneid, Eckhard and Knabner, Peter and Radu, Adrian Florin}, doi = {10.1007/s00211-003-0509-2}, faupublication = {yes}, journal = {Numerische Mathematik}, pages = {353-370}, peerreviewed = {Yes}, title = {{A} priori error estimates for a mixed finite element discretization of the {Richards}' equation}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_SchneidKnRadu_APrioriErrorEstimatesForAMixedRichardsEquation}, volume = {98}, year = {2004} } @book{faucris.123750484, abstract = {Transport und Sorption gelöster Stoffe in porösen Medien sind von fundamentaler Bedeutung in verschiedenen Gebieten, von analytischer Chemie und chemischer Verfahrenstechnik zu Bodenkunde und Hydrologie. Die vorliegende Arbeit entwickelt ein allgemeines mathematisches Modell in Form eines Systems aus nichtlinearen parabolischen und gewöhnlichen Differentialgleichungen. Die typische Form von Adsorptionsisothermen hat Singularitäten in den Nichtlinearitäten des Systems zur Folge, sogenannte Degeneration. Nach grundsätzlichen Fragen (eindeutige Existenz, Stabilität, Grenzverhalten) werden für eine Raumdimension die qualitativen Eigenschaften von Lösungen untersucht. Nichtlineare Struktur und Degeneration bedingen laufende Wellen und die (physikalisch korrekte) Existenz scharfer Konzentrationsfronten. Numerische Simulationen schließen die Arbeit ab.}, author = {Knabner, Peter}, faupublication = {no}, isbn = {9783631437186}, peerreviewed = {unknown}, publisher = {Peter Lang}, series = {Methoden und Verfahren der mathematischen Physik}, title = {{Mathematische} {Modelle} für {Transport} und {Sorption} gelöster {Stoffe} in porösen {Medien}}, year = {1991} } @incollection{faucris.121994664, address = {Heidelberg}, author = {Marheineke, Nicole}, booktitle = {Studien- und Berufsplaner Mathematik}, faupublication = {yes}, pages = {140-142}, peerreviewed = {unknown}, publisher = {Springer Spektrum}, title = {{Problemlösungskompetenz} {Mathematische} {Modellierung}}, volume = {5. ed.}, year = {2015} } @incollection{faucris.116444064, author = {Grün, Günther and Rumpf, Martin and Lenz, Martin}, booktitle = {Proceedings of the Third International Symposium on Finite Volumes for Complex Applications}, editor = {R. Herbin, D. Krönger}, faupublication = {no}, isbn = {1-9039-9634-1}, pages = {567-574}, peerreviewed = {unknown}, publisher = {Hermes Penton Science}, title = {{A} finite volume scheme for surfactant driven thin film flow}, year = {2002} } @incollection{faucris.123194984, author = {Lu, Y. and Marheineke, Nicole and Mohring, Jan}, booktitle = {PMS&A - Modelling, Simulation and Application}, faupublication = {yes}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Interpolation} strategy for {BT}-based parametric {MOR} of gas pipeline-networks}, year = {2016} } @article{faucris.223661270, abstract = {We study travelling wave solutions for the model developed in Part 1 of this paper. We develop and discuss a condition characterizing their existence. The possibility of finiteness is investigated. We consider the convergence to various limit cases and point out their different qualitative behaviour. Numerical examples are discusse}, author = {van Duijn, CJ Hans and Knabner, Peter}, doi = {10.1007/BF00618542}, faupublication = {no}, journal = {Transport in Porous Media}, keywords = {Travelling wave, self-sharpening front, mathematical analysis, mathematical model, ion exchange}, pages = {199-225}, peerreviewed = {Yes}, title = {{Travelling} {Waves} in the {Transport} of {Reactive} {Solutes} through {Porous} {Media}: {Adsorption} and {Binary} {Ion} {Exchange} - {Part} 2*}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1992}, volume = {8}, year = {1992} } @inproceedings{faucris.124084004, author = {Hoffmann, Joachim and Kräutle, Serge and Knabner, Peter}, booktitle = {International Workshop on Modelling Reactive Transport in Porous Media}, faupublication = {yes}, peerreviewed = {unknown}, title = {{Computation} of the {MoMaS} {Benchmark}-{Problem} with a {Parallel} {Global}-{Implicit} 2-{D} solver based on a {Reformulation} of the {PDE}-{ODE} {System}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2008/2008_HoffmannKraeutleKn_ComputOfMoMasBenchmProblWthParallGlobImplic2DsolvBasOnAReform}, venue = {Strasbourg}, year = {2008} } @incollection{faucris.117974824, abstract = {A two—scale model for liquid—solid phase transitions with equiaxed dendritic microstructure for binary material with slow solute diffusion is presented. The model consists of a macroscopic energy transport equation, coupled with local cell problems describing the evolution of the microstructure and the microsegregation. It is derived by an asymptotic expansion of a sharp interface model with Gibbs—Thomson effect. A discretization of the model leading to a two—scale method for such problems is presented, and a numerical example is given}, address = {Berlin, Heidelberg}, author = {Eck, Christof and Knabner, Peter}, booktitle = {High Performance Scientific And Engineering Computing}, doi = {10.1007/978-3-642-55919-8_26}, editor = {Michael Breuer, Franz Durst, Christoph Zenger}, faupublication = {yes}, isbn = {978-3-540-42946-3}, pages = {237-244}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes in Computational Science and Engineering}, title = {{A} {Two}—{Scale} {Method} for {Liquid}—{Solid} {Phase} {Transitions} with {Dendritic} {Microstructure}}, url = {http://link.springer.com/chapter/10.1007%2F978-3-642-55919-8_26}, volume = {21}, year = {2002} } @article{faucris.215926552, abstract = {It has been shown that hematopoietic stem cells migrate in vitro and in vivo following the gradient of a chemotactic factor produced by stroma cells. In this paper, a quantitative model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results. Our investigations represent a first step in the process of elucidating the mechanism underlying the homing of hematopoietic stem cells quantitativel}, author = {Kettemann, Anita and Neuss-Radu, Maria}, doi = {10.1007/s00285-007-0132-4}, faupublication = {no}, journal = {Journal of Mathematical Biology}, keywords = {Chemotaxis equations; Homing; Hematopoietic stem cells; Nonlinear boundary conditions}, pages = {579-610}, peerreviewed = {Yes}, title = {{Derivation} and analysis of a system modeling the chemotactic movement of hematopoietic stem cells}, volume = {56}, year = {2008} } @article{faucris.119872764, abstract = {We perform the periodic homogenization (i. e. e ¿ 0) of the non-stationary Nernst-Planck-Poisson system using two-scale convergence, where e is a suitable scale parameter. The objective is to investigate the influence of variable choices of scaling in e of the microscopic system of partial differential equations on the structure of the (upscaled) limit model equations. Due to the specific nonlinear coupling of the underlying equations, special attention has to be paid when passing to the limit in the electric drift term. As a direct result of the homogenization procedure, various classes of upscaled model equations are obtaine}, author = {Ray, Nadja and Eck, Christof and Muntean, Adrian and Knabner, Peter}, faupublication = {yes}, journal = {The Preprint-Series of the Institute of Applied Mathematics}, keywords = {Homogenization; two-scale convergence; porous media; Nernst-Planck-Poisson system; colloidal transport}, pages = {1-32}, peerreviewed = {No}, title = {{Variable} {Choices} of {Scaling} in the {Homogenization} of a {Nernst}-{Planck}-{Poisson} {Problem}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2011/2011_RayEckMunteanKn_VarChoiOfScalInTheHomogeniOfNernstPlanPoiProblem}, volume = {344}, year = {2011} } @incollection{faucris.117979004, abstract = {Mathematical models provide the starting point for the simulation of complex processes, which arise in the natural and engineering sciences. Characteristic properties of the considered systems are represented by model parameters or coefficients. These have to be determined by experiments. If the coefficients are not measured directly, as direct measurements are not possible or do not lead to satisfying results, numerical identification procedures have to be applie}, address = {Berlin, Heidelberg}, author = {Knabner, Peter and Igler, B. A.}, booktitle = {Lectures on Applied Mathematics}, doi = {10.1007/978-3-642-59709-1_12}, editor = {Hans-Joachim Bungartz, Ronald H. W. Hoppe, Christoph Zenger}, faupublication = {yes}, isbn = {978-3-642-64094-0}, pages = {157-175}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Structural} {Identification} of {Nonlinear} {Coefficient} {Functions} in {Transport} {Processes} through {Porous} {Media}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2000/2000_KnIgler_StructIdentifOfNonlinCoeffFunctInTranspProcessThroughPM}, year = {2000} } @article{faucris.119221344, abstract = {In this paper we present a rigorous error analysis for the Lagrange-Galerkin method applied to convection-dominated diffusion problems. We prove new error estimates in which the constants depend on norms of the data and not of the solution and do not tend to infinity in the hyperbolic limit. This is in contrast to other results in this field. For the time discretization, uniform convergence with respect to the diffusion parameter of order O(k/t) is shown for initial values in L and O(k) for initial values in H. For the spatial discretization with linear finite elements, we verify uniform convergence of order O(h + min{h, h/k}) for data in H. By interpolation of Banach spaces, suboptimal convergence rates are derived under less restrictive assumptions. The analysis is heavily based on a priori estimates, uniform in the diffusion parameter, for the solution of the continuous and the semidiscrete problem. They are derived in a Lagrangian framework by transforming the Eulerian coordinates completely into subcharacteristic coordinates. Finally, we illustrate the error estimates by some numerical result}, author = {Bause, Markus and Knabner, Peter}, doi = {10.1137/S0036142900367478}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {ε-uniform convergence; Characteristics; Convection-diffusion equation; Finite element method; Lagrange-Galerkin scheme}, pages = {1954-1984}, peerreviewed = {Yes}, title = {{Uniform} error analysis for {Lagrange}-{Galerkin} approximations of convection-dominated problems}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_BauseKn_UniformErrAnalysisForLangrGalerkinApproxOfConveDominaProb}, volume = {39}, year = {2002} } @incollection{faucris.222586691, abstract = {An algorithm for the numerical simulation of saturated-unsaturated flow through porous media is described and some aspects of its performance are outline}, address = {Boston}, author = {Knabner, Peter}, booktitle = {Large Scale Scientific Computing}, doi = {10.1007/978-1-4684-6754-3_6}, editor = {P. Deuflhard B. Engquist}, faupublication = {no}, isbn = {978-1-4684-6756-7}, pages = {83-93}, peerreviewed = {unknown}, publisher = {Birkhäuser}, series = {Progress in Scientific Computing}, title = {{Finite} {Element} {Simulation} of {Saturated}-{Unsaturated} {Flow} through {Porous} {Media}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1987/1987_Kn_FinitElementSimulOfSaturUnsaturFlowThroughPM}, volume = {7}, year = {1987} } @article{faucris.108978584, author = {Grün, Günther}, doi = {10.1137/130908208}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {two-phase flow; Cahn-Hilliard equation; diffuse interface model; convergence of finite-element schemes}, pages = {3036-3061}, peerreviewed = {Yes}, title = {{On} convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities}, volume = {51}, year = {2013} } @article{faucris.209240734, author = {Reuter, Balthasar and Rupp, Andreas and Aizinger, Vadym and Knabner, Peter}, doi = {10.1016/j.camwa.2018.12.020}, faupublication = {yes}, journal = {Computers & Mathematics With Applications}, keywords = {Darcy flow; Coupled model; Hydrostatic equations; Discrete energy stability analysis; Three-dimensional shallow water equations with free surface; Local discontinuous Galerkin method}, pages = {2291-2309}, peerreviewed = {Yes}, title = {{Discontinuous} {Galerkin} method for coupling hydrostatic free surface flows to saturated subsurface systems}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_ReuterRuppAizingerKn_DiscontGalerkinMethForCouplHydrostFreeSurfaceFlowsToSaturSubsurfSyst}, volume = {77}, year = {2019} } @masterthesis{faucris.110910184, author = {Marheineke, Nicole}, faupublication = {no}, peerreviewed = {automatic}, school = {Friedrich-Alexander-Universität Erlangen-Nürnberg}, title = {{Modified} {FEM} for {Fibre}-{Fluid} {Interactions}}, year = {2001} } @article{faucris.121945384, abstract = {The paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber. Independent of the choice of the drag model, a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow. Its construction as a centered differentiable Gaussian field complies thereby with the requirements of the stochastic k-ε turbulence model and Kolmogorov's universal equilibrium theory on local isotropy. © 2006 Society for Industrial and Applied Mathematics.}, author = {Marheineke, Nicole and Wegener, Raimund}, doi = {10.1137/050637182}, faupublication = {no}, journal = {SIAM Journal on Applied Mathematics}, keywords = {Cosserat rod; Diflerentiable gaussian fields; Double-velocity correlations; Fiber-fluid interaction; Kolmogorov's energy spectrum; Turbulence modeling}, pages = {1703-1726}, peerreviewed = {Yes}, title = {{Fiber} dynamics in turbulent flows: {General} modeling framework}, volume = {66}, year = {2006} } @incollection{faucris.223662715, abstract = {Dissolved organic Carbon(DOC) in soils and aquifers is considered as a sorbent and carrier for organic contaminants. The knowledge of the interaction between hydophobic organic chemicals (HOC) and soil or aquifer materials in the presence of mobile carriers, such as DOC, plays an important role in understanding of the transport behavior of the HOC. Several studies focused on the influence of the chemical characteristics of the contaminant on solubility enhancement by DOC [6]. Enhanced transport of pesticides and hydrophobic organic chemicals mediated by the presence of DOC in soils and other porous media has been documented by a number of studies [14]. In these studies, it is generally assumed that DOC is an non-reactive component of the aqueous phase with respect to interactions with bulk soil material (BSM). This inhanced mobility or leaching of the contaminant may have impact on risk assessment studies. Recently, there is experimental evidence that also the carrier DOC itself and the HOC-DOC compounds undergo adsorption to the soil matrix [3]. This process reduces again the mobility of the contaminant such that the overall result is not clea}, address = {Dordrecht}, author = {Knabner, Peter and Kögel-Knabner, Ingrid and Totsche, K. U.}, booktitle = {Contaminated Soil '93}, doi = {10.1007/978-94-011-2018-0_31}, faupublication = {no}, isbn = {978-94-010-4892-7}, keywords = {Porous Medium, Mass Concentration, Sorption Site, Risk Assessment Study, Total Dissolve Organic Carbon}, pages = {163-172}, peerreviewed = {unknown}, publisher = {Springer}, series = {Soil and Environment}, title = {{CARRIER} {INFLUENCED} {TRANSPORT} {OF} {POLLUTANTS} {IN} {POROUS} {MEDIA}: {MATHEMATICAL} {MODELLING}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1993/1993_KnKoegelknabTotsche_CarrInfluencedTransOfPollutantsInPM}, volume = {2}, year = {1993} } @article{faucris.215928610, abstract = {Mathematical modeling and simulation of fluid-structure interaction problems are in the focus of research already for a longer period. However, taking into account also chemical reactions, leading to structural changes, including changes of mechanical properties of the solid phase, is rather new but for many applications is highly important area. This paper formulates a model system for reactive flow and transport in a vessel system, the penetration of chemical substances into the solid wall. Inside the wall, reactions take place that lead to changes of volume and of the mechanical properties of the wall. Numerical algorithms are developed and used to simulate the dynamics of such a mechano-chemical fluid-structure interaction problem. As a proof of concept scenario, plaque formation in blood vessels is chosen. The arbitrary Lagrangian Eulerian approach (ALE) is chosen to solve the systems numerically. Temporal discretization of the fully coupled monolithic model is accomplished by backward Euler scheme and spatial discretization by stabilized finite elements. The numerical approach is verified by numerical tests, and effective methods to maintain mesh qualities under large deformations are described. For realistic system parameters, the simulations show that the plaque formation in blood vessel is a long-time effect. The time scale of the formation is in the simulation of comparable order as in reality. Copyright (c) 2016 John Wiley & Sons, Ltd.}, author = {Yang, Yifan and Richter, Thomas and Jäger, Willi and Neuss-Radu, Maria}, doi = {10.1002/fld.4345}, faupublication = {yes}, journal = {International Journal For Numerical Methods in Fluids}, keywords = {mechano-chemical fluid-structure interaction;growth modeling;monolithic ALE framework;harmonic and biharmonic extension;finite elements;local projection stabilization;simulation of plaque formation}, pages = {199-220}, peerreviewed = {Yes}, title = {{An} {ALE} approach to mechano-chemical processes in fluid-structure interactions}, volume = {84}, year = {2017} } @article{faucris.117785624, abstract = {In this paper we consider the Lagrange-Galerkin finite element approximation by continuous piecewise linears in space of the following problem: Given ω ⊂ R, 1 ≤ d ≤ 3, find u(x, t) and v(x, t) such that ∂u + ∂v - ∇.(D ∇u) + q.∇u = f in ω × (0, T], dv = k(φ(u) - v) in ω × (0, T], u(x, 0) = g(x), v(x, 0) = g(x) ∀ x ∈ ω, with periodic boundary conditions. Here k ∈ R and the spatial differential operator is uniformly elliptic, but φ ∈ C(R) ∩ C (-∞, 0] ∪ (0, ∞) is a monotonically increasing function satisfying φ(0) = 0, which is only locally Hölder continuous, with exponent p ∈ (0, 1) at the origin; e.g., ∈(s) := [s] . We obtain error bounds which improve on those in the literatur}, author = {Barrett, John W. and Knabner, Peter}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {Convection-dominated flow; Degenerate parabolic systems; Error analysis; Finite elements; Lagrange-galerkin; Modified method of characteristics; Porous medium}, pages = {1862-1882}, peerreviewed = {Yes}, title = {{An} improved error bound for a {Lagrange}-{Galerkin} method for contaminant transport with non-lipschitzian adsorption kinetics}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1998/1998_BarrettKn_ImprErrBoundForLagrGalerkMethForContTranspNonLipAdsKinet}, volume = {35}, year = {1998} } @inproceedings{faucris.207760690, author = {Burger, Martin and et al.}, author_hint = {Burger M, Chu S, Markowich P, Schönlieb C}, booktitle = {PAMM Special Issue: ICIAM 07}, doi = {10.1002/pamm.200700802}, faupublication = {no}, note = {WWU::69510}, pages = {1011209-1011210}, peerreviewed = {No}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{Cahn}-{Hilliard} inpainting and the {Willmore} functional}, venue = {Zurich}, year = {2008} } @article{faucris.123847944, abstract = {The semismooth Newton method was introduced in a paper by Qi and Sun (Math. Program. 58:353-367, 1993) and the subsequent work by Qi (Math. Oper. Res. 18:227-244, 1993). This method became the basis of many solvers for certain classes of nonlinear systems of equations defined by a nonsmooth mapping. Here we consider a particular system of equations that arises from the discretization of a reactive transport model in the subsurface including mineral precipitation-dissolution reactions. The model is highly complicated and uses a coupling of PDEs, ODEs, and algebraic equations, together with some complementarity conditions arising from the equilibrium conditions of the minerals. The aim is to show that this system, though quite complicated, usually satisfies the convergence criteria for the semismooth Newton method, and can therefore be solved by a locally quadratically convergent method. This gives a theoretical sound approach for the solution of this kind of applications, whereas the geoscientists community most frequently applies algorithms involving some kind of trial-and-error strategies. © 2010 Springer Science+Business Media, LL}, author = {Buchholzer, Hannes and Kanzow, Christian and Knabner, Peter and Kräutle, Serge}, doi = {10.1007/s10589-010-9379-6}, faupublication = {yes}, journal = {Computational Optimization and Applications}, keywords = {Complementarity problems; Mineral precipitation-dissolution; Quadratic convergence; Reactive transport; Semismooth Newton method}, pages = {193-221}, peerreviewed = {Yes}, title = {{The} semismooth {Newton} method for the solution of reactive transport problems including mineral precipitation-dissolution reactions}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2011/2011_BuchholzerKanzowKnKraeutle_TheSemismoothNewtonMethodForSolutionIncludeMineral}, volume = {50}, year = {2011} } @article{faucris.117817084, abstract = {Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphat}, author = {Elbinger, Tobias and Gahn, Markus and Hante, Falk and Voll, Lars and Leugering, Günter and Knabner, Peter and Neuss-Radu, Maria}, doi = {10.3389/fbioe.2016.00013}, faupublication = {yes}, journal = {Frontiers in Bioengineering and Biotechnology}, peerreviewed = {Yes}, title = {{Model}-{Based} {Design} of {Biochemical} {Microreactors}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2016/2016_ElbingerGahnNeusRaduHanteVollLeugering_ModelBasedDesignOfBioMicroReact}, year = {2016} } @article{faucris.116668684, author = {Grün, Günther and Becker, Jürgen}, faupublication = {no}, journal = {Proceedings in Applied Mathematics and Mechanics}, pages = {47-50}, peerreviewed = {Yes}, title = {{Some} remarks on modelling and simulation of thin-film flow}, volume = {4}, year = {2004} } @article{faucris.222788602, author = {Schulz, Raphael and Ray, Nadja and Zech, Simon and Rupp, Andreas and Knabner, Peter}, doi = {10.1007/s11242-019-01321-y}, faupublication = {yes}, journal = {Transport in Porous Media}, peerreviewed = {Yes}, title = {{Beyond} {Kozeny}-{Carman}: {Predicting} the {Permeability} in {Porous} {Media}}, year = {2019} } @article{faucris.121082104, abstract = {We present a cross-platform scaling investigation for an OpenMP parallelization of UTBEST3D – a coastal and regional ocean code based on the discontinuous Galerkin finite element method. The study is conducted for a real life application on an unstructured computational mesh of the Northwest Atlantic with realistic topography and well resolved coast line on a broad selection of current computing platforms. Four numerical setups of increasing physical and computational complexity are used for comparison: barotropic with no vertical eddy viscosity, barotropic with an algebraic eddy viscosity parametrization, baroclinic with an algebraic eddy viscosity, and baroclinic withparabolic systems arising from flow problems in porous media with slow adsorption. The

characteristic difficulty of these problems comes from their monotone but non-Lipschitzian

nonlinearity. For a model problem of this type, optimal-order pointwise error estimates

are derived for the spatial semi-discretization by the finite element Galerkin method. The

proof is based on linearization through a parabolic duality argument in L∞(L∞) spaces and

corresponding sharp L1 estimates for regularized parabolic Green functions.

p) the travelling wave may exist. For

We have implemented a particular variant of the phase-field method (PFM) into the computational core of a pore-scale multiphase flow simulator, namely PMFS-PFM, for the numerical simulations of incompressible flows of two immiscible fluid phases. The implementation was discussed in a previous paper for rectangular prism-shaped fully-connected domains, e.g., for simulating two-phase flow in a 2D slit or a 3D duct (Alpak et al., 2016). In this paper, we discuss the recent developments on PMFS-PFM. The main components of the new work are (I) implementation of support for inactive cells in PMFS-PFM by extending the original finite-volume method-based discretizations of the underlying partial differential equations (PDEs) in order to solve realistic 3D pore-scale flow problems on rock volumes stemming from imaging, and (II) enhancing the performance of the simulator through implementation of modern sparse linear solvers and distributed parallel computing. It has been shown that the simulations performed on complex pore-scale domains are consistent with the physics of the immiscible two-phase displacement. The parallel scalability of the code is reasonably well varying between 50% and 86% on the investigated test cases of varying complexity. Results indicate that the more disconnected the pore-scale domains, the lower the parallel efficiency. It has been noted that there is a possibility of improving the parallel efficiency by exploring various grid subdivisions.

0, alpha is an element of (1, n/n-2] and distributed with period epsilon. On the boundary of the balls, we have the following nonlinear restrictions u(epsilon) >= 0, partial derivative(nu)u(epsilon) >= -epsilon(-gamma) sigma (x, u(epsilon)), u(epsilon)(partial derivative(nu)u(epsilon) + epsilon(-gamma) sigma(x, u(epsilon)) = 0, gamma = alpha (n - 1) - n. The weak convergence of the solutions s to the solution of an effective problem is given. In the critical case alpha = n/n-2, the effective equation contains a nonlinear term which has to be determined as a solution of a functional equation. Furthermore, a corrector result with respect to the energy norm is proved. (C) 2012 Elsevier Ltd. All rights reserved.}, author = {Neuss-Radu, Maria and Jäger, Willi and Shaposhnikova, Tatiana A.}, doi = {10.1016/j.nonrwa.2012.01.027}, faupublication = {no}, journal = {Nonlinear Analysis-Real World Applications}, keywords = {Perforated domains;Small holes;Variational inequality;Nonlinear restrictions;Functional equation}, month = {Jan}, pages = {367-380}, peerreviewed = {Yes}, title = {{Homogenization} of a variational inequality for the {Laplace} operator with nonlinear restriction for the flux on the interior boundary of a perforated domain}, volume = {15}, year = {2014} } @article{faucris.113577244, author = {Grün, Günther and Bertsch, Michiel and Dal Passo, Roberta and Garcke, Harald}, faupublication = {no}, journal = {Advances in Differential Equations}, pages = {417-440}, peerreviewed = {unknown}, title = {{The} thin viscous flow equation in higher space dimensions}, volume = {3}, year = {1998} } @article{faucris.116433284, author = {Grün, Günther and Eck, Christof and Fontelos, Marco Antonio and Klingbeil, Fabian and Vantzos, Orestis}, doi = {10.4171/IFB/211}, faupublication = {yes}, journal = {Interfaces and Free Boundaries}, pages = {259-290}, peerreviewed = {Yes}, title = {{A} phase-field model for electrowetting}, volume = {11}, year = {2009} } @article{faucris.215926246, abstract = {Boundary layers are used in the homogenization of elliptic problems with periodically oscillating coefficients, for example when we want to improve the macroscopic approximation given by homogenization in the neighborhood of the boundary of a domain. For problems with a special geometry the boundary layers are defined on a semi-infinite strip ]0, 1[n-1 ×]0, ∞[, and their energies decrease exponentially with respect to the second variable. In our paper, we show that in general this decay property does not hold, i.e., we cannot get uniform exponential decay of the boundary layer}, author = {Neuss-Radu, Maria}, faupublication = {no}, journal = {Asymptotic Analysis}, pages = {313-328}, peerreviewed = {Yes}, title = {{A} result on the decay of the boundary layers in the homogenization theory}, url = {https://www.scopus.com/record/display.uri?eid=2-s2.0-0040305966&origin=inward}, volume = {23}, year = {2000} } @article{faucris.117784304, abstract = {[1] A new systematic approach for the efficient computation of the transport and reaction of a multispecies multireaction system is developed. The objective of this approach is to reduce the number of coupled nonlinear differential equations drastically, while splitting errors are avoided. The reduction mechanism is able to handle both kinetic reactions and heterogeneous equilibrium reactions and mobile and immobile species. It leads to a formulation of the nonlinear system with a Jacobian that has very few nonzero entries. Applications of the reduction mechanism to reaction networks, including a biodegradation problem which is modeled by the Monod approach, are given. Two numerical examples demonstrate the speed up of the presented reduction mechanism. Copyright 2005 by the American Geophysical Unio}, author = {Kräutle, Serge and Knabner, Peter}, doi = {10.1029/2004WR003624}, faupublication = {yes}, journal = {Water Resources Research}, pages = {1-17}, peerreviewed = {Yes}, title = {{A} new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2005/2005_KraeutleKn_ANewNumericalReductionSchemeForFullyCoupledMultiCompTransReactionProbInPM}, volume = {41}, year = {2005} } @article{faucris.117788484, abstract = {This paper deals with the analytical investigations of a nonstationary Darcy-Nernst-Planck-Poisson-type system which is a mathematical model describing electrolyte solutions. For this nonlinear fully coupled system of partial differential equations we proof existence and uniqueness of global weak solutions. Furthermore, we establish physical properties for the number densities such as nonnegativity and boundedness in L∞ ((0, T) × ω). © 2012 WILEY-VCH Verlag GmbH & Co. KGa}, author = {Herz, Matthias and Ray, Nadja and Knabner, Peter}, doi = {10.1002/gamm.201210013}, faupublication = {yes}, journal = {GAMM-Mitteilungen}, keywords = {Darcy-Nernst-Planck-Poisson system; Electro-hydrodynamics; Fixed point approach; Global existence; Moser iteration}, pages = {191-208}, peerreviewed = {unknown}, title = {{Existence} and uniqueness of a global weak solution of a {Darcy}-{Nernst}-{Planck}-{Poisson} system}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2012/2012_HerzRayKn_ExistAndUniqueOfGlobWeakSolutionDarcyNernPlanPoissSystem}, volume = {35}, year = {2012} } @article{faucris.106902664, abstract = {Biodegradable collagen matrices have become a promising alternative to synthetic polymers as drug delivery systems for sustained release. Previously, a mathematical model describing water penetration, matrix swelling and drug release by diffusion from dense collagen matrices was introduced and tested (cf. Radu et al. in J. Pharm. Sci. 91:964-972, 2002). However, enzymatic matrix degradation influences the drug release as well. Based on experimental studies (cf. Metzmacher in Enzymatic degradation and drug release behavior of dense collagen implants. Ph.D. thesis, LMU University of Munich, 2005), a mathematical model is presented here that describes drug release by collagenolytic matrix degradation. Existence and uniqueness of a solution of the model equations is reviewed. A mixed Raviart-Thomas finite element discretization for solving the coupled system of partial and ordinary differential equations is proposed and analyzed theoretically. The model is verified by a comparison of numerically calculated and experimentally measured data and, in particular, investigated by a parameter sensitivity study. For illustration, some concentration profiles of a twodimensional simulation are shown. © The Author(s) 200}, author = {Radu, F. A. and Bause, Markus and Knabner, Peter and Friess, W. and Metzmacher, I.}, doi = {10.1007/s00791-008-0118-9}, faupublication = {yes}, journal = {Computing and Visualization in Science}, pages = {409-420}, peerreviewed = {unknown}, title = {{Numerical} simulation of drug release from collagen matrices by {Enzymatic} degradation}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2009/2009_RaduBauseKnFriessMetzmacher_NumericalSimulationOfDrugReleaFromCollagenMetricsByEnzymDegrad}, volume = {12}, year = {2009} } @article{faucris.117796624, abstract = {The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion. © 2009 The American Physical Societ}, author = {Suciu, Nicolae and Vamo̧s, Cǎlin and Radu, Adrian Florin and Vereecken, Harry and Knabner, Peter}, doi = {10.1103/PhysRevE.80.061134}, faupublication = {yes}, journal = {Physical Review E}, peerreviewed = {Yes}, title = {{Persistent} memory of diffusing particles}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2009/2009_SuciuVamosRaduVereeckenKn_PersistMemoryOfDiffParticles}, volume = {80}, year = {2009} } @incollection{faucris.117975044, abstract = {For the Richards equation, a nonlinear elliptic-parabolic partial differential equation modelling saturated-unsaturated flow in porous media, we present a hybrid mixed finite element discretization. The efficiency of the algorithm is improved by local time step and grid adaption. The adaption algorithms are based on rigorous error estimators, whose derivation is indicated. Examples elucidate the performance of the algorit}, address = {Berlin, Heidelberg}, author = {Knabner, Peter and Schneid, Eckhard}, booktitle = {High Performance Scientific And Engineering Computing}, doi = {10.1007/978-3-642-55919-8_4}, editor = {Michael Breuer, Franz Durst, Christoph Zenger}, faupublication = {yes}, isbn = {978-3-540-42946-3}, pages = {37-44}, peerreviewed = {unknown}, publisher = {Springer}, series = {Lecture Notes in Computational Science and Engineering}, title = {{Adaptive} {Hybrid} {Mixed} {Finite} {Element} {Discretization} of {Instationary} {Variably} {Saturated} {Flow} in {Porous} {Media}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_KnSchneid_AdaptHybridMixedFiniteElementDiscretiOfInstatVariablySaturFlowInPM}, volume = {21}, year = {2002} } @incollection{faucris.116677484, author = {Arne, Walter and Rivero-Rodriguez, Javier and Pérez-Saborid, Miguel and Marheineke, Nicole and Wegener, Raimund}, booktitle = {Progress in Industrial Mathematics at ECMI 2014}, faupublication = {yes}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Homotopy} method for viscous {Cosserat} rod model describing electrospinning}, year = {2016} } @article{faucris.108977484, abstract = {By methods from nonequilibrium thermodynamics, we derive a diffuse interface

model for two-phase flow of incompressible fluids with dissolved noninteracting

polymers. The polymers are modeled by dumbbells subjected to general elastic springforce

potentials including in particular Hookean and finitely extensible, nonlinear elastic

(FENE) potentials. Their density and orientation are described by a Fokker-Planck-type

equation which is coupled to a Cahn-Hilliard and a momentum equation for phase-field

and gross velocity/pressure. Henry-type energy functionals are used to describe different

solubility properties of the polymers in the different phases or at the liquid-liquid interface.

Taking advantage of the underlying energetic/entropic structure of the system, we

prove existence of a weak solution globally in time for the case of FENE-potentials. As

a by-product in the case of Hookean spring potentials, we derive a macroscopic diffuse interface

model for two-phase flow of Oldroyd-B-type liquids.}, author = {Grün, Günther and Metzger, Stefan}, doi = {10.1142/S0218202516500196}, faupublication = {yes}, journal = {Mathematical Models & Methods in Applied Sciences}, pages = {823-866}, peerreviewed = {Yes}, title = {{On} micro-macro-models for two-phase flow with dilute polymeric solutions -- modeling and analysis}, volume = {26}, year = {2016} } @article{faucris.117797064, abstract = {We present here the definition of the reactive transport benchmark of Groupement Mathematical Modeling and Numerical Simulation for Nuclear Waste Management Problems. The aim of this benchmark is to propose a challenging test for numerical methods used for reactive transport modeling in porous media. In order to focus on numerical methods, the problem presented here is of quite a small size, both from a hydrodynamical and from a geochemical point of view. Though the chemical coefficients used in this benchmark are not taken from a real chemical system, they are realistic, and the test case is quite challenging. © Springer Science + Business Media B.V. 200}, author = {Carrayrou, Jérôme and Kern, Michel and Knabner, Peter}, doi = {10.1007/s10596-009-9157-7}, faupublication = {yes}, journal = {Computational Geosciences}, keywords = {Benchmark; MoMaS; Porous media; Reactive transport}, pages = {385-392}, peerreviewed = {Yes}, title = {{Reactive} transport benchmark of {MoMaS}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2010/2010_CarrayrouKernKn_ReactiveTransportBenchmarkOfMoMaS}, volume = {14}, year = {2010} } @incollection{faucris.108971544, author = {Arne, Walter and Marheineke, Nicole and Meister, Andreas and Wegener, Raimund}, booktitle = {Progress in Industrial Mathematics at ECMI 2012}, faupublication = {yes}, pages = {99-107}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Numerical} treatment on non-stationary viscous {Cosserat} rod in a two-dimensional {Eulerian} framework}, year = {2014} } @article{faucris.117792664, abstract = {We investigate the influence of aggregation and deposition on the colloidal dynamics in a saturated porous medium. On the pore scale, the aggregation of colloids is modeled by the Smoluchowski equation. Essentially, the colloidal mass splits into different size clusters and we treat clusters as different species involved in a diffusion-reaction mechanism. This modeling procedure allows for different material properties to be varied between the different species, specifically the diffusion rate, which changes due to size as described by the Stokes-Einstein relation, and the deposition rate. The periodic homogenization procedure is applied to obtain a macroscopic model. The resulting model is illustrated by numerical computations that capture the colloidal transport with and without aggregatio}, author = {Krehel, Oleh and Muntean, Adrian and Knabner, Peter}, doi = {10.1016/j.advwatres.2015.10.005}, faupublication = {yes}, journal = {Advances in Water Resources}, keywords = {Aggregation; Colloidal transport; Deposition; Multiscale coefficients; Periodic homogenization}, pages = {209-216}, peerreviewed = {Yes}, title = {{Multiscale} modeling of colloidal dynamics in porous media including aggregation and deposition}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_KrehelMunteanKn_MultiscModelOfColloidDynamInPMinclAggregDepos}, volume = {86}, year = {2015} } @article{faucris.215927610, abstract = {In this paper, we develop multiscale methods appropriate for the homogenization of processes in domains containing thin heterogeneous layers. Our model problem consists of a nonlinear reaction-diffusion system defined in such a domain, and properly scaled in the layer region. Both the period of the heterogeneities and the thickness of the layer are of order epsilon. By performing an asymptotic analysis with respect to the scale parameter epsilon we derive an effective model which consists of reaction-diffusion equations in two domains separated by an interface together with appropriate transmission conditions across the interface. These conditions are determined by solving local problems on the standard periodicity cell in the layer. Our asymptotic analysis is based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers. For the derivation of the transmission conditions, we develop a new method based on test-functions of boundary-layer type.

In this
paper, we develop multiscale methods appropriate for the homogenization
of processes in domains containing thin heterogeneous layers. Our model
problem consists of a nonlinear reaction-diffusion system defined in
such a domain, and properly scaled in the layer region. Both the period
of the heterogeneities and the thickness of the layer are of order
$\varepsilon.$ By performing an asymptotic analysis with respect to the
scale parameter $\varepsilon$ we derive an effective model which
consists of the reaction-diffusion equations on two domains separated by
an interface together with appropriate transmission conditions across
this interface. These conditions are determined by solving local
problems on the standard periodicity cell in the layer. Our asymptotic
analysis is based on weak and strong two-scale convergence results for
sequences of functions defined on thin heterogeneous layers. For the
derivation of the transmission conditions, we develop a new method based
on test functions of boundary layer type.

Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

In this
paper, we develop multiscale methods appropriate for the homogenization
of processes in domains containing thin heterogeneous layers. Our model
problem consists of a nonlinear reaction-diffusion system defined in
such a domain, and properly scaled in the layer region. Both the period
of the heterogeneities and the thickness of the layer are of order
$\varepsilon.$ By performing an asymptotic analysis with respect to the
scale parameter $\varepsilon$ we derive an effective model which
consists of the reaction-diffusion equations on two domains separated by
an interface together with appropriate transmission conditions across
this interface. These conditions are determined by solving local
problems on the standard periodicity cell in the layer. Our asymptotic
analysis is based on weak and strong two-scale convergence results for
sequences of functions defined on thin heterogeneous layers. For the
derivation of the transmission conditions, we develop a new method based
on test functions of boundary layer type.

Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

In this
paper, we develop multiscale methods appropriate for the homogenization
of processes in domains containing thin heterogeneous layers. Our model
problem consists of a nonlinear reaction-diffusion system defined in
such a domain, and properly scaled in the layer region. Both the period
of the heterogeneities and the thickness of the layer are of order
$\varepsilon.$ By performing an asymptotic analysis with respect to the
scale parameter $\varepsilon$ we derive an effective model which
consists of the reaction-diffusion equations on two domains separated by
an interface together with appropriate transmission conditions across
this interface. These conditions are determined by solving local
problems on the standard periodicity cell in the layer. Our asymptotic
analysis is based on weak and strong two-scale convergence results for
sequences of functions defined on thin heterogeneous layers. For the
derivation of the transmission conditions, we develop a new method based
on test functions of boundary layer type.

Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

},
author = {Neuss-Radu, Maria and Jäger, Willi},
doi = {10.1137/060665452},
faupublication = {no},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {Homogenization; Thin heterogeneous layer; Nonlinear reaction-diffusion systems; Two-scale convergence; Transmission conditions},
pages = {687-720},
peerreviewed = {Yes},
title = {{Effective} transmission conditions for reaction-diffusion processes in domains separated by an interface},
volume = {39},
year = {2007}
}
@incollection{faucris.223183641,
abstract = {Sorption of organic chemicals in soils is considered here as an
equilibrium of environmental chemicals in three phases: dissolved,
sorbed to dissolved macromolecules, and sorbed to the bulk soil matrix.
In the present study we demonstrate the binding of organic chemicals to
water-soluble soil organic matter using an experimental acidic herbicide
as an example. Sorption isotherms are determined for the sorption of
organic chemicals to bulk soil materials (Ap horizons) and to
water-soluble organic matter. From these data an effective equilibrium
sorption isotherm to bulk soil can be calculated which takes into
account the sorption of organic chemicals to dissolved organic carbon.
This is compared to a more traditional two-phase model of the same type,
where the mobile sorbent is ignore},
address = {Dordrecht},
author = {Kögel-Knabner, Ingrid and Knabner, Peter and Deschauer, Helmut},
booktitle = {Contaminated Soil '90},
doi = {10.1007/978-94-011-3270-1_72},
editor = {F. Arendt, M. Hinsenveld, W. J. Van Den Brink},
faupublication = {no},
isbn = {978-94-010-5443-0},
keywords = {Bulk Soil, Sorption Isotherm, Elution Curve, Acidic Herbicide, Quinolinecarboxylic Acid},
pages = {323-329},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Enhanced} {Leaching} of {Organic} {Chemicals} in {Soils} due to {Binding} to {Dissolved} {Organic} {Carbon}?},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1990/1990_KoegelknKnDeschauer_EnhancLeachOfOrganChemicInSoilsDueToBindToDissolvOrganCarbon},
year = {1990}
}
@article{faucris.117783644,
author = {Knabner, Peter and Tapp, Christoph and Thiele, Kathrin},
doi = {10.1016/S1464-1909(01)00013-2},
faupublication = {yes},
journal = {Physics and Chemistry of the Earth, Part B},
pages = {319-324},
peerreviewed = {Yes},
title = {{Adaptivity} in the finite volume discretization of variable density flows in porous media},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2001/2001_KnTappThiele_AdaptivityInTheFiniteVolumeDiscretizationOfVarDensFlowsinPM},
volume = {26},
year = {2001}
}
@article{faucris.121687984,
author = {Grün, Günther},
faupublication = {no},
journal = {Annales de l'Institut Henri Poincaré - Analyse Non Linéaire},
pages = {255-269},
peerreviewed = {Yes},
title = {{Droplet} spreading under weak slippage: the waiting time phenomenon},
volume = {21},
year = {2004}
}
@article{faucris.121684684,
author = {Grün, Günther and Rumpf, Martin},
faupublication = {no},
journal = {Numerische Mathematik},
pages = {113-152},
peerreviewed = {Yes},
title = {{Nonnegativity} preserving convergent schemes for the thin film equation},
volume = {87},
year = {2000}
}
@article{faucris.223067314,
abstract = {In this paper we consider the non‐characteristic Cauchy problem Read More: https://epubs.siam.org/doi/abs/10.1137/060665452?casa_token=8mgAtl16CJsAAAAA:AWP6wryXXeEfybHMoDiXWuWmHe7alFKC8xGQjLUJd95i5m5GH-gAlKRp0Saq2DLQBPwY30O71Is

The paper considers the simultaneous identification of parameters and source term appearing in elliptic Partial Differential Equations from knowledge of the reference potential, which satisfies Dirichlet boundary conditions. Identifying the unknowns means complying with the equation error criterion. The adaptive method consists of transforming both the potential and the unknown parameters into time-dependent quantities, which evolve according to given laws. In the continuum case, the time decay property of a Lyapunov function is shown to hold.

Next, the finite-dimensional (Galerkin) approximation of the evolution equations is considered and, given the existence of a solution, the pertaining convergence results are stated s. t., the time-dependent potential approaches the reference potential and the unknowns asymptotically comply with the time-independent equation.

Then, the discrete identification problem is stated. The finite element method is used to obtain the discrete equation error criterion. A Eulerian numerical scheme for all discretized evolution equations is provided. The sequences of the corresponding solutions i. e., the potential and the unknowns, are shown to converge. The detailed proof is provided.

Finally, the above method is applied to the identification of hydraulic conductivity, leakage parameter and source term of a two-layered aquifer in the Aachtal (Germany) area. The numerical results obtained from noiseless and noisy potential data are provided.

A Helmholtz free-energy-driven, thermodynamically-based diffuse-interface/phase-field method is used for the effective simulation of a large number of advecting interfaces, while honoring the interfacial tension. The advective Cahn–Hilliard (mass balance, energy dissipation) and Navier–Stokes (momentum balance, incompressibility) equations are coupled to each other within the phase-field framework. Wettability on rock-fluid interfaces is accounted for via an energy-penalty-based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass balance equation is semi-implicit in time using a convex–concave splitting of the energy term. The momentum balance equation is split from the incompressibility constraint by a projection method and linearized with a Picard splitting. Mass and momentum balance equations are coupled to each other via operator splitting and solved sequentially.

We discuss the mathematical model and its DG discretization and briefly introduce nonlinear and linear solution strategies. Numerical validation tests show optimal convergence rates for the DG discretization indicating the correctness of the numerical scheme and of its implementation. Physical validation tests demonstrate the consistency of the phase distribution and velocity fields simulated within our framework. Finally, two-phase flow simulations on two real pore-scale images demonstrate the utility of the pore-scale simulator. The direct pore-scale numerical simulation methodology rigorously takes into account the flow physics by directly acting on pore-scale images of rocks without remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problem},
author = {Frank, Florian and Liu, Chen and Alpak, Faruk Omer and Berg, Steffen and Rivière, Béatrice},
doi = {10.2118/182607-PA},
faupublication = {no},
journal = {Spe Journal},
keywords = {phase-field method; diffuse interface; digital rock; oil and gas; Cahn–Hilliard; contact angle},
pages = {1–18},
peerreviewed = {Yes},
title = {{Direct} numerical simulation of flow on pore-scale images using the phase-field method},
url = {https://www.onepetro.org/journal-paper/SPE-182607-PA},
volume = {23},
year = {2018}
}
@article{faucris.119190764,
abstract = {In the course of miniaturization of electronic and microfluidic devices, reliable predictions of the stability of ultrathin films have a strategic role for design purposes. Consequently, efficient computational techniques that allow for a direct comparison with experiment become increasingly important. Here we demonstrate, for the first time, that the full complex spatial and temporal evolution of the rupture of ultrathin films can be modelled in quantitative agreement with experiment. We accomplish this by combining highly controlled experiments on different film-rupture patterns with computer simulations using novel numerical schemes for thin-film equations. For the quantitative comparison of the pattern evolution in both experiment and simulation we introduce a novel pattern analysis method based on Minkowski measures. Our results are fundamental for the development of efficient tools capable of describing essential aspects of thin-film flow in technical systems.},
author = {Becker, Jürgen and Grün, Günther and Seemann, Ralf and Manz, Hubert and Jacobs, Karin and Mecke, Klaus and Blossey, Ralf},
doi = {10.1038/nmat788},
faupublication = {no},
journal = {Nature Materials},
month = {Jan},
pages = {59-63},
peerreviewed = {Yes},
title = {{Complex} dewetting scenarios captured by thin-film models},
volume = {2},
year = {2003}
}
@article{faucris.123794264,
abstract = {This article addresses 8 testcases proposed in the MoMaS benchmark for phase (dis)appearance. It involves slightly compressible, compositional two-phase flow with the components hydrogen and water, where hydrogen can appear in gas form or dissolved in water. The simplest example illustrates phase transition. Other problem set-ups include one and two-dimensional heterogeneous porous media and discontinuous initial conditions. Generalized mixed hybrid finite elements (MHFE) are used, which offer flexibility in the choice of primary variables. In the most general case, a complementarity formulation for the local problems together with a semi-smooth Newton algorithm is used. The ability of the method to solve all of the compositional benchmark problems is shown. This set of simple examples indicates that in the view of risk assessment of underground nuclear waste repositories the phenomenon of phase transition cannot be ignored and can take place even after the sources of hydrogen are extinguished. Moreover it can be useful for a numerical code to enable several choices of primary variables in order to cover a large variety of situations. © 2014 Elsevier Lt},
author = {Marchand, Estelle and Knabner, Peter},
doi = {10.1016/j.advwatres.2014.07.005},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {Compositional modeling; Gas-liquid flow with phase transitions; Multiphase flow in porous medium domains; Nuclear and chemical waste storage},
pages = {74-96},
peerreviewed = {Yes},
title = {{Results} of the {MoMaS} benchmark for gas phase appearance and disappearance using generalized {MHFE}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2014/2014_MarchandKn_ResulOfMoMaBenchmForGasPhasAppearAndDisappearUsingGeneralMHFE},
volume = {73},
year = {2014}
}
@masterthesis{faucris.209591150,
author = {Prechtel, Alexander},
faupublication = {no},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{La} pollution d'un site par des hydrocarbures - aspects de la modélisation hydrogéologique et étude géostatistique},
year = {1998}
}
@article{faucris.123853444,
abstract = {Modeling carrier-influenced transport needs to take into account the reactivity of the carrier itself. This paper presents a mathematical model of reactive solute transport with sorption to mobile and immobile sorbents. The mobile sorbent is also considered to be reactive. To justify the assumptions and generality of our modeling approach, experimental findings are reviewed and analyzed. A transformation of the model in terms of total concentrations of solute and mobile sorbents is presented which simplifies the mathematical formulations. Breakthrough data on dissolved organic carbon are presented to exemplify the need to take into account the reactivity of the mobile sorbent. Data on hexachtorobiphenyl and cadmium are presented to demonstrate carrier-introduced increased mobility, whereas data on anthracene and pyrene are presented to demonstrate carrier- introduced reduced mobility. The experimental conditions leading to the different findings are pointed out. The sorption processes considered in the model are both equilibrium and nonequilibrium processes, allowing for different sorption sites and nonlinear isotherms and rate functions. Effective isotherms, which describe the sorption to the immobile sorbent in the presence of a mobile sorbent and rate functions, are introduced and their properties are discusse},
author = {Knabner, Peter and Totsche, Kai Uwe and Kögel-Knabner, Ingrid},
doi = {10.1029/95WR02994},
faupublication = {yes},
journal = {Water Resources Research},
pages = {1611-1622},
peerreviewed = {Yes},
title = {{The} modeling of reactive solute transport with sorption to mobile and immobile sorbents. 1. {Experimental} evidence and model development},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1996/1996_KnTotscheKoegelkn_ModelOfReactSolutTranspWthsorptToMobImmobSorbP1},
volume = {32},
year = {1996}
}
@article{faucris.122614404,
abstract = {A parameter identification problem for the hydraulic properties of porous media is considered. Numerically, this inverse problem is solved by minimizing an output least-squares functional. The unknown hydraulic properties which are nonlinear coefficients of a partial differential equation are approximated by spline functions. The identification is embedded into a multi-level algorithm and coupled with a linear sensitivity analysis to describe the ill-posedness of the inverse problem. © 2002 Elsevier Science B.V. All rights reserve},
author = {Bitterlich, Sandro and Knabner, Peter},
doi = {10.1016/S0377-0427(02)00430-2},
faupublication = {yes},
journal = {Journal of Computational and Applied Mathematics},
keywords = {Adjoint method; Direct method; Inverse problem; Multi-level algorithm; Parameter identification; Sensitivity analysis},
pages = {153-173},
peerreviewed = {Yes},
title = {{An} efficient method for solving an inverse problem for the {Richards} equation},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_BitterlichKn_AnEfficientMethodForSolvAnInversProbForRichardsEquation},
volume = {147},
year = {2002}
}
@incollection{faucris.123987424,
author = {Grün, Günther and Becker, Jürgen and Rumpf, Martin},
booktitle = {Proceedings of the 5th World Congress on Computational Mechanics},
editor = {H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner},
faupublication = {no},
isbn = {3-9501554-0-6},
peerreviewed = {unknown},
publisher = {Vienna Technical University},
title = {{On} space-time adaptive convergent finite element schemes for a general class of lubrication-type equations},
year = {2002}
}
@article{faucris.116621164,
abstract = {This work deals with the modeling and simulation of non-Newtonian jet dynamics as it occurs in fiber spinning processes. Proceeding from a three-dimensional instationary boundary value problem of upper-convected Maxwell equations, we present a strict systematic derivation of a one-dimensional viscoelastic string model by using asymptotic analysis in the slenderness ratio of the jet. The model allows for the unrestricted motion and shape of the jet's curve, and its deduction extends the hitherto existing uniaxial asymptotic approaches. However, the system of partial differential equations with algebraic constraint has a varying character (hyperbolic, hyperbolic-elliptic, parabolic deficiency). Its applicability range turns out to be limited depending on the physical parameters and the boundary conditions (i.e. singular perturbation). Numerical results are discussed for the hyperbolic regime of gravitational inflow-outflow set-ups which become relevant in drawing and extrusion processes. The simulations are performed with a normal form total upwind scheme in space and an implicit time-integration ensuring convergence of first order.},
author = {Marheineke, Nicole and Liljegren-Sailer, Björn and Lorenz, Maike and Wegener, Raimund},
doi = {10.1142/S021820251650010X},
faupublication = {yes},
journal = {Mathematical Models & Methods in Applied Sciences},
keywords = {asymptotic expansions; Curved viscoelastic jets; normal form total upwind scheme; slender body theory; upper-convected Maxwell model},
pages = {569-600},
peerreviewed = {Yes},
title = {{Asymptotics} and numerics for the upper-convected {Maxwell} model describing transient curved viscoelastic jets},
volume = {26},
year = {2016}
}
@article{faucris.223318418,
abstract = {We investigate the following problem: To influence a heat conduction process in such a way that the conductor melts in a prescribed manner. Since we treat a linear auxiliary problem, it suffices to deal with a linear-quadratic defect minimization problem with linear restrictions, where we use splines or polynomials as approximation spaces. In case of exact controllability we derive various order of convergence estimates, which we discuss for some numerical examples.

Subject Classifications: AMS(MOS): 65P05; CR: G1.8.

and dissolution and precipitation of immobile species present inside a porous medium. The

transport of mobile species in the pores is modeled by a system of semilinear parabolic

partial differential equations. The reactions amongst the mobile species are assumed to

be reversible, i.e. both forward and backward reactions are considered. These reversible

reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial

differential equations. This system of equations for the mobile species is complemented

by flux boundary conditions at the outer boundary. Furthermore, the dissolution and

precipitation of immobile species on the surface of the solid parts are modeled by mass

action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution

term. The model is posed at the pore (micro) scale. The contribution of this paper is two-

fold: first we show the existence of a unique positive global weak solution for the coupled

systems and then we upscale (homogenize) the model from the micro scale to the macro

scale. For the existence of solution, some regularization techniques, Schaefer’s fixed point

theorem and Lyapunov type arguments have been used whereas the concepts of two-scale

convergence and periodic unfolding are used for the homogenization.

},
author = {Elbinger, Tobias and Gahn, Markus and Neuss-Radu, Maria and Knabner, Peter},
booktitle = {2nd International Conference on Multi-scale Computational Methods for Solids and Fluids},
editor = {A. Ibrahimbegovic, J.-M. Ghidaglia, A. Serdarevic, E. Ilic- Georgijevic, M. Hrasnica, S. Dolarevic, N. Ademovic},
faupublication = {yes},
keywords = {Multiscale Finite Element Methods; Decoupling Algorithm; Newton’s Method},
pages = {44-47},
peerreviewed = {unknown},
title = {{Simulation} of {Processes} in {Living} {Cells}: {Towards} the {Modelling} of {Metabolic} {Channeling}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_Proc_ElbingerGahnNeussraduKn_SimulOfProcessInLivCellsTowardsModellingOfMetaboliChannel},
year = {2015}
}
@book{faucris.221486868,
author = {Jäger, Willi and Neuss-Radu, Maria},
doi = {10.1007/978-3-540-28396-6_19},
faupublication = {no},
isbn = {354028379X},
pages = {531-553},
peerreviewed = {unknown},
publisher = {Springer Berlin Heidelberg},
title = {{Multiscale} analysis of processes in complex media},
year = {2007}
}
@article{faucris.221485234,
author = {Jäger, Willi and Neuss-Radu, Maria and Shaposhnikova, Tatiana A.},
doi = {10.1134/S1064562411020219},
faupublication = {no},
journal = {Doklady Mathematics},
pages = {204-208},
peerreviewed = {Yes},
title = {{Scale} limit of a variational inequality modeling diffusive flux in a domain with small holes and strong adsorption in case of a critical scaling},
volume = {83},
year = {2011}
}
@article{faucris.216552999,
abstract = {The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar SDE. We focus here on a toy model with similar properties to the one arising in the context of an Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with nonglobally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cutoff or localization, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence.},
author = {Wacker, Philipp Konstantin and Blömker, Dirk and Schillings, Claudia},
doi = {10.1137/17M1132367},
faupublication = {yes},
journal = {SIAM Journal on Numerical Analysis},
keywords = {Ensemble Kalman filter;stochastic differential equations;Bayesian inverse problems},
month = {Jan},
pages = {2537-2562},
peerreviewed = {Yes},
title = {{A} {STRONGLY} {CONVERGENT} {NUMERICAL} {SCHEME} {FROM} {ENSEMBLE} {KALMAN} {INVERSION}},
volume = {56},
year = {2018}
}
@article{faucris.219826314,
author = {Lieu, Alice and et al.},
author_hint = {Lieu Alice, Gabard Gwénaël, Bériot Hadrien},
doi = {10.1016/j.jcp.2016.05.045},
faupublication = {no},
journal = {Journal of Computational Physics},
keywords = {Comparison, Helmholtz problems, High-order finite element method, Wave-based discontinuous Galerkin method},
pages = {105--125},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{A} {Comparison} of {High}-order {Polynomial} and {Wave}-based {Methods} for {Helmholtz} {Problems}},
volume = {321},
year = {2016}
}
@article{faucris.204146981,
abstract = {Mathematical models of virus dynamics have not previously acknowledged spatial resolution at the intracellular level despite substantial arguments that favor the consideration of intracellular spatial dependence. The replication of the hepatitis C virus (HCV) viral RNA (vRNA) occurs within special replication complexes formed from membranes derived from endoplasmatic reticulum (ER). These regions, termed membranous webs, are generated primarily through specific interactions between nonstructural virus-encoded proteins (NSPs) and host cellular factors. The NSPs are responsible for the replication of the vRNA and their movement is restricted to the ER surface. Therefore, in this study we developed fully spatio-temporal resolved models of the vRNA replication cycle of HCV. Our simulations are performed upon realistic reconstructed cell structuresnamely the ER surface and the membranous websbased on data derived from immunostained cells replicating HCV vRNA. We visualized 3D simulations that reproduced dynamics resulting from interplay of the different components of our models (vRNA, NSPs, and a host factor), and we present an evaluation of the concentrations for the components within different regions of the cell. Thus far, our model is restricted to an internal portion of a hepatocyte and is qualitative more than quantitative. For a quantitative adaption to complete cells, various additional parameters will have to be determined through further in vitro cell biology experiments, which can be stimulated by the results described in the present study.},
author = {Knodel, Markus and et al.},
author_hint = {Knodel MM, Reiter S, Targett-Adams P, Grillo A, Herrmann E, Wittum G},
doi = {10.3390/v9100282},
faupublication = {no},
journal = {Viruses-Basel},
keywords = {computational virology;hepatitis C virus (HCV);viral dynamics;within-host viral modelling;mathematical models of viral RNA cycle;3D spatio-temporal resolved mathematical models;realistic geometries;(surface) partial differential equations;Finite Volumes;massively parallel multigrid solvers},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{3D} {Spatially} {Resolved} {Models} of the {Intracellular} {Dynamics} of the {Hepatitis} {C} {Genome} {Replication} {Cycle}},
volume = {9},
year = {2017}
}
@article{faucris.223042992,
abstract = {The existence of travelling wave solutions with compact support is shown for a system of nonlinear non-local degenerate diffusion equations of population dynamics. Besides the behaviour of an important class of particular solutions of this type is investigated analytically and numerically.

0, when the thin layer reduces to an interface Sigma between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface Sigma. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.}, author = {Gahn, Markus and Neuss-Radu, Maria and Knabner, Peter}, doi = {10.3934/nhm.2018028}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, keywords = {Homogenization;nonlinear transmission conditions;reaction-diffusion systems;effective interface conditions;unfolding and averaging operator for thin heterogeneous layer;weak and strong two-scale convergence}, pages = {609-640}, peerreviewed = {Yes}, title = {{EFFECTIVE} {INTERFACE} {CONDITIONS} {FOR} {PROCESSES} {THROUGH} {THIN} {HETEROGENEOUS} {LAYERS} {WITH} {NONLINEAR} {TRANSMISSION} {AT} {THE} {MICROSCOPIC} {BULK}-{LAYER} {INTERFACE}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_GahnNeussraduKn_EffectInterfCondForProcThrouThinHeterogenLayerWthNonlinearTransmissAtTeMicroscBulklayerInterface}, volume = {13}, year = {2018} } @article{faucris.109093424, abstract = {This work deals with the curling behavior of slender viscous jets in rotational spinning processes. In terms of slender-body theory, an instationary incompressible viscous Cosserat rod model is formulated which differs from the approach of Ribe et al., in the incompressibility approximation and reduces to the string model of Marheineke and Wegener for a vanishing slenderness parameter. Focusing exclusively on viscous and rotational effects on the jet in the exit plane near the spinning nozzle, the stationary two-dimensional scenario is described by a two-point boundary value problem of a system of first-order ordinary differential equations for jet's center-line, tangent, curvature, velocity, inner shear and traction force and couple. The numerical analysis shows that the rod model covers the string model in an inertia-dominated jet regime. Beyond that it overcomes the limitations of the string model studied by Götz et al. and enables even the handling of the viscous-inertial jet regime. Thus, the rod model shows its applicability for the simulation of industrially relevant parameter ranges and enlarges the domain of validity with respect to the string approach. © 2010 World Scientific Publishing Company.}, author = {Arne, Walter and Marheineke, Nicole and Meister, Andreas and Wegener, Raimund}, doi = {10.1142/S0218202510004738}, faupublication = {no}, journal = {Mathematical Models & Methods in Applied Sciences}, keywords = {asymptotic Cosserat models; boundary value problem; Curved viscous fibers; existence of numerical solutions; rotational spinning process}, pages = {1941-1965}, peerreviewed = {Yes}, title = {{Numerical} analysis of {Cosserat} rod and string models for viscous jets in rotational spinning processes}, volume = {20}, year = {2010} } @article{faucris.205334100, author = {Burger, Martin and et al.}, author_hint = {Schuster T, Hahn B, Burger M}, doi = {10.1088/1361-6420/aab0f5}, faupublication = {no}, journal = {Inverse Problems}, peerreviewed = {Yes}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{Dynamic} inverse problems: modelling-regularization-numerics {Preface}}, volume = {34}, year = {2018} } @inproceedings{faucris.222404057, abstract = {We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers (Formula Presented) and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.}, author = {Burger, Martin and Korolev, Yury and Schönlieb, Carola Bibiane and Stollenwerk, Christiane}, booktitle = {Lecture Notes in Computer Science}, date = {2019-06-30/2019-07-04}, doi = {10.1007/978-3-030-22368-7_38}, editor = {Jan Lellmann, Jan Modersitzki, Martin Burger}, faupublication = {yes}, isbn = {9783030223670}, keywords = {First order regularization; Image denoising; Total generalized variation; Total variation}, note = {CRIS-Team Scopus Importer:2019-07-16}, pages = {485-497}, peerreviewed = {unknown}, publisher = {Springer Verlag}, title = {{A} {Total} {Variation} {Based} {Regularizer} {Promoting} {Piecewise}-{Lipschitz} {Reconstructions}}, venue = {Hofgeismar}, volume = {11603 LNCS}, year = {2019} } @incollection{faucris.116454624, address = {Heidelberg}, author = {Grün, Günther and Giacomelli, Lorenzo and Dal Passo, Roberta}, booktitle = {Geometric analysis and nonlinear partial differential equations}, editor = {S. Hildebrandt, H. Karcher}, faupublication = {no}, pages = {637-648}, peerreviewed = {unknown}, publisher = {Springer-Verlag}, title = {{Waiting} time phenomena for degenerate parabolic equations -- a unifying approach}, year = {2003} } @article{faucris.217582803, abstract = {We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012), pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown.}, author = {Brunner, Fabian and Fischer, Julian and Knabner, Peter}, doi = {10.1137/15M1035379}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {advection-diffusion problem;mixed finite element methods;suboptimal convergence;optimal convergence}, month = {Jan}, pages = {2359-2378}, peerreviewed = {Yes}, title = {{Analysis} of a {Modified} {Second}-{Order} {Mixed} {Hybrid} ${BDM_1}$ {Finite} {Element} {Method} for {Transport} {Problems} in {Divergence} {Form}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2016/2016_BrunnerFischerKn_AnalysisOfModifSecondOrderMixBDMInDivergenceForm}, volume = {54}, year = {2016} } @incollection{faucris.120500644, abstract = {In many liquid-solid phase transitions, a specific dendritic microstructure of the phase interface is observed. In this contribution we present two-scale models capable to describe the evolution of equiaxed microstructure. The models are based either on a sharp interface model or on a phase field model for phase transitions in binary alloys. In both cases, a formal asymptotic expansion in terms of a scale parameter ε for the microstructure is carried out, with solute diffusivity scaling proportional to ε^{2}. In the limit ε→0 we obtain a two scale-model consisting of a macroscopic heat transport equation and, at each point of the macroscopic domain, of a local cell problem modeling the microsegregation and the evolution of the microstructur},
address = {Berlin, Heidelberg},
author = {Eck, Christof and Knabner, Peter},
booktitle = {Multiscale Problems in Science and Technology},
doi = {10.1007/978-3-642-56200-6_7},
editor = {Nenad Antonić, C. J. van Duijn, Willi Jäger, Andro Mikelić},
faupublication = {yes},
isbn = {978-3-540-43584-6},
pages = {175-187},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Two}-{Scale} {Models} for {Liquid}-{Solid} {Phase} {Transitions} in {Binary} {Material} with {Equiaxed} {Microstructure}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_EckKn_2ScaleModelsForLiquidSolidPhaseTransitInBinaryMatWithEquiaxMicrostruct},
year = {2002}
}
@inproceedings{faucris.123856744,
author = {Barrett, John W. and Knabner, Peter},
booktitle = {Mathematical Modelling of Flow through Porous Media},
editor = {Alain Bourgeat et al.},
faupublication = {yes},
pages = {75–99},
publisher = {Singapore: World Scientific Publishers},
title = {{Analysis} and {Finite} {Element} {Approximation} of {Transport} of {Reactive} {Solutes} in {Porous} {Media}},
venue = {St. Etienne},
year = {1995}
}
@article{faucris.215237342,
author = {Burger, Martin and et al.},
author_hint = {Berendsen J., Burger M., Pietschmann J.},
doi = {10.1016/j.na.2017.03.010},
faupublication = {no},
journal = {Nonlinear Analysis - Theory Methods & Applications},
keywords = {Cross-diffusion; Size exclusion; Nonlocal interaction},
pages = {10-39},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{On} a cross-diffusion model for multiple species with nonlocal interaction and size exclusion},
volume = {159},
year = {2017}
}
@article{faucris.221482408,
abstract = {

0, when the thin layer reduces to an interface Sigma between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface Sigma. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.}, author = {Gahn, Markus and Neuss-Radu, Maria and Knabner, Peter}, doi = {10.3934/nhm.2018028}, faupublication = {yes}, journal = {Networks and Heterogeneous Media}, keywords = {Homogenization;nonlinear transmission conditions;reaction-diffusion systems;effective interface conditions;unfolding and averaging operator for thin heterogeneous layer;weak and strong two-scale convergence}, pages = {609-640}, peerreviewed = {Yes}, title = {{EFFECTIVE} {INTERFACE} {CONDITIONS} {FOR} {PROCESSES} {THROUGH} {THIN} {HETEROGENEOUS} {LAYERS} {WITH} {NONLINEAR} {TRANSMISSION} {AT} {THE} {MICROSCOPIC} {BULK}-{LAYER} {INTERFACE}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2018/2018_GahnNeussraduKn_EffectInterfCondForProcThrouThinHeterogenLayerWthNonlinearTransmissAtTeMicroscBulklayerInterface}, volume = {13}, year = {2018} } @article{faucris.109093424, abstract = {This work deals with the curling behavior of slender viscous jets in rotational spinning processes. In terms of slender-body theory, an instationary incompressible viscous Cosserat rod model is formulated which differs from the approach of Ribe et al., in the incompressibility approximation and reduces to the string model of Marheineke and Wegener for a vanishing slenderness parameter. Focusing exclusively on viscous and rotational effects on the jet in the exit plane near the spinning nozzle, the stationary two-dimensional scenario is described by a two-point boundary value problem of a system of first-order ordinary differential equations for jet's center-line, tangent, curvature, velocity, inner shear and traction force and couple. The numerical analysis shows that the rod model covers the string model in an inertia-dominated jet regime. Beyond that it overcomes the limitations of the string model studied by Götz et al. and enables even the handling of the viscous-inertial jet regime. Thus, the rod model shows its applicability for the simulation of industrially relevant parameter ranges and enlarges the domain of validity with respect to the string approach. © 2010 World Scientific Publishing Company.}, author = {Arne, Walter and Marheineke, Nicole and Meister, Andreas and Wegener, Raimund}, doi = {10.1142/S0218202510004738}, faupublication = {no}, journal = {Mathematical Models & Methods in Applied Sciences}, keywords = {asymptotic Cosserat models; boundary value problem; Curved viscous fibers; existence of numerical solutions; rotational spinning process}, pages = {1941-1965}, peerreviewed = {Yes}, title = {{Numerical} analysis of {Cosserat} rod and string models for viscous jets in rotational spinning processes}, volume = {20}, year = {2010} } @article{faucris.205334100, author = {Burger, Martin and et al.}, author_hint = {Schuster T, Hahn B, Burger M}, doi = {10.1088/1361-6420/aab0f5}, faupublication = {no}, journal = {Inverse Problems}, peerreviewed = {Yes}, support_note = {Author relations incomplete. You may find additional data in field 'author_hint'}, title = {{Dynamic} inverse problems: modelling-regularization-numerics {Preface}}, volume = {34}, year = {2018} } @inproceedings{faucris.222404057, abstract = {We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers (Formula Presented) and, in particular, establish topological equivalence. Our numerical experiments show that the proposed regularizer can achieve similar performance as total generalized variation while having the advantage of a very intuitive interpretation of its free parameter, which is just a local estimate of the norm of the gradient. It also provides a natural approach to spatially adaptive regularization.}, author = {Burger, Martin and Korolev, Yury and Schönlieb, Carola Bibiane and Stollenwerk, Christiane}, booktitle = {Lecture Notes in Computer Science}, date = {2019-06-30/2019-07-04}, doi = {10.1007/978-3-030-22368-7_38}, editor = {Jan Lellmann, Jan Modersitzki, Martin Burger}, faupublication = {yes}, isbn = {9783030223670}, keywords = {First order regularization; Image denoising; Total generalized variation; Total variation}, note = {CRIS-Team Scopus Importer:2019-07-16}, pages = {485-497}, peerreviewed = {unknown}, publisher = {Springer Verlag}, title = {{A} {Total} {Variation} {Based} {Regularizer} {Promoting} {Piecewise}-{Lipschitz} {Reconstructions}}, venue = {Hofgeismar}, volume = {11603 LNCS}, year = {2019} } @incollection{faucris.116454624, address = {Heidelberg}, author = {Grün, Günther and Giacomelli, Lorenzo and Dal Passo, Roberta}, booktitle = {Geometric analysis and nonlinear partial differential equations}, editor = {S. Hildebrandt, H. Karcher}, faupublication = {no}, pages = {637-648}, peerreviewed = {unknown}, publisher = {Springer-Verlag}, title = {{Waiting} time phenomena for degenerate parabolic equations -- a unifying approach}, year = {2003} } @article{faucris.217582803, abstract = {We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012), pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown.}, author = {Brunner, Fabian and Fischer, Julian and Knabner, Peter}, doi = {10.1137/15M1035379}, faupublication = {yes}, journal = {SIAM Journal on Numerical Analysis}, keywords = {advection-diffusion problem;mixed finite element methods;suboptimal convergence;optimal convergence}, month = {Jan}, pages = {2359-2378}, peerreviewed = {Yes}, title = {{Analysis} of a {Modified} {Second}-{Order} {Mixed} {Hybrid} ${BDM_1}$ {Finite} {Element} {Method} for {Transport} {Problems} in {Divergence} {Form}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2016/2016_BrunnerFischerKn_AnalysisOfModifSecondOrderMixBDMInDivergenceForm}, volume = {54}, year = {2016} } @incollection{faucris.120500644, abstract = {In many liquid-solid phase transitions, a specific dendritic microstructure of the phase interface is observed. In this contribution we present two-scale models capable to describe the evolution of equiaxed microstructure. The models are based either on a sharp interface model or on a phase field model for phase transitions in binary alloys. In both cases, a formal asymptotic expansion in terms of a scale parameter ε for the microstructure is carried out, with solute diffusivity scaling proportional to ε

In this
article we obtain rigorously the homogenization limit for a
fluid-structure-reactive flow system. It consists of cell tissue and
intercellular liquid, transporting solutes. The cell tissue is assumed
to be linearly elastic and deforming with a viscous nonstationary flow.
The elastic moduli of the tissue change with cumulative concentration
value. In the limit, when the scale parameter goes to zero, we obtain
the quasi-static Biot system, coupled with the upscaled reactive flow.
Effective Biot's coefficients depend on the reactant concentration. In
addition to the weak two-scale convergence results, we prove convergence
of the elastic and viscous energies. This then implies a strong
two-scale convergence result for the fluid-structure variables. Next we
establish the regularity of the solutions for the upscaled equations. To
the best of our knowledge, ours is the only known study of the
regularity of solutions to the quasi-static Biot system. The regularity
is used to prove the uniqueness for the upscaled model.

Read More: https://epubs.siam.org/doi/abs/10.1137/100808393

Read More: https://epubs.siam.org/doi/abs/10.1137/100808393

In this
article we obtain rigorously the homogenization limit for a
fluid-structure-reactive flow system. It consists of cell tissue and
intercellular liquid, transporting solutes. The cell tissue is assumed
to be linearly elastic and deforming with a viscous nonstationary flow.
The elastic moduli of the tissue change with cumulative concentration
value. In the limit, when the scale parameter goes to zero, we obtain
the quasi-static Biot system, coupled with the upscaled reactive flow.
Effective Biot's coefficients depend on the reactant concentration. In
addition to the weak two-scale convergence results, we prove convergence
of the elastic and viscous energies. This then implies a strong
two-scale convergence result for the fluid-structure variables. Next we
establish the regularity of the solutions for the upscaled equations. To
the best of our knowledge, ours is the only known study of the
regularity of solutions to the quasi-static Biot system. The regularity
is used to prove the uniqueness for the upscaled model.

Read More: https://epubs.siam.org/doi/abs/10.1137/100808393

},
author = {Jäger, Willi and Mikelic, Andro and Neuss-Radu, Maria},
faupublication = {no},
journal = {SIAM Journal on Mathematical Analysis},
pages = {1390–1435},
peerreviewed = {Yes},
title = {{Homogenization} limit of a model system for interaction of flow, chemical reactions, and mechanics in cell tissues},
volume = {43},
year = {2011}
}
@article{faucris.117799924,
abstract = {The objective of this research is to investigate the upscaling of flow and transport models including both an evolving solid-liquid interface and a quite general potentially even oscillating interaction potential. Starting from a comprehensive pore-scale model, formal, two-scale, asymptotic expansion in a level set framework is applied. In doing so, the interplay between flow, transport, interaction potential, and evolving geometry becomes evident. As a result of an averaging procedure, a fully coupled micro-macro model is established with new main variables. Moreover, time-and spacedependent coefficient functions are explicitly characterized by means of supplementary, fully coupled cell problems. The theoretical results obtained are complemented by the numerical computations of a heterogeneous multiscale scenario with a focus on the development of anisotropic transport. Because of the general framework considered in this research, numerous application are expected, such as in biology or colloidal dynamic},
author = {Ray, Nadja and Elbinger, Tobias and Knabner, Peter},
doi = {10.1137/140990292},
faupublication = {yes},
journal = {SIAM Journal on Applied Mathematics},
keywords = {Evolving porous medium; Flow; Interaction potential; Level set; Transport; Two-scale asymptotic expansion; Upscaling},
pages = {2170-2192},
peerreviewed = {Yes},
title = {{Upscaling} the flow and transport in an evolving porous medium with general interaction potentials},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_RayElbingerKn_UpscalingFlowTransInAnEvolvingPMWithGeneralInterPotentials},
volume = {75},
year = {2015}
}
@article{faucris.116635904,
abstract = {This work deals with the behavior of viscoelastic jets under gravitational forces described by an asymptotic upper convected Maxwell (UCM) model, system of partial differential equations. Considering fiber spinning, we show that the one-dimensional model equations in general allow for the simulation of drawing processes with and without die swell effect. But, as the model is of hyperbolic type and the run of the characteristics crucially depend on the physical parameters, the existence regimes of the stationary solutions associated to certain boundary conditions turn out to be limited. We investigate the regimes for gravitational uniaxial and 2d spinning scenarios numerically.},
author = {Marheineke, Nicole and Lorenz, Maike and Wegener, Raimund},
doi = {10.1186/2190-5983-4-2},
faupublication = {yes},
journal = {Journal of Mathematics in Industry},
keywords = {boundary value problem; die swell; existence of solutions; fiber spinning; nonlinear viscoelasticity; UCM fluid},
peerreviewed = {unknown},
title = {{On} simulations of spinning processes with a stationary one-dimensional upper convected {Maxwell} model},
volume = {4},
year = {2014}
}
@article{faucris.123862464,
abstract = {In this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration, backward Euler time discretization, and possibly regularization and relaxation of the following degenerate parabolic equation arising in a model of reactive solute transport in porous media: find u(x,t) such that u + [(u)] - δu = f in ω (0,T], u = 0 on ω (0,T] u(',0) = g(.) in ω for known data ω R, 1 ≤ d ≤ 3, f, g, and a monotonically increasing C(R) C (-,0) (0,) satisfying (0) = 0, which is only locally Hölder continuous with exponent p (0,1) at the origin; e.g., (s) [s] . This lack of Lipschitz continuity at the origin limits the regularity of the unique solution u and leads to difficulties in the finite element error analysi},
author = {Barrett, John W. and Knabner, Peter},
doi = {10.1137/S0036142993258191},
faupublication = {yes},
journal = {SIAM Journal on Numerical Analysis},
keywords = {Degenerate parabolic problem; Error analysis; Finite elements; Porous medium},
pages = {455-479},
peerreviewed = {Yes},
title = {{Finite} element approximation of the transport of reactive solutes in porous media. part {II}: {Error} estimates for equilibrium adsorption processes},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_BarrettKn_FinEleApprOfTranspOfReactSolutInPMPart2},
volume = {34},
year = {1997}
}
@article{faucris.119270844,
author = {Grün, Günther and Becker, Jürgen and Lenz, Martin and Rumpf, Martin},
faupublication = {no},
journal = {Applications of Mathematics},
pages = {517-543},
peerreviewed = {Yes},
title = {{Numerical} methods for fourth order nonlinear diffusion problems},
volume = {47},
year = {2002}
}
@article{faucris.113569984,
author = {Grün, Günther and Abels, Helmut and Garcke, Harald},
faupublication = {yes},
journal = {Mathematical Models & Methods in Applied Sciences},
pages = {1150013-1150052},
peerreviewed = {Yes},
title = {{Thermodynamically} consistent diffuse interface models for incompressible two-phase flows with different densities},
volume = {22},
year = {2012}
}
@incollection{faucris.122390444,
author = {Knabner, Peter and Barrett, John W. and Kappmeier, H.},
booktitle = {Computational Methods in Water Resources},
faupublication = {yes},
pages = {299-308},
peerreviewed = {unknown},
title = {{Lagrange}-{Galerkin} {Approximation} of {Advection}-{Dominated} {Nonlinear} {Contaminant} {Transport} in {Porous} {Media}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1994/1994_KnBarrettKappmeier_LagrGalApprxOfAdvectDominNonlinContamTrspInPM},
volume = {1},
year = {1994}
}
@article{faucris.108994644,
author = {Grün, Günther and Fontelos, Marco Antonio and Kindelan, U. and Klingbeil, Fabian},
faupublication = {yes},
journal = {Journal of Adhesion Science and Technology},
pages = {1805-1824},
peerreviewed = {Yes},
title = {{Numerical} simulation of static and dynamic electrowetting},
volume = {26},
year = {2012}
}
@unpublished{faucris.222788298,
author = {Ray, Nadja and Schulz, Raphael},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/222788298},
peerreviewed = {Yes},
title = {{Derivation} of an {Effective} {Dispersion} {Model} for {Electroosmotic} {Flow} {Involving} {Free} {Boundaries} in a {Thin} {Strip}},
year = {2019}
}
@book{faucris.106243104,
abstract = {Dieses Lehrbuch bietet eine Einführung in Diskretisierungsmethoden für partielle Differentialgleichungen. Im Mittelpunkt steht das Finite-Element-Verfahren, aber es werden auch Finite-Differenzen- und Finite-Volumen-Verfahren behandelt. Basierend auf einer mathematisch präzisen Darstellung von Verfahren und ihrer Theorie spannt der Text den Rahmen bis hin zur Finite-Element-Implementierung. Dies beinhaltet eine Einführung in moderne Entwicklungen wie Multilevel- oder adaptive Verfahren. Das Spektrum der behandelten Differentialgleichungen reicht von linearen elliptischen Randwertaufgaben bis zu - auch konvektionsdominierten - nichtlinearen parabolischen Problemen. Diese werden jeweils durch Modelle aus einem spezifischen Anwendungsgebiet illustriert. Das Lehrbuch entspricht im Umfang etwa einer einsemestrigen Veranstaltung mit Ergänzungen und wendet sich an Studierende der Mathematik und der Ingenieur- oder Naturwissenschaften nach dem Vordiplo},
address = {Berlin-Heidelberg},
author = {Knabner, Peter and Angermann, Lutz},
doi = {10.1007/978-3-642-57181-7},
edition = {1},
faupublication = {yes},
isbn = {978-3-540-66231-0},
peerreviewed = {unknown},
publisher = {Springer},
series = {Springer-Lehrbuch Masterclass},
title = {{Numerik} partieller {Differentialgleichungen}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2000/2000_BUCH_KnAngermann_NumerikPartiellerDifferentialgleichEineAnwendungsorientEinfuehrung},
year = {2000}
}
@article{faucris.207687060,
abstract = {Modeling the droplet nucleation process requires a molecular-scale approach to describe the interfacial tension (IFT) of spherical interfaces. Density gradient theory (DGT), also referred to as square gradient theory in some publications has been widely used to compute the IFT of many pure and mixed systems at the molecular scale. However, the application of DGT to droplet interfaces is limited by its setup in open systems, in which a stable droplet cannot be achieved. In this article, we propose a mass-conserved DGT model in a closed system, i.e. with no-flux boundary conditions, where no mass exchange is allowed with the outside environment. As opposed to the traditional approach, this model enforces a canonical ensemble and guarantees energy dissipation. The proposed model has been successfully applied to systems with planar as well as spherical interfaces, especially for droplet's IFT calculation in the nucleation process. By extending the DGT model from open to closed systems, we demonstrate the potential of DGT as an inhomogeneous model for a wider range of academic and industrial applications.Read More: https://epubs.siam.org/doi/abs/10.1137/100808393

We propose a modified HSS scheme (“inexact HSS”, “IHSS”) that exploits the highly parallel fine-scale solver more extensively and only approximates the coarse-scale solution in every iteration thus resulting in a significant speedup. The tolerance of the coarse-scale solver is adapted in every IHSS cycle, controlled by the residual norm of the fine-scale system. Anderson acceleration is employed in the repeated solving of the fine-scale system to stabilize the scheme. We investigate the applicability of IHSS to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn–Hilliard equation, discuss its hybrid parallel implementation for large-scale simulations, and compare the performance of a widely used iterative solver with and without IHS},
author = {Thiele, Christopher and Araya-Polo, Mauricio and Alpak, Faruk Omer and Rivière, Béatrice and Frank, Florian},
doi = {10.1016/j.camwa.2017.06.025},
faupublication = {no},
journal = {Computers & Mathematics With Applications},
keywords = {discontinuous Galerkin method; sparse linear solver; hierarchical scale separation; parallel computing; p-multigrid method; modal basis},
pages = {1769–1778},
peerreviewed = {Yes},
title = {{Inexact} hierarchical scale separation: a two-scale approach for linear systems from discontinuous {Galerkin} discretizations},
url = {https://www.sciencedirect.com/science/article/pii/S0898122117303735?via=ihub},
volume = {74},
year = {2017}
}
@article{faucris.123423564,
abstract = {In this article a method is proposed for the efficient simulation of reactive systems or reactive transport problems including (equilibrium) mineral precipitation-dissolution. The difficulty lies in the fact that for larger systems it is usually a priorily not known for which mineral the fluid is saturated and for which it is undersaturated. Currently wide-spread algorithms use some trial-and-error strategy requiring repeated computations, or an approximation of equilibrium precipitation-dissolution by a kinetic description. In this article we propose to formulate the problem as a so-called complementarity problem (CP) and to solve it with the semismooth Newton method, a solution strategy well known in the field of optimization theory. The CP formulation of the mineral reactions is then combined with a reformulation for the full multicomponent reactive transport problem which leads to a reduction of the number of unknowns. The reactive transport problem is tackled in the sense of a one step (global implicit) method. © 2010 Elsevier Ltd.},
author = {Kräutle, Serge},
doi = {10.1016/j.advwatres.2010.10.004},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {Reactive transport; Minerals; Complementarity problems; Semismooth Newton method},
pages = {137-151},
peerreviewed = {Yes},
title = {{The} semismooth {Newton} method for multicomponent reactive transport with minerals},
url = {http://www.sciencedirect.com/science/article/pii/S0309170810001934},
volume = {34},
year = {2011}
}
@book{faucris.122729244,
abstract = {Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in - search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as nume- cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monograph},
address = {New York},
author = {Knabner, Peter and Angermann, Lutz},
doi = {10.1007/b97419},
edition = {1},
faupublication = {yes},
isbn = {978-0-387-95449-3},
peerreviewed = {unknown},
publisher = {Springer},
series = {Texts in Applied Mathematics},
title = {{Numerical} {Methods} for {Elliptic} and {Parabolic} {Partial} {Differential} {Equations}},
volume = {44},
year = {2003}
}
@article{faucris.116633484,
abstract = {In [N. Marheineke and R. Wegener, SIAM J. Appl. Math., 66 (2006), pp. 1703-1726], an aerodynamic force concept for a general air drag model based on a stochastic k-ε description for a turbulent flow field is derived. The turbulence effects on the dynamics of a long, slender, elastic fiber are specifically modeled by a correlated random Gaussian force and in its asymptotic limit on a macroscopic fiber scale by Gaussian white noise with flow-dependent amplitude. The present paper states quantitative similarity estimates and numerical comparisons for the choice of a Taylor drag model in a given application. © 2007 Society for Industrial and Applied Mathematics.},
author = {Marheineke, Nicole and Wegener, Raimund},
doi = {10.1137/06065489X},
faupublication = {no},
journal = {SIAM Journal on Applied Mathematics},
keywords = {Air drag; ARMA process; Fiber-turbulence interaction scales; Flexible fibers; K-ε turbulence model; Random gaussian aerodynamic force; Stochastic differential equations; White noise},
pages = {1-23},
peerreviewed = {Yes},
title = {{Fiber} dynamics in turbulent flows: {Specific} taylor drag},
volume = {68},
year = {2007}
}
@article{faucris.204938499,
abstract = {This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy-Nernst-Planck-Poisson system. The scheme uses the second order backward difference formula (BDF2) in time and the mixed finite element method with Raviart-Thomas elements in space. In the first step, we show that the solution of the underlying weak continuous problem is also a solution of a third problem for which an existence result is already established. Thereby a stability estimate arises, which provides an L-∞ bound of the concentrations / masses of the system. This bound is used as a level for a cut-off operator that enables a proper formulation of the fully discrete scheme. The error analysis works without semi-discrete intermediate formulations and reveals convergence rates of optimal orders in time and space. Numerical simulations validate the theoretical results for lowest order finite element spaces in two dimensions.},
author = {Frank, Florian and Knabner, Peter},
doi = {10.1051/m2an/2017002},
faupublication = {yes},
journal = {Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique},
keywords = {Stokes / Darcy-Nernst-Planck-Poisson system;mixed finite elements;backward difference formula;error analysis;porous media},
pages = {1883-1902},
peerreviewed = {Yes},
title = {{Convergence} analysis of a {BDF2}/mixed finite element discretization of a {Darcy}–{Nernst}–{Planck}–{Poisson} system},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2017/2017_FrankKn_ConvergenceAnalysisOfBDFDarcyNernstPlanckPoissonSystem},
volume = {51},
year = {2017}
}
@article{faucris.123790304,
abstract = {The Global Random Walk algorithm (GRW) performs a simultaneous tracking on a fixed grid of huge numbers of particles at costs comparable to those of a single-trajectory simulation by the traditional Particle Tracking (PT) approach. Statistical ensembles of GRW simulations of a typical advection-dispersion process in groundwater systems with randomly distributed spatial parameters are used to obtain reliable estimations of the input parameters for the upscaled transport model and of their correlations, input-output correlations, as well as full probability distributions of the input and output parameter},
author = {Suciu, Nicolae and Vamo̧s, Cǎlin and Vereecken, Harry and Knabner, Peter},
faupublication = {yes},
journal = {Mathematics and its Applications : Annals of the Academy of Romanian Scientists},
keywords = {Groundwater contamination; Monte carlo methods; Probabilistic particle methods; Transport processes},
pages = {218-234},
peerreviewed = {unknown},
title = {{Global} {Random} {Walk} simulations for sensitivity and uncertainty analysis of passive transport models},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2011/2011_SuciuVamosVereeckenKn_GlobalRandomWalkSimuForSensAndUncertAnalysisOfPassiveTransModels},
volume = {3},
year = {2011}
}
@inproceedings{faucris.215226191,
author = {Tenbrinck, Daniel and Lozes, Francois and Elmoataz, Abderrahim},
booktitle = {International Conference on Scale Space and Variational Methods in Computer Vision},
faupublication = {no},
pages = {601-612},
peerreviewed = {Yes},
title = {{Solving} {Minimal} {Surface} {Problems} on {Surfaces} and {Point} {Clouds}},
venue = {Bordeaux},
year = {2015}
}
@article{faucris.116669124,
author = {Grün, Günther},
faupublication = {no},
journal = {Zeitschrift für Analysis und ihre Anwendungen},
pages = {987-998},
peerreviewed = {Yes},
title = {{On} {Bernis}' interpolation inequalities in multiple space dimensions},
volume = {20},
year = {2001}
}
@incollection{faucris.116679464,
abstract = {In some processes for spinning synthetic fibers the filaments are exposed to highly turbulent flows to achieve a high degree of stretching. The quality of the resulting fabric is thus determined essentially by the turbulent fiber-fluid interactions. Due to the required fine resolution, direct numerical simulations fail. Therefore we model the flow fluctuations as random field in *ε* turbulence description and describe the interactions in the context of slender-body theory as one-way-coupling with a corresponding stochastic aerodynamic drag force on the fibers. Hereby we exploit the special covariance structure of the random field, namely isotropy, homogeneity and decoupling of space and time. In this work we will focus on the construction and efficient simulation of the turbulent fluctuations assuming constant flow parameters and give an outlook on applications.},
author = {Hübsch, Florian and Marheineke, Nicole and Wegener, Raimund},
booktitle = {Progress in Industrial Mathematics at ECMI 2012},
doi = {10.1007/978-3-319-05365-3_16},
editor = {Magnus Fontes, Michael Günther, Nicole Marheineke},
faupublication = {yes},
isbn = {978-3-319-05364-6},
pages = {119-126},
peerreviewed = {unknown},
publisher = {Springer},
series = {Mathematics in Industry},
title = {{Efficient} simultaion of random fields for fiber-fluid interactions in isotropic turbulence},
volume = {19},
year = {2014}
}
@incollection{faucris.224028659,
abstract = {A two-dimensional numerical test case study was performed in order to
investigate the influence of sorption inhomogeneities on the migration
behaviour of contaminants. The sorption processes are modelled with a
nonlinear kinetic-Freundlich reaction. Thereby a kinetic approach of
first order is supposed. In a first test case homogeneous sorption
parameters are used, and in a second case the transport path is divided
into regions of low and high sorption. For the latter the numerical
solution using the TRAPIC code requires a finer discretization in the
region of the sorption inhomogeneities. The nonlinearities caused by the
kinetic-Freundlich reaction are resolved using Newton's method. Besides
the well known retardation of the contaminants caused by sorption
processes the simulations yield the following results: The distribution
of the concentration of the sorbed contaminants is discontinuous in the
region of the sorption inhomogeneities. The nonlinear behaviour of the
kinetic-Freundlich reaction counteracts the spreading effects of the
contaminant caused by the diffusive and dispersive transpor},
author = {Lührmann, Ludger and Fein, Eckhard and Knabner, Peter},
booktitle = {Environmental Sciences (B3110)},
faupublication = {yes},
isbn = {92-64-16099-X},
pages = {317-326},
peerreviewed = {unknown},
publisher = {INIS},
title = {{A} case study on the influence of sorption inhomogeneities on the migration of contaminants},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_LuehrmannFeinKn_CaseStudyOnTheInfluenOfSorptInhomogenOnMigrationOfContaminants},
volume = {29},
year = {1998}
}
@article{faucris.117783864,
abstract = {A model for transport of solutes in a porous medium participating in a dissolution-precipitation reaction, in general not in equilibrium, is studied. Ignoring diffusion-dispersion the initial value problem for piecewise constant initial states is studied, which e.g. for ionic species include a change of the ionic composition of the solution. The mathematical solution, nearly explicitly found by the method of characteristics up to the (numerical) solution of an integral equation for the position of the dissolution front, exhibits a generalized expanding plateau-structure determined by the dissolution front and the water flow (or salinity) fron},
author = {Knabner, Peter and Van Duijn, C. J. (Hans) and Schotting, Ruud J.},
doi = {10.1016/S0309-1708(97)00038-9},
faupublication = {yes},
journal = {Advances in Water Resources},
pages = {1-16},
peerreviewed = {Yes},
title = {{An} analysis of crystal dissolution fronts in flows through porous media part 2: incompatible boundary conditions},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1998/1998_DuijnKnSchotting_AnalysCrystDissolFroInFlowThrPMPart2IncompBoundCondi.pdf},
volume = {22},
year = {1998}
}
@article{faucris.116639644,
abstract = {We present and investigate techniques for optimizing particle dispersions in all kinds of Reynolds number flows. In particular, we show the range of application, power and efficiency of space mapping approaches that are based on a hierarchy of models ranging from a complex (fine, accurate, costly) to simple (coarse, rough, cheap) model. Space mapping turns out to be a reasonable approximation to optimal control and a competitive alternative to instantaneous control regarding speed and memory demands when dealing with complex, non-stationary problems. Moreover it allows the easy and efficient treatment of stochastic design problems. To control random particle dynamics in a turbulent flow, we suggest a Monte-Carlo aggressive space mapping algorithm which yields very convincing numerical results. © 2011 Springer Science+Business Media, LLC.},
author = {Marheineke, Nicole and Pinnau, René and Reséndiz, Edgar},
doi = {10.1007/s11081-011-9150-6},
faupublication = {yes},
journal = {Optimization and Engineering},
keywords = {Instantaneous control; k-ε{lunate} turbulence model; Navier-Stokes equations; Optimal control; Particle dynamics; Space mapping technique; Stochastic control; Tracking problem},
pages = {101-120},
peerreviewed = {Yes},
title = {{Space} mapping-focused control techniques for particle dispersions in fluids},
volume = {13},
year = {2012}
}
@inproceedings{faucris.122721544,
author = {Knabner, Peter and Korotov, Sergey and Summ, Gerhard},
booktitle = {17th GAMM-Seminar Leipzig on Construction of Grid Generation Algorithms},
editor = {Hackbusch W, Langer U},
faupublication = {yes},
title = {{Conditions} for the {Invertibility} of the {Isoparametric} {Mapping} for {3D} {Multilinear} {Finite} {Elements}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2001/2001_SummKnKorotov_CondForTheInvertibOfTheIsoparametMappFor3DmultilinFinitElem},
venue = {Leipzig},
year = {2001}
}
@inproceedings{faucris.224029850,
author = {Lührmann, Ludger and Klotz, Dietmar and Knabner, Peter},
booktitle = {Uranium Mining and Hydrogeology II},
date = {1998-09-15/1998-09-17},
faupublication = {yes},
peerreviewed = {unknown},
title = {{Modellierung} des kolloidgetragenen {Stofftransports} angewandt auf {Eu}-{Humat}-{Migrationsexperimente}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1998/1998_LuehrmannKlotzKnNoseck_ModellierDesKolloidgetrStofftrnspAngewAufEuHumatMigrationsex},
venue = {Freiberg},
year = {1998}
}
@incollection{faucris.122059124,
author = {Marheineke, Nicole},
booktitle = {Progress in Industrial Mathematics at ECMI 2002},
editor = {A. D. Fitt, A. Buikis, R. Ciegis},
faupublication = {no},
pages = {81-90},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Modified} {FEM} for fibre-fluid interactions},
year = {2004}
}
@article{faucris.116659884,
author = {Vibe, Alexander and Marheineke, Nicole},
doi = {10.1080/23324309.2016.1181090},
faupublication = {yes},
journal = {Journal of Computational and Theoretical Transport},
peerreviewed = {Yes},
title = {{Impact} of weak particle inertia on dilute suspensions of ellipsoids},
year = {2016}
}
@inproceedings{faucris.203704185,
address = {Colmar, Alsace, France},
author = {Schönwetter, Dominik and Ditter, Alexander and Kleinert, Bruno and Hendricks, Arne and Aizinger, Vadym and Köstler, Harald and Fey, Dietmar},
booktitle = {Proceedings of the 5th International Conference on Simulation and Modeling Methodologies, Technologies and Applications},
date = {2015-07-21/2015-07-23},
doi = {10.5220/0005566603770382},
faupublication = {yes},
isbn = {978-989-758-120-5},
keywords = {Crisis modeling and simulation; Environmental modeling; Multiscale simulation},
note = {UnivIS-Import:2018-09-06:Pub.2015.tech.IMMD.IMMD3.tsunam},
pages = {377-382},
peerreviewed = {unknown},
publisher = {SciTePress},
title = {{Tsunami} and {Storm} {Surge} {Simulation} {Using} {Low} {Power} {Architectures} - {Concept} and {Evaluation}},
venue = {Colmar},
year = {2015}
}
@article{faucris.106897164,
abstract = {We consider mixed finite element discretization for a class of degenerate parabolic problems including the Richards' equation. After regularization, time discretization is achieved by an Euler implicit scheme, while mixed finite elements are employed for the discretization in space. Based on the results obtained in (Radu et al. RANA Preprint 02-06, Eindhoven University of Technology, 2002), this paper considers a simple iterative scheme to solve the emerging nonlinear elliptic problems. © 2003 Elsevier B.V. All rights reserve},
author = {Pop, Iuliu Sorin and Radu, Adrian Florin and Knabner, Peter},
doi = {10.1016/j.cam.2003.04.008},
faupublication = {yes},
journal = {Journal of Computational and Applied Mathematics},
keywords = {Degenerate parabolic problems; Euler implicit scheme; Linearization; Mixed finite elements; Regularization; Richards' equation},
pages = {365-373},
peerreviewed = {Yes},
title = {{Mixed} finite elements for the {Richards}' equation: {Linearization} procedure},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_PopRaduKn_MixedFiniteElementsForRichardsEquationLinearProcedure},
volume = {168},
year = {2004}
}
@article{faucris.219793959,
author = {Class, H. and Knabner, Peter and Pop, I. S. and Radu, F. A.},
doi = {10.1007/s10596-019-9814-4},
faupublication = {yes},
journal = {Computational Geosciences},
note = {CRIS-Team WoS Importer:2019-06-07},
pages = {203-205},
peerreviewed = {unknown},
title = {{Multiphase}, multicomponent flow in deformable porous media: modelling and simulation ({Dedicated} to {Prof}. {Dr}.-{Ing}. {Rainer} {Helmig} on the occasion of his 60th birthday)},
volume = {23},
year = {2019}
}
@inproceedings{faucris.219833179,
author = {Lieu, Alice and et al.},
author_hint = {Lieu Alice, Gabard Gwénaël, Bériot Hadrien},
booktitle = {22nd International Congress on Sound and Vibration 2015},
date = {2015-07-12/2015-07-16},
faupublication = {no},
peerreviewed = {unknown},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{A} performance study of high-order finite elements and wave-based discontinuous {Galerkin} methods for a convected {Helmholtz} problem},
venue = {Florence, Italy},
year = {2015}
}
@misc{faucris.221482707,
author = {Antonic, Nenad and Capasso, Vincenzo and Jäger, Willi and Mikelic, Andro and Neuss-Radu, Maria and Pop, Iuliu Sorin},
doi = {10.1016/j.nonrwa.2013.09.008},
faupublication = {no},
month = {Jan},
peerreviewed = {automatic},
title = {{Multiscale} {Problems} in {Science} and {Technology}. {Challenges} to {Mathematical} {Analysis} and {Perspectives}},
year = {2014}
}
@article{faucris.216729740,
author = {Grün, Günther and Grillmeier, Hubertus},
doi = {10.1090/mcom/3372},
faupublication = {yes},
journal = {Mathematics of Computation},
pages = {1021-1059},
peerreviewed = {Yes},
title = {{Nonnegativity} preserving convergent schemes for the stochastic porous-medium equation},
year = {2018}
}
@article{faucris.117799044,
abstract = {We analyze mathematically and numerically a model derived in part 1 of this paper [Knabner et el, this issue]. The model deals with carrier- influenced transport and besides advective and dispersive transport and equilibrium and nonequilibrium sorption takes into account both carrier facilitation and cosorption. Guidelines are derived, whether the overall mobility is enhanced or reduced and various other properties of the model elucidated, in particular for varying carrier concentrations. We indicate how to modify existing numerical codes for the usual adsorption model and discuss simulations for experimental data set},
author = {Totsche, Kai Uwe and Knabner, Peter and Kögel-Knabner, Ingrid},
doi = {10.1029/95WR02993},
faupublication = {yes},
journal = {Water Resources Research},
pages = {1623-1634},
peerreviewed = {Yes},
title = {{The} modeling of reactive solute transport with sorption to mobile and immobile sorbents. 2. {Model} discussion and numerical simulation},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1996/1996_KnTotscheKoegelkn_ModelOfReactSolutTranspWthsorptToMobImmobSorbP2},
volume = {32},
year = {1996}
}
@article{faucris.119486444,
abstract = {Upscaling transport in porous media including both biomass development and simultaneous structural changes in the solid matrix is extremely challenging. This is because both affect the medium's porosity as well as mass transport parameters and flow paths. We address this challenge by means of a multiscale model. At the pore scale, the local discontinuous Galerkin (LDG) method is used to solve differential equations describing particularly the bacteria's and the nutrient's development. Likewise, a sticky agent tightening together solid or bio cells is considered. This is combined with a cellular automaton method (CAM) capturing structural changes of the underlying computational domain stemming from biomass development and solid restructuring. Findings from standard homogenization theory are applied to determine the medium's characteristic time- and space-dependent properties. Investigating these results enhances our understanding of the strong interplay between a medium's functional properties and its geometric structure. Finally, integrating such properties as model parameters into models defined on a larger scale enables reflecting the impact of pore scale processes on the larger scale. (C) 2017 Elsevier Ltd. All rights reserved.},
author = {Ray, Nadja and Rupp, Andreas and Prechtel, Alexander},
doi = {10.1016/j.advwatres.2017.04.001},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {Soil structure;Biomass development;Transport properties;Mathematical modeling;Cellular automaton;Multiscale methods},
pages = {393-404},
peerreviewed = {Yes},
title = {{Discrete}-continuum multiscale model for transport, biomass development and solid restructuring in porous media},
volume = {107},
year = {2017}
}
@article{faucris.123685144,
abstract = {In this paper we present and investigate a stochastic model and its associated Fokker-Planck equation for the lay-down of fibers on a conveyor belt in the production process of nonwoven materials. The model is based on a stochastic differential equation taking into account the motion of the fiber under the influence of turbulence. A reformulation as a stochastic Hamiltonian system and an application of the stochastic averaging theorem lead to further simplifications of the model. Finally, the model is used to compute the distribution of functionals of the process that are important for the quality assessment of industrial fabrics. © 2007 Society for Industrial and Applied Mathematics.},
author = {Götz, Thomas and Klar, Axel and Marheineke, Nicole and Wegener, Raimund},
doi = {10.1137/06067715X},
faupublication = {no},
journal = {SIAM Journal on Applied Mathematics},
keywords = {Fiber dynamics; Fokker-planck equations; Stochastic averaging; Stochastic hamiltonian system},
pages = {1704-1717},
peerreviewed = {Yes},
title = {{A} stochastic model and associated {Fokker}-{Planck} equation for the fiber lay-down process in nonwoven production processes},
volume = {67},
year = {2007}
}
@article{faucris.121947584,
abstract = {Considering the dynamics of long slender elastic fibers in turbulent flows, there exists a stochastic aerodynamic force concept for a general drag model based on a k- ε{lunate} turbulence description in literature. In this paper we generalize the concept and formulate an air drag model that is universally valid for all Reynolds number regimes and incident flow directions. The associated turbulent force overcomes the hitherto existing limitations and allows the simulation of all kind of fibers (flexible, stiff, light, heavy) immersed in turbulent flows. Moreover, the comparison of the numerical results with PIV-measurements shows very convincing agreements in the industrial application of technical textile manufacturing. © 2010 Elsevier Ltd.},
author = {Marheineke, Nicole and Wegener, Raimund},
doi = {10.1016/j.ijmultiphaseflow.2010.10.001},
faupublication = {yes},
journal = {International Journal of Multiphase Flow},
keywords = {Aerodynamic drag; Data interpolation; Dimensional analysis; Fiber-fluid interactions; Long slender fibers; Numerical simulations; Stochastic partial differential algebraic equation; Turbulence modeling},
pages = {136-148},
peerreviewed = {Yes},
title = {{Modeling} and application of a stochastic drag for fibers in turbulent flows},
volume = {37},
year = {2011}
}
@incollection{faucris.214979280,
abstract = {In this work we investigate the computation of nonlinear eigenfunctions via the extinction profiles of gradient flows. We analyze a scheme that recursively subtracts such eigenfunctions from given data and show that this procedure yields a decomposition of the data into eigenfunctions in some cases as the 1-dimensional total variation, for instance. We discuss results of numerical experiments in which we use extinction profiles and the gradient flow for the task of spectral graph clustering as used, eg, in machine learning applications.

Moreover, effects and knowledge deduced from the model are transfered to
scales which are more relevant for applications. The quality of these
averaged models is of general interest, since simulations for the
field-scale that resolve the pore-scale are not applicable for
economical reasons. Thus, this book compares parameterizations of
diffusivities with mathematically rigorous results and gives suggestions
to improve the formulas that can be found in the literature.

Last but not least, it is imperative to apply a proper numerical method
to implement the model in silico. The local discontinuous Galerkin (LDG)
method seems to be suitable for this task, since it is locally
mass-conservative and is stable for discontinuous data - that might, for
example, originate from the discrete movement of the geometry or from
the sharp boundaries between the different phases. Additionally, this
method has no problems with complicated transfer conditions. These
aspects are demonstrated in a mathematically rigorous way, and the
method is improved upon by reducing the linear system of equations
resulting from the discretization. This is a real enhancement, since it
does not diminish the order of convergence but decreases the
computational costs.},
author = {Rupp, Andreas},
faupublication = {yes},
keywords = {microaggregate development; discontinuous Galerkin; upscaling; diffusion; homogenization; Darcy’s equation},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Simulating} {Structure} {Formation} in {Soils} across {Scales} using {Discontinuous} {Galerkin} {Methods}},
year = {2019}
}
@article{faucris.214175734,
abstract = {Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertex-based slope limiters for tensorvalued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics. (C) 2019 Elsevier Inc. All rights reserved.},
author = {Hajduk, Hennes and Kuzmin, Dmitri and Aizinger, Vadym},
doi = {10.1016/j.jcp.2019.01.032},
faupublication = {yes},
journal = {Journal of Computational Physics},
note = {CRIS-Team WoS Importer:2019-03-22},
pages = {308-325},
peerreviewed = {Yes},
title = {{New} directional vector limiters for discontinuous {Galerkin} methods},
volume = {384},
year = {2019}
}
@article{faucris.209915729,
author = {Brinkmann, Eva Maria and Burger, Martin and Grah, Joana Sarah},
doi = {10.1007/s10851-018-0861-6},
faupublication = {yes},
journal = {Journal of Mathematical Imaging and Vision},
keywords = {Denoising; Helmholtz decomposition; (Higher-order) total variation (TV) regularisation; Variational methods; Natural differential operators; Sparse regularisation},
peerreviewed = {Yes},
title = {{Unified} {Models} for {Second}-{Order} {TV}-{Type} {Regularisation} in {Imaging}: {A} {New} {Perspective} {Based} on {Vector} {Operators}},
url = {https://arxiv.org/pdf/1802.01895.pdf},
year = {2018}
}
@incollection{faucris.223308773,
address = {Mannheim},
author = {Hoffmann, Karl-Heinz and Knabner, Peter},
booktitle = {Approximatiion in Theorie und Praxis},
editor = {Prof. Dr. Günter Meinardus},
faupublication = {no},
isbn = {3411015675},
peerreviewed = {unknown},
publisher = {Bibliographisches Institut},
title = {{Freie} {Randwertprobleme}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1979/1979_HoffmannKn_FreieRandwertprobleme},
year = {1979}
}
@article{faucris.223591174,
abstract = {We develop a model for solute transport in porous media, which takes
into account equilibrium and nonequilibrium multiple-site adsorption.
Binary ion exchange is included. Adsorption rate formulations and their
isotherms are reviewed and their mathematical properties investigated.
This forms the basis of the study of travelling-wave solutions in the
second part of the pape},
author = {van Duijn, CJ Hans and Knabner, Peter},
doi = {10.1007/BF00617116},
faupublication = {no},
journal = {Transport in Porous Media},
keywords = {Travelling wave; self-sharpening front; mathematical analysis; mathematical model; adsorption; ion exchange},
pages = {167-194},
peerreviewed = {Yes},
title = {{Travelling} {Waves} in the {Transport} of {Reactive} {Solutes} through {Porous} {Media}: {Adsorption} and {Binary} {Ion} {Exchange} - {Part} 1},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1992/1992_DuijnKn_TravellingWavesThroughPMAdsorptionPart1},
volume = {8},
year = {1992}
}
@article{faucris.204147514,
abstract = {Exploring biophysical properties of virus-encoded components and their requirement for virus replication is an exciting new area of interdisciplinary virological research. To date, spatial resolution has only rarely been analyzed in computational/biophysical descriptions of virus replication dynamics. However, it is widely acknowledged that intracellular spatial dependence is a crucial component of virus life cycles. The hepatitis C virus-encoded NS5A protein is an endoplasmatic reticulum (ER)-anchored viral protein and an essential component of the virus replication machinery. Therefore, we simulate NS5A dynamics on realistic reconstructed, curved ER surfaces by means of surface partial differential equations (sPDE) upon unstructured grids. We match the in silico NS5A diffusion constant such that the NS5A sPDE simulation data reproduce experimental NS5A fluorescence recovery after photobleaching (FRAP) time series data. This parameter estimation yields the NS5A diffusion constant. Such parameters are needed for spatial models of HCV dynamics, which we are developing in parallel but remain qualitative at this stage. Thus, our present study likely provides the first quantitative biophysical description of the movement of a viral component. Our spatio-temporal resolved ansatz paves new ways for understanding intricate spatial-defined processes central to specfic aspects of virus life cycles.},
author = {Knodel, Markus and et al.},
author_hint = {Knodel MM, Nagel A, Reiter S, Vogel A, Targett-Adams P, McLauchlan J, Herrmann E, Wittum G},
doi = {10.3390/v10010028},
faupublication = {no},
journal = {Viruses-Basel},
keywords = {computational virology;hepatitis C virus (HCV);viral dynamics;within-host viral modelling;parameter estimation;3D spatio-temporal resolved mathematical models;realistic geometries;(surface) partial differential equations;Finite Volumes;massively parallel multigrid solvers},
month = {Jan},
peerreviewed = {Yes},
support_note = {Author relations incomplete. You may find additional data in field 'author_hint'},
title = {{Quantitative} {Analysis} of {Hepatitis} {C} {NS5A} {Viral} {Protein} {Dynamics} on the {ER} {Surface}},
volume = {10},
year = {2018}
}
@article{faucris.123853224,
abstract = {We prove optimal order convergence of an upwind-mixed hybrid finite element scheme for linear parabolic advection-diffusion-reaction problems. It was introduced in [Radu et al., Adv. Water Resources, 34(2011), pp. 47-61] and is based on an Euler-implicit mixed hybrid finite element discretization of the problem in fully mass conservative form using the Raviart-Thomas mixed finite element of lowest order on triangular meshes. Optimal order convergence in time and space is obtained for the fully discrete formulation. The scheme provides the same order of convergence as the standard upwind-mixed method, while it is more efficient since a local elimination of variables is possible with our choice of the upwind weights. The theoretical findings are sustained by a numerical experiment. © 2014 Society for Industrial and Applied Mathematic},
author = {Brunner, Fabian and Radu, Adrian Florin and Knabner, Peter},
doi = {10.1137/130908191},
faupublication = {yes},
journal = {SIAM Journal on Numerical Analysis},
keywords = {A priori error estimates; Advection-diffusion problem; Advection-dominance; Mixed finite element method; Upwind weighting},
pages = {83-102},
peerreviewed = {Yes},
title = {{Analysis} of an upwind-mixed hybrid finite element method for transport problems},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2014/2014_BrunnerRaduKn_AnalysisOfAnUpwindMixedHyTransProblems},
volume = {52},
year = {2014}
}
@incollection{faucris.123750044,
abstract = {Surfactants occur already in undisturbed biological processes in soils but within the development of remediation techniques these substances are of large interest because of their interaction with hydrophobic substances. The standard models for (un-)saturated water flow and solute transport are extended to include the influence of the surfactant transport on the flow of the water phase. Two effects of surfactants on (un-)saturated water flow are included: the modification of the interfacial tension between water and air and the swelling of clay minerals due to sorption of surfactants. Simulations are presented, which exhibit the feedback of surfactant transport on water flow and content. An identification algorithm for hydraulic soil properties has been developed that supplements the simulation too},
address = {Berlin, Heidelberg},
author = {Knabner, Peter and Bitterlich, Sandro and Iza Teran, Rodrigo and Prechtel, Alexander and Schneid, Eckhard},
booktitle = {Mathematics - Key Technology for the Future},
doi = {10.1007/978-3-642-55753-8_12},
editor = {Willi Jäger, Hans-Joachim Krebs},
faupublication = {yes},
isbn = {978-3-642-62914-3},
pages = {152-161},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Influence} of {Surfactants} on {Spreading} of {Contaminants} and {Soil} {Remediation}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2003/2003_KnBitterlichTeranPrechtelSchneid_InfluOfSurfactOnSpreadOfContaminSoilRemed},
year = {2003}
}
@article{faucris.215925573,
abstract = {We give a characterization of relatively compact sets $F$ in $L^p(\Omega,B)$ for $p\in [1,\infty)$, $B$ a Banach-space, and $\Omega \subset \R^n$. This is a generalization of the results obtained in \cite{Simon} for the space $L^p((0,T),B)$ with $T>0$, first to rectangles $\Omega =(a,b) \subset \R^n$ and, under additional conditions, to arbitrary open and bounded subsets of $\R^n$. An application of the main compactness result to a problem arising in homogenization of processes on periodic surfaces is given.

0}, the boundary layers are defined on the semi-infinite strip ]0, 1[(n-1)x]0, infinity[, and their energies decrease exponentially with respect to the second variable. In [11] we have shown that the property of uniform exponential decay of the boundary layers does not hold in general.In our contribution, we improve the result of (11) by showing that for general domains the optimal decay of the energy of the boundary layers is polynomial of degree -1.},
author = {Neuss-Radu, Maria},
faupublication = {no},
month = {Jan},
pages = {243-250},
peerreviewed = {unknown},
title = {{The} failure of uniform exponential decay for boundary layers},
year = {2002}
}
@article{faucris.118023884,
abstract = {We prove the existence and uniqueness of time-global solutions for multi-species multi-reaction advection-diffusion-dispersion problems with mass action kinetics in the space W-p(2,1)([0, T] x Omega). The reaction terms of mass action kinetics may contain polynomial expressions of arbitrarily high order. The difficulty to obtain an a priori estimate for the semilinar system of PDEs is tackled with a special Lyapunov function.},
author = {Kräutle, Serge},
doi = {10.11948/2011034},
faupublication = {yes},
journal = {JOURNAL OF APPLIED ANALYSIS AND COMPUTATION},
keywords = {Existence of solutions;A priori estimate;Parabolic equations;Reactive transport;Mass action kinetics},
month = {Jan},
pages = {497-515},
peerreviewed = {unknown},
title = {{Existence} of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media},
volume = {1},
year = {2011}
}
@article{faucris.117788044,
abstract = {We derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures. Special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region. The main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system, a suitable regularity of the exact solutions, and a spectrum estimate for the operator of the Allen-Cahn equation. The spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model. This is done by a technique introduced in [X. Feng and A. Prohl, Math. Comp., 73 (2004), pp. 541-567] for phase transitions of pure materials with linear constitutive relations. © 2010 Society for Industrial and Applied Mathematic},
author = {Eck, Christof and Jadamba, Baasansuren and Knabner, Peter},
doi = {10.1137/050637984},
faupublication = {yes},
journal = {SIAM Journal on Numerical Analysis},
keywords = {A priori error estimate; Finite element method; Phase field model; Thermodynamically consistent model},
pages = {4429-4445},
peerreviewed = {Yes},
title = {{Error} estimates for a finite element discretization of a phase field model for mixtures},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2010/2010_EckJadambaKn_ErrorEstimatesForFiniteElementDiscreOfAPhaseFieldModelForMix},
volume = {47},
year = {2010}
}
@incollection{faucris.116677924,
author = {Cibis, Thomas and Leithäuser, Christian and Marheineke, Nicole and Wegener, Raimund},
booktitle = {Progress in Industrial Mathematics at ECMI 2014},
faupublication = {yes},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Homogenization} strategies for fiber curtains and bundles in air flows},
year = {2016}
}
@incollection{faucris.122532124,
author = {Panda, Satyananda and Marheineke, Nicole and Wegener, Raimund},
booktitle = {Progress in Industrial Mathematics at ECMI 2006},
editor = {L. L. Bonilla, M. Moscoso, G. Platero, J. M. Vega},
faupublication = {no},
pages = {685-690},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Dynamics} of curved viscous fibers},
year = {2007}
}
@article{faucris.117786944,
abstract = {A two-scale model for liquid-solid phase transitions with equiaxed dendritic microstructure in binary material in the case of slow solute diffusion is presented. The model consists of a macroscopic energy transport equation and, for each point of the macroscopic domain, a local cell problem describing the evolution of the microstructure and the microsegregation. It is derived by formal homogenization of a sharp interface model, including the Gibbs-Thomson law and kinetic undercooling. Based on the two-scale model, a numerical two-scale method for the simulation of phase transitions with dendritic microstructure is developed, and numerical examples are presented. © 2002 Elsevier Science (USA},
author = {Eck, Christof and Knabner, Peter and Korotov, Sergey},
doi = {10.1006/jcph.2002.7018},
faupublication = {yes},
journal = {Journal of Computational Physics},
keywords = {Crystal growth; Gibbs-Thomson law; Homogenization; Kinetic undercooling; Multiscale model; Stefan problem},
pages = {58-80},
peerreviewed = {Yes},
title = {{A} two-scale method for the computation of solid-liquid phase transitions with dendritic microstructure},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2002/2002_EckKnKorotov_ATwoScaleMethodForTheComputationOfSolidLiquidPhaseTransitions},
volume = {178},
year = {2002}
}
@article{faucris.122619024,
abstract = {This article deals with the determination of nonlinear coefficient functions in partial differential equations in the field of soil science. We consider two examples to illustrate the numerical determination of nonlinear coefficient functions. In detail, these are the determination of the sorption characteristic of a chemical and the determination of the unsaturated hydraulic properties of a porous medium from measurements obtained from suitable column experiments. This inverse problem is treated by minimizing a least square functional. To cope with the ill-posedness, we apply a parametrization of the unknown nonlinear coefficient function, which is defined by an appropriate interpolation. The parametrization does not use a priori assumptions, which are not justified by physical properties. This kind of parametrization permits a hierarchical approach in the number of the degrees of freedom used. According to the hierarchical structure, we integrate the determination of the coefficient functions into a multi-level procedure. The investigation of the stability of the parametrization is based on the singular values of the sensitivity matri},
author = {Bitterlich, Sandro and Knabner, Peter},
doi = {10.1080/10682760310001597482},
faupublication = {yes},
journal = {Inverse Problems in Science and Engineering},
keywords = {Multi-level identification; Sorption characteristics; Stability analysis; Unsaturated hydraulic properties},
pages = {361-378},
peerreviewed = {Yes},
title = {{Numerical} methods for the determination of material properties in soil science},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_BitterlichKn_NumericMethForDeterminOfMaterPropertInSoilScience},
volume = {12},
year = {2004}
}
@article{faucris.106887924,
abstract = {A theory of a posteriori error estimation for initial-value problems for nonlinear systems of partial differential equations is presented. The theoretical conclusions arc applied to a specific problem arising in the theory of flows in porous media. The paper is finished by an instructive numerical example. © 1998 Elsevier Science B.},
author = {Angermann, Lutz and Knabner, Peter and Thiele, Kathrin},
doi = {10.1016/S0168-9274(97)00084-6},
faupublication = {yes},
journal = {Applied Numerical Mathematics},
pages = {179-191},
peerreviewed = {Yes},
title = {{An} error estimator for a finite volume discretization of density driven flow in porous media},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1998/1998_AngermannKnThiele_ErrorEstimForFiniteVolDiscretOfDensDrivFlowInPM},
volume = {26},
year = {1998}
}
@article{faucris.221484951,
author = {Jäger, Willi and Neuss-Radu, Maria and Shaposhnikova, Tatiana A.},
doi = {10.1007/s10958-011-0603-4},
faupublication = {no},
journal = {Journal of Mathematical Sciences},
pages = {446-459},
peerreviewed = {unknown},
title = {{Homogenization} of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain - the influence of the scaling on the nonlinearity in the effective sink-source term},
volume = {179},
year = {2011}
}
@article{faucris.216548158,
abstract = {We
study the two-dimensional snake-like pattern that arises in phase
separation of alloys described by spinodal decomposition in the
Cahn-Hilliard model. These are somewhat universal patterns due to an
overlay of eigenfunctions of the Laplacian with a similar wave-number.
Similar structures appear in other models like reaction-diffusion
systems describing animal coats' patterns or vegetation patterns in
desertification.
Our main result studies random functions given by cosine Fourier
series with independent Gaussian coefficients, that dominate the
dynamics in the Cahn-Hilliard model. This is not a cosine process, as
the sum is taken over domains in Fourier space that not only grow and
scale with a parameter of order 1/ε,
but also move to infinity. Moreover, the model under consideration is
neither stationary nor isotropic.
To study the pattern size of nodal domains we consider the density of
zeros on any straight line through the spatial domain. Using a theorem
by Edelman and Kostlan and weighted ergodic theorems that ensure the
convergence of the moving sums, we show that the average distance of
zeros is asymptotically of order ε with a precisely given constan},
author = {Wacker, Philipp Konstantin and Bianchi, Luigi Amedeo and Blömker, Dirk},
doi = {10.1137/15M1052081},
faupublication = {no},
journal = {SIAM Journal on Applied Mathematics},
pages = {1292 - 1319},
peerreviewed = {Yes},
title = {{Pattern} size in {Gaussian} fields from spinodal decomposition},
url = {https://arxiv.org/pdf/1510.01967.pdf},
volume = {77},
year = {2017}
}
@article{faucris.110901824,
abstract = {Inverse methods are increasingly used to estimate the hydraulic properties of unsaturated soils. The method generally uses a weighted least-squares approach in which numerically simulated data are fitted to measured data. In this study we used inverse methods to estimate the unsaturated soil hydraulic properties from continuous and multi step column outflow experiments. The method employs piecewise polynomial functions to obtain a free-form parameterization of the hydraulic properties, rather than fixed functional forms typical of the van Gen-uchten-Mualem and Brooks-Corey-Burdine models. For the polynomial functions we used quadratic B-splines and piecewise cubic Hermite interpolation. The method leads to local parameterizations that can also be hierarchic, depending on the invoked number of degrees of freedom. Since a suitable number of degrees of freedom cannot be defined a priori, we embedded the estimation method into a multilevel procedure, which also included a stability analysis, based on singular value decomposition of the sensitivity matrix. The optimization procedure was made more stable by imposing monotonicity constraints on the hydraulic functions. Tests with synthetic and measured data from column outflow experiments show the validity and robustness of the method. © Soil Science Society of Americ},
author = {Bitterlich, Sandro and Durner, Wolfgang and Iden, Sascha and Knabner, Peter},
doi = {10.2113/3.3.971},
faupublication = {yes},
journal = {Vadose Zone Journal},
pages = {971-981},
peerreviewed = {Yes},
title = {{Inverse} estimation of the unsaturated soil hydraulic properties from column outflow experiments using free-form parameterizations},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2004/2004_BitterlichDurnerIdenKn_InverseEstimationOfTheUnsaturatedSoilHydraulPropFromColumnOutlowExperiments},
volume = {3},
year = {2004}
}
@incollection{faucris.208631534,
author = {Neuß, Nicolas},
booktitle = {Abschlussbericht SFB 359},
doi = {10.1007/978-3-540-28396-6_21},
editor = {Universität Heidelberg},
faupublication = {no},
isbn = {9783540283799},
pages = {567-577},
peerreviewed = {unknown},
publisher = {Springer Berlin Heidelberg},
title = {{High}-accuracy approximation of effective coefficients},
year = {2007}
}
@book{faucris.217114270,
abstract = {Dieses umfangreiche Vorlesungsskript war (zum Teil) Grundlage für meine Vorlesungen "Mathematik für die Informationswirtschaft" am KIT Karlsruhe und ist Grundlage für meine Vorlesungen "Mathematik für Ingenieure" an der FAU Erlangen-Nürnberg.

The

conjugate gradient boundary iteration

(CGBI) is a domain decom-

position method for elliptic partial differential equations [1] [2] [3] [6

] [7].

In the context of fractional step methods [10], a Navier-Stokes solver based

on CGBI and a characteristic’s method was constructed. In this article,

numerical results for the incompressible flow past a backward facing step

and past a cylinder (both modelled in 2d) are presented.

},
author = {Kräutle, Serge and Wielage, Kerstin},
booktitle = {The Navier-Stokes Equations: Theory and Numerical Methods},
editor = {Rodolfo Salvi},
faupublication = {no},
isbn = {9780824706722},
pages = {247-255},
peerreviewed = {unknown},
series = {Lecture Notes in Pure and Applied Mathematics},
title = {{Numerical} results for the {CGBI} method to viscous channel flow},
volume = {223},
year = {2001}
}
@article{faucris.224152083,
abstract = {We investigate the influence of multiscale aggregation and deposition on the
colloidal dynamics in a saturated porous medium. At the pore scale, the
aggregation of colloids is modeled by the Smoluchowski equation. Essentially,
the colloidal mass is distributed between different size clusters. We treat
these clusters as different species involved in a diffusion-advection-reaction
mechanism. This modeling procedure allows for different material properties to
be varied between the different species, specifically the rates of diffusion,
aggregation, deposition as well as the advection velocities. We apply the
periodic homogenization procedure to give insight into the effective
coefficients of the upscaled model equations. Benefiting from direct access to
microstructural information, we capture by means of 2D numerical simulations
the effect of aggregation on the deposition rates recovering this way both the
blocking and ripening regimes reported in the literatu},
author = {Krehel, Oleh and Muntean, Adrian and Knabner, Peter},
doi = {10.1016/j.advwatres.2015.10.005},
faupublication = {yes},
journal = {Advances in Water Resources},
pages = {209-216},
peerreviewed = {Yes},
title = {{Multiscale} {Modeling} of {Colloidal} {Dynamics} in {Porous} {Media}: {Capturing} {Aggregation} and {Deposition} {Effects}},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2015/2015_KrehelMunteanKn_MultiScaleOfCollDyninPMCapturingAggregAndDeposiEffects},
volume = {86},
year = {2015}
}
@article{faucris.124247464,
abstract = {In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton's method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method. © Springer Science + Business Media B.V. 200},
author = {Hoffmann, Joachim and Kräutle, Serge and Knabner, Peter},
doi = {10.1007/s10596-009-9173-7},
faupublication = {yes},
journal = {Computational Geosciences},
keywords = {Complementarity problems; Numerical simulation; Porous media; Reactive transport; Size reduction},
pages = {421-433},
peerreviewed = {Yes},
title = {{A} parallel global-implicit 2-{D} solver for reactive transport problems in porous media based on a reduction scheme and its application to the {MoMaS} benchmark problem},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2010/2010_HoffmannKraeutleKn_AParallelGlobalimplicit2DSolToTheMoMaSBenchmarkProblem},
volume = {14},
year = {2010}
}
@incollection{faucris.109102224,
address = {Heidelberg},
author = {Wegener, Raimund and Marheineke, Nicole and Hietel, Dietmar},
booktitle = {Mathematik im Fraunhofer Institut},
editor = {H. Neunzert, D. Praetzel-Wolters},
faupublication = {yes},
pages = {105-165},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Virtuelle} {Produktion} von {Filamenten} und {Vliesstoffen}},
year = {2015}
}
@article{faucris.123855864,
abstract = {Accurate identification of interactions of reactive solutes with porous media constituents is necessary for reliable risk assessment studies and the development of efficient sanitation strategies. Standard parameter estimation procedures bear a number of unsolved problems with respect to uniqueness and identifiability. This paper presents a new approach for the identification of nonlinear interaction parameters of column outflow experiments. The procedure requires no a priori assumptions on the shape of the underlying interaction process functions. Employing experimental data sets on cadmium and anthracene breakthrough as case studies, possible applications of the new approach will be shown, and its features will be discussed. Error analysis based on singular value decomposition of the sensitivity matrix quantifies the identification error. Identification procedures without a priori shape information are superior to fixed parametrizations in diagnostic investigations, especially in cases without reliable a priori knowledge on the sorptive interactions. © Springer 200},
author = {Knabner, Peter and Igler, B. A. and Totsche, Kai Uwe and DuChateau, Paul},
doi = {10.1007/s10596-005-9008-0},
faupublication = {yes},
journal = {Computational Geosciences},
keywords = {Adaptation; Breakthrough experiment; Identification; Inverse problems; Output least squares minimization; Sorption},
pages = {203-217},
peerreviewed = {Yes},
title = {{Unbiased} identification of nonlinear sorption characteristics by soil column breakthrough experiments},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_IglerTotscheKn_UnbiasIdentOfNonlinSorptCharactBySoilColumBreakthrExperim},
volume = {9},
year = {2005}
}
@article{faucris.117798824,
author = {Knabner, Peter and Mikelic, Andro and Pop, Iuliu Sorin},
doi = {10.1007/s10596-013-9348-0},
faupublication = {yes},
journal = {Computational Geosciences},
pages = {443-445},
peerreviewed = {Yes},
title = {{Special} issue "{Mathematics} of {Porous} {Media}," dedicated to {Professor} {C}.{J}. van {Duijn} on the occasion of his 60th anniversary},
url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_2013/2013_KnMikelicPop_SpecialIssueProfDuijn60thAnniversary},
volume = {17},
year = {2013}
}
@article{faucris.121151844,
author = {Grün, Günther},
faupublication = {no},
journal = {Zeitschrift für Analysis und ihre Anwendungen},
pages = {541-573},
peerreviewed = {Yes},
title = {{Degenerate} parabolic equations of fourth order and a plasticity model with nonlocal hardening},
volume = {14},
year = {1995}
}
@incollection{faucris.223190217,
abstract = {Dissolved organic matter (DOM) can act as sorbent for exogenous organic
chemicals in soils, possibly leading to enhanced leaching of these
compounds. The distribution of exogeous organic chemicals in soils is
considered here as an equilibrium of environmental chemicals in a three
compartment system: dissolved, sorbed to dissolved macromolecules, and
sorbed to the bulk soil matrix. In the present study the distribution of
a new acidic herbicide in the three compartment system is investigated.
Sorption isotherms are determined for the sorption of organic chemicals
to bulk soil materials (Ap horizons) and to water-soluble soil organic
matter. From these data an effective equilibrium sorption isotherm to
bulk soil can be calculated which takes into account the sorption of
organic chemicals to dissolved organic matter. A markable influence of
DOM on sorption isotherms of the experimental herbicide is found only
for high amounts of DOM (100 mg C LThe one-dimensional scheme derived is able to handle shock solutions. In the non-equilibrium case a more severe restriction on the timestep has to be regarde}, address = {Braunschweig, Wiesbaden}, author = {Barrett, John W. and Kappmeier, H. and Knabner, Peter}, booktitle = {Modeling and Computation in Environmental Sciences}, doi = {10.1007/978-3-322-89565-3_4}, editor = {Rainer Helmig, Willi Jäger, Wolfgang Kinzelbach, Peter Knabner, Gabriel Wittum}, faupublication = {yes}, pages = {36-48}, peerreviewed = {unknown}, publisher = {Vieweg+Teubner}, series = {Notes on Numerical Fluid Mechanics}, title = {{Lagrange}-{Galerkin} {Approximation} {For} {Advection}-{Dominated} {Contaminant} {Transport} {With} {Nonlinear} {Equilibrium} {Or} {Non}-equilibrium {Adsorption}}, url = {https://www1.am.uni-erlangen.de/research/publications/Jahr_1997/1997_BarrettKappmeierKn_LagrangeGalerkinApproxWithNonlinearNonequilibAdsorption}, volume = {59}, year = {1997} } @article{faucris.221486263, author = {Jäger, Willi and Mikelic, Andro and Neuss-Radu, Maria}, doi = {10.1007/s00205-008-0118-4}, faupublication = {no}, journal = {Archive for Rational Mechanics and Analysis}, pages = {331-374}, peerreviewed = {Yes}, title = {{Analysis} of differential equations modelling the reactive flow through a deformable system of cells}, volume = {192}, year = {2009} }