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.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 aggregation.}, 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}, volume = {86}, year = {2015} } @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.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 Society.}, 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}, volume = {80}, year = {2009} } @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) front.}, 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}, volume = {22}, year = {1998} } @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 reserved.}, 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}, volume = {27}, year = {2004} } @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 discussed.}, 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}, volume = {32}, year = {1996} } @article{faucris.119171184, abstract = {Abstract We identify sufficient conditions under which evolution equations for probability density functions (PDF) of random concentrations are equivalent to Fokker-Planck equations. The novelty of our approach is that it allows consistent PDF approximations by densities of computational particles governed by Itô processes in concentration-position spaces. Accurate numerical solutions are obtained with a global random walk (GRW) algorithm, stable, free of numerical diffusion, and insensitive to the increase of the total number of computational particles. The system of Itô equations is specified by drift and diffusion coefficients describing the PDF transport in the physical space, provided by up-scaling procedures, as well as by drift and mixing coefficients describing the PDF transport in concentration spaces. Mixing models can be obtained similarly to classical PDF approaches or, alternatively, from measured or simulated concentration time series. We compare their performance for a GRW-PDF numerical solution to a problem of contaminant transport in heterogeneous groundwater systems.}, author = {Suciu, Nicolae and Radu, Adrian Florin and Attinger, Sabine and Schüler, Lennart and Knabner, Peter}, doi = {10.1016/j.cam.2015.01.030}, faupublication = {yes}, journal = {Journal of Computational and Applied Mathematics}, keywords = {Mixing; PDF methods; Porous media; Random walk}, pages = {241-252}, peerreviewed = {Yes}, title = {{A} {Fokker}-{Planck} approach for probability distributions of species concentrations transported in heterogeneous media}, volume = {289}, year = {2015} } @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} } @incollection{faucris.111089484, author = {Knabner, Peter and Frank, Florian and Hoffmann, Joachim and Kräutle, Serge and Oßmann, Stephan and Prechtel, Alexander}, booktitle = {KORA - Synopse "Systemanalyse, Modellierung und Prognose der Wirkungen natürlicher Schadstoffminderungsprozesse - eine rezente Synopse"}, faupublication = {yes}, pages = {195-233}, peerreviewed = {unknown}, series = {Gemeinsame Mitteilungen des Dresdner Grundwasserforschungszentrum e.V. und seiner Partner}, title = {{Entwicklung}, {Zuverlässigkeit} und {Effizienz} reaktiver {Mehrkomponententransportmodelle}}, volume = {5}, year = {2008} } @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 Vordiplom.}, 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}}, year = {2000} } @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 reserved.}, 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}, volume = {168}, year = {2004} } @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}}, volume = {1}, year = {1994} } @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. 2009.}, 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}}, volume = {14}, year = {2010} } @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://epubs.siam.org/doi/abs/10.1137/16M1109266}, volume = {49}, year = {2017} } @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}}, venue = {Strasbourg}, year = {2008} } @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 Press.}, 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}, volume = {52}, year = {1994} } @article{faucris.123788104, abstract = {The conditions for the invertibility of the isoparametric mapping for hexahedral finite elements are discussed. The mapping gives a reference element onto a global element for trilinear finite elements. An algorithm that checks the positivity of the Jacobian determinant depending on the position of the vertices is also presented.}, author = {Knabner, Peter and Korotov, Sergey and Summ, Gerhard}, doi = {10.1016/S0168-874X(02)00196-8}, faupublication = {yes}, journal = {Finite Elements in Analysis and Design}, keywords = {Hexahedral finite elements; Invertibility; Isoparametric mapping}, pages = {159-172}, peerreviewed = {Yes}, title = {{Conditions} for the invertibility of the isoparametric mapping for hexahedral finite elements}, volume = {40}, year = {2003} } @article{faucris.110982124, abstract = {Drug release from collagen matrices is in most cases governed by diffusion from swollen matrices but also enzymatic matrix degradation or hydrophobic drug/ collagen interactions may contribute. To reduce water uptake and to prolong the release, insoluble collagen matrices have been chemically or dehydrothermally cross-linked. Assuming Fickian diffusion a one-dimensional model was developed and tested that allows description of water penetration, swelling and drug release and that may be expanded considering a subsequent erosion process or interactions. Swelling is described by a volume balance. For dry collagen matrices crosslinked by thermal treatment the existence of a moving front separating the polymer from a gel phase was considered, and a convective term induced by the volume expansion was incorporated. The resulting moving boundary problem was solved using a method based on biquadratic finite elements in both space and time that is stable, shows high accuracy, and is suitable for solving problems with a singularity at the initial time point. The model was verified for insoluble collagen matrices at different crosslinking degrees for both chemical and thermal treatment. For constant diffusion coefficients a close form of the solution was derived yielding equivalent results to the numerical approach. © 2002 Wiley-Liss, Inc.}, author = {Radu, Adrian Florin and Bause, Markus and Knabner, Peter and Lee, Geoffrey and Friess, Wolfgang}, doi = {10.1002/jps.10098}, faupublication = {yes}, journal = {Journal of Pharmaceutical Sciences}, keywords = {Collagen matrices; Drug release; Finite element method; Moving boundary; Numerical simulation}, note = {UnivIS-Import:2015-03-09:Pub.2002.nat.dchph.lphte.modeli}, pages = {964-972}, peerreviewed = {Yes}, title = {{Modeling} of {Drug} {Release} {From} {Collagen} {Matrices}}, volume = {91}, year = {2002} } @book{faucris.123797124, abstract = {Das Buch bietet detailliert und verständlich ausgearbeitete Lösungsvorschläge zu den 430 Aufgaben aus dem Buch "Mathematik für Ingenieure und Naturwissenschaftler, Lineare Algebra und Analysis in R". Zahlreiche dieser Lösungsvorschläge werden gesondert besprochen und analysiert, und bei einigen Aufgaben werden verschiedene Lösungswege vorgelegt.

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.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 reserved.},
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},
volume = {56},
year = {2002}
}
@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}},
year = {2016}
}
@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},
year = {2005}
}
@article{faucris.123784364,
abstract = {Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.},
author = {Schüler, Lennart and Suciu, Nicolae and Knabner, Peter and Attinger, Sabine},
doi = {10.1016/j.advwatres.2016.06.012},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {Global random walk; Heterogeneity; Mixing; PDF method; Solute transport; Variance},
pages = {55-67},
peerreviewed = {Yes},
title = {{A} time dependent mixing model to close {PDF} equations for transport in heterogeneous aquifers},
volume = {96},
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 regions.},
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},
venue = {Berlin},
volume = {77},
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 analysis.},
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 = {http://epubs.siam.org/doi/abs/10.1137/S0036142993258191},
volume = {34},
year = {1997}
}
@article{faucris.110912604,
abstract = {We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices, for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices, which ensure that the isoparametric transformation is bijective.},
author = {Knabner, Peter and Summ, Gerhard},
doi = {10.1007/PL00005454},
faupublication = {yes},
journal = {Numerische Mathematik},
pages = {661-681},
peerreviewed = {Yes},
title = {{The} invertibility of the isoparametric mapping for pyramidal and prismatic finite elements},
volume = {88},
year = {2001}
}
@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 = {http://www.mso.math.fau.de/fileadmin/am1/users/knabner/publicationen/ConsVelApproxDensFlow_CompMethWR_96.pdf},
volume = {1},
year = {1996}
}
@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},
volume = {178},
year = {2002}
}
@inproceedings{faucris.111587784,
address = {Erlangen},
author = {Suciu, Nicolae and Vamos, Calin and Knabner, Peter and Rüde, Ulrich},
booktitle = {Simulationstechnique. 18th Symposium in Erlangen. September 2005},
faupublication = {yes},
isbn = {3-936150-41-9},
note = {UnivIS-Import:2015-04-16:Pub.2005.tech.IMMD.lsinfs.biased},
pages = {562-567},
publisher = {SCS Publishing House},
title = {{Biased} {Global} {Random} {Walk}, a {Cellular} {Automaton} for {Diffusion}},
venue = {Erlangen},
year = {2005}
}
@article{faucris.122161424,
author = {Knabner, Peter},
faupublication = {yes},
journal = {The Preprint-Series of the Institute of Applied Mathematics},
peerreviewed = {No},
title = {{Open} {Questions} and {Research} {Directions} in {Parameter} {Identification} for {Multicomponent} {Reactive} {Transport} in {Porous} {Media}},
volume = {315},
year = {2007}
}
@article{faucris.215929949,
abstract = {In this paper, we are dealing with the mathematical modeling and homogenization of nonlinear reaction-diffusion processes in a porous medium that consists of two components separated by an interface. One of the components is connected, and the other one is disconnected and consists of periodically distributed inclusions. At the interface, the fluxes are given by nonlinear functions of the concentrations on both sides of the interface. Thus, the concentrations may be discontinuous across the interface. For the derivation of the effective (homogenized) model, we use the method of two-scale convergence. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we give a new approach which involves the boundary unfolding operator and a compactness result for Banach-space-valued functions. The model is motivated by metabolic and regulatory processes in cells, where biochemical species are exchanged between organelles and cytoplasm through the organellar membranes. In this context the nonlinearities are given by kinetics corresponding to multispecies enzyme catalyzed reactions, which are generalizations of the classical Michaelis-Menten kinetics to multispecies reactions.},
author = {Gahn, Markus and Neuss-Radu, Maria and Knabner, Peter},
doi = {10.1137/15M1018484},
faupublication = {yes},
journal = {SIAM Journal on Applied Mathematics},
keywords = {nonlinear reaction-diffusion equations;nonlinear transmission conditions;homogenization;weak and strong two-scale convergence;kinetics for multisubstrate enzymatic reactions},
month = {Jan},
pages = {1819-1843},
peerreviewed = {Yes},
title = {{HOMOGENIZATION} {OF} {REACTION}-{DIFFUSION} {PROCESSES} {IN} {A} {TWO}-{COMPONENT} {POROUS} {MEDIUM} {WITH} {NONLINEAR} {FLUX} {CONDITIONS} {AT} {THE} {INTERFACE}},
volume = {76},
year = {2016}
}
@inproceedings{faucris.121307164,
author = {Knabner, Peter and Schneid, Eckhard},
booktitle = {Numerical Modelling in Continuum Mechanics, Part II},
faupublication = {yes},
pages = {337–343},
title = {{Numerical} {Solution} of {Unsteady} {Saturated}/{Unsaturated} {Flow} {Through} {Porous} {Media}},
year = {1997}
}
@article{faucris.123745204,
abstract = {Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdif-fusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.},
author = {Suciu, Nicolae and Attinger, Sabine and Radu, Adrian Florin and Vamo̧s, Cǎlin and Vanderborght, Jan and Vereecken, Harry and Knabner, Peter},
doi = {10.1515/auom-2015-0054},
faupublication = {yes},
journal = {Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica},
keywords = {Porous media; Random fields; random walk; transport},
pages = {167-186},
peerreviewed = {unknown},
title = {{Solute} transport in aquifers with evolving scale heterogeneity},
url = {https://www.scopus.com/error.uri},
volume = {23},
year = {2015}
}
@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 literature.},
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}},
venue = {Kopenhagen},
volume = {XVI},
year = {2006}
}
@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 Ltd.},
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},
volume = {35},
year = {2012}
}
@article{faucris.204435300,
abstract = {In this paper, we consider diffusion and reaction of mobile chemical species,

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.

p) the travelling wave may exist. For *p* ≤ 1, the travelling wave always exists, whereas for 1 < *p* ≤ 2 it depends on the values of the other adsorption parameters and whether a lower bound of the upstream concentration (at *x* = −∞) is exceeded. For *p* ≥ 2, the existence of the travelling wave requires that the upstream concentration does not exceed an (specified) upper bound. Besides illustrating some waves we show that two different rate functions that have the Freundlich isotherm as their limit for an infinite rate parameter result in qualitatively different travelling waves.},
author = {Van Duijn, C. J. (Hans) and Knabner, Peter and van der Zee, Sjoerd E. A. T. M.},
doi = {10.1016/0309-1708(93)90001-V},
faupublication = {no},
journal = {Advances in Water Resources},
keywords = {Transport; travelling wave; porous media; Langmuir isotherm; Freundlich isotherm; non-linear adsorption},
pages = {97-105},
peerreviewed = {Yes},
title = {{Travelling} waves during the transport of reactive solute in porous media: {Combination} of {Langmuir} and {Freundlich} isotherms},
url = {http://www.sciencedirect.com/science/article/pii/030917089390001V},
volume = {16},
year = {1993}
}
@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}},
venue = {Singapore},
year = {1998}
}
@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 applied.},
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 = {http://link.springer.com/chapter/10.1007%2F978-3-642-59709-1_12},
year = {2000}
}
@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 literature.},
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 = {http://epubs.siam.org/doi/pdf/10.1137/S0036142996301512},
volume = {35},
year = {1998}
}
@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-Verlag.},
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},
volume = {109},
year = {2008}
}
@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 matrix.},
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},
volume = {12},
year = {2004}
}
@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, Weinheim.},
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},
volume = {93},
year = {2013}
}
@book{faucris.123798224,
abstract = {Das Buch beinhaltet alle Themen einer Mathematikvorlesung, die für Ingenieure in den beiden ersten Semestern an deutschen Universitäten relevant sind: Lineare Algebra und Analysis in einer Raumdimension. Eine detaillierte Darstellung und zahlreiche kreative und teils ausgefallene Beispiele, die an jeder Stelle zu finden sind, zeichnen dieses Buch aus. Zusätzliches Übungsmaterial wird per Video im Internet bereitgestellt. Da die meisten Aussagen bewiesen bzw. mit einer Beweisidee versehen werden, ist es auch für Studierende des Lehramtes und der Mathematik (Bachelor) als hervorragende Ergänzung geeignet.

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.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}, volume = {26}, year = {2001} } @article{faucris.204143727, author = {Aizinger, Vadym and Rupp, Andreas and Schütz, Jochen and Knabner, Peter}, doi = {10.1007/s10596-017-9682-8}, faupublication = {yes}, journal = {Computational Geosciences}, keywords = {Mixed discontinuous Galerkin method; Stability and error analysis; Local discontinuous Galerkin method; Instationary Darcy problem}, pages = {179-194}, peerreviewed = {Yes}, title = {{Analysis} of a mixed discontinuous {Galerkin} method for instationary {Darcy} flow}, volume = {22}, year = {2018} } @incollection{faucris.117981864, abstract = {The underlying physical processes and their specific properties are discussed, yielding a model for the spread of hydrophobic contaminants, which allows for numerical simulation. In the case of convection dominance the contaminant transport is approximated by means of a Lagrange-Galerkin approach. A numerical scheme is presented for transport with equilibrium sorption. Nonlinear sorption isotherms are identified with soil column experiments. The efficiency of the identification algorithm is mainly determined by the choice of an appropiate initial value and the numerical scheme to compute the gradient.}, address = {Berlin, Heidelberg}, author = {Knabner, Peter and Igler, B. A. and Kappmeier, H. and Schneid, Eckhard and Hempfling, Rüdiger}, booktitle = {Mathematik Schlüsseltechnologie für die Zukunft}, doi = {10.1007/978-3-642-60550-5_20}, editor = {Karl-Heinz Hoffmann, Willi Jäger, Thomas Lohmann, Hermann Schunck}, faupublication = {yes}, pages = {231-241}, peerreviewed = {unknown}, publisher = {Springer}, title = {{Trägerbeeinflußter} und lösungsvermittelter {Transport} von {Umweltchemikalien} in porösen {Medien}}, url = {http://link.springer.com/chapter/10.1007%2F978-3-642-60550-5_20}, year = {1997} } @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}}, volume = {5}, year = {2008} } @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 groundwater.}, 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}, volume = {23}, year = {2015} } @incollection{faucris.117818184, abstract = {We consider a two-scale homogenized model for processes in living cells including reaction and diffusion inside and on the surface of organelles. The resulting equations are fully coupled and the number of biochemical species in applications is large. We developed numerical schemes in order to rcomputation time by means of better scaling on parallel machines and reduction of the problemsize. A combination of those schemes provides a suitable numerical framework for our problem.

},
author = {Elbinger, Tobias and Gahn, Markus and Neuß, 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}},
year = {2015}
}
@article{faucris.117795524,
abstract = {In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newton's method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes. © Springer-Verlag 2004.},
author = {Bause, Markus and Knabner, Peter},
doi = {10.1007/s00791-004-0139-y},
faupublication = {yes},
journal = {Computing and Visualization in Science},
pages = {61-78},
peerreviewed = {unknown},
title = {{Numerical} simulation of contaminant biodegradation by higher order methods and adaptive time stepping},
volume = {7},
year = {2004}
}
@inproceedings{faucris.123751584,
author = {Knabner, Peter and Summ, Gerhard},
booktitle = {Scientific Computing in Chemical Engineering II},
editor = {al. FK},
faupublication = {yes},
pages = {110-117},
peerreviewed = {unknown},
title = {{Hybrid} {Mixed} {Discretization} {Methods} for {Combustion} {Problems} in {Porous} {Media}},
url = {http://www.mso.math.fau.de/fileadmin/am1/users/knabner/publicationen/HybMixDiscCombPorMed_SCCE_99.pdf},
year = {1999}
}
@article{faucris.117797504,
author = {Knabner, Peter and Otto, Felix},
doi = {10.1016/S0362-546X(98)00352-6},
faupublication = {yes},
journal = {Nonlinear Analysis - Theory Methods & Applications},
keywords = {Degenerate parabolic systems; Flow and transport in porous media; Stability; Uniqueness},
pages = {381-403},
peerreviewed = {Yes},
title = {{Solute} transport in porous media with equilibrium and nonequilibrium multiple-site adsorption: {Uniqueness} of weak solutions},
volume = {42},
year = {2000}
}
@article{faucris.117793544,
abstract = {We consider colloidal dynamics and single-phase fluid flow within a saturated porous medium in two space dimensions. A new approach in modeling pore clogging and porosity changes on the macroscopic scale is presented. Starting from the pore scale, transport of colloids is modeled by the Nernst-Planck equations. Here, interaction with the porous matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid-liquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy's law and a modified averaged convection-diffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i. e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view. © 2012 Springer Science+Business Media B.V.},
author = {Ray, Nadja and van Noorden, Tycho and Frank, Florian and Knabner, Peter},
doi = {10.1007/s11242-012-0068-z},
faupublication = {yes},
journal = {Transport in Porous Media},
keywords = {Colloidal transport; Evolving microstructure; Level set function; Mixed finite elements; Multiscale coefficients; Particle-surface interaction; Periodic homogenization; Pore clogging; Pore scale modeling; Porosity changes},
pages = {669-696},
peerreviewed = {Yes},
title = {{Multiscale} {Modeling} of {Colloid} and {Fluid} {Dynamics} in {Porous} {Media} {Including} an {Evolving} {Microstructure}},
volume = {95},
year = {2012}
}
@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 Mathematics.},
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},
volume = {52},
year = {2014}
}
@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},
volume = {33},
year = {2017}
}
@article{faucris.217583344,
abstract = {The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore-scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore-scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm-liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE-ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright (c) 2016 John Wiley & Sons, Ltd.},
author = {Schulz, Raphael and Knabner, Peter},
doi = {10.1002/mma.4211},
faupublication = {yes},
journal = {Mathematical Methods in the Applied Sciences},
keywords = {biofilm growth;porous media;fluid-solid interactions;evolving microstructure;formal homogenization;weak solutions},
pages = {2930-2948},
peerreviewed = {Yes},
title = {{Derivation} and analysis of an effective model for biofilm growth in evolving porous media},
volume = {40},
year = {2017}
}
@incollection{faucris.123320824,
abstract = {

An extension of the Lagrange-Galerkin approach is developed for advection-dominated problems with nonlinear adsorption, either being in equilibrium or in non-equilibrium, possibly with isotherms of the Freundlich type. The scheme should be feasable also for the hyperbolic limit case. The basic problems to deal with are the possibility of non-unique characteristics due to the nonlinear isotherms and the incorporation of the time-dependent non-equilibrium adsorption kinetics, avoiding a strong restriction on the CFL-number.

The 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 regarded.

}, 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 = {http://link.springer.com/chapter/10.1007%2F978-3-322-89565-3_4}, volume = {59}, year = {1997} } @incollection{faucris.106884184, abstract = {The reliable prediction of the fate of contaminants in the subsurface is a demanding task for modelers in the environmental sciences. We present a simulation tool that is capable of handling a variety of complex scenarios that are of interest for site remediation or natural attenuation. The modular model components include the Richards equation for (un-) saturated fluid flow which is discretized by locally mass conserving hybrid mixed finite elements. The transport equations contain (nonlinear) terms for equilibrium or kinetic sorption processes and biodegradation reactions. Microbial processes may be described by first order reactions leading to linear reaction networks or a multiplicative dual Monod model including electron donator, electron acceptor and biomass. Simulation of carrier facilitation or surfactant transport (including associated permeability changes and effects on surface tension of the fluid) are feasible as well. The systems of equations are treated with Newton's method and adaptive techniques are applied to improve the efficiency of the implementation. We present an example of coupled surfactant migration and fluid flow. Current and future work includes the application to column experiments to investigate the potential of contaminated sites for natural attenuation and the extension of the model to geochemical multicomponent transport. © 2002 Elsevier B.V. All rights reserved.}, author = {Prechtel, Alexander and Knabner, Peter}, booktitle = {Computational Methods in Water Resources}, doi = {10.1016/S0167-5648(02)80125-3}, editor = {S. Majid Hassanizadeh, Ruud J. Schotting, William G. Gray and George F. Pinder}, faupublication = {yes}, isbn = {978-0-444-50975-8}, pages = {687-694}, peerreviewed = {unknown}, series = {Computational Methods in Water Resources, Proceedings of the XIV International Conference on Computational Methods in Water Resources (CMWR XIV)}, title = {{Accurate} and efficient simulation of coupled water flow and nonlinear reactive transport in the saturated and vadose zone-application to surfactant enhanced and intrinsic bioremediation}, volume = {47}, year = {2002} } @article{faucris.119193844, abstract = {This paper is a prequel to that of Marchand et al. (Comput Geosci 16:691-708, 2012), where an efficient and accurate hybrid-mixed finite element approximation for a system of time-dependent nonlinear conservation equations has been formulated, implemented, and tested, which are general enough to represent most of the existing formulations for two-component liquid-gas flow in porous medium with phase exchange, also allowing for any (dis)appearance of one of the phases. Temperature variation is neglected, but capillary effects are included by extended Darcy's law, and Fickian diffusion is taken into account. The efficiency and stability of the numerical method of Lake (1989) relies on an equivalent reformulation of the otherwise commonly used model in terms of new principal variables and subsequent static (flash) equations allowing more generally for any (dis)appearance of one of the phases without the need of variable switching or unphysical quantities. In particular, the formulation in terms of complementarity conditions allows for an efficient and stable solution by the semismooth Newton's method. © 2013 Springer Science+Business Media Dordrecht.}, author = {Marchand, Estelle and Müller, Torsten and Knabner, Peter}, doi = {10.1007/s10596-013-9341-7}, faupublication = {yes}, journal = {Computational Geosciences}, keywords = {Complementarity problems; Compositional multiphase flow; Nonlinear PDEs}, pages = {431-442}, peerreviewed = {Yes}, title = {{Fully} coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. {Part} {I}: {Formulation} and properties of the mathematical model}, volume = {17}, year = {2013} } @article{faucris.117782324, abstract = {The application of natural attenuation as a site remediation strategy depends essentially on the reliable prediction of the migration of the contaminant plume. We present a one-dimensional simulation tool that is capable of handling a variety of complex scenarios predicted to be of interest in site remediation problems. The implementation of the different components is organized in a modular structure that facilitates arbitrary extensions of the incorporated models and enables the model components to be combined. Efficient, robust numerical techniques (e.g. hybrid mixed finite elements) are embedded in a menu driven, user-friendly environment to serve hydrogeologists or engineers without profound knowledge of the mathematical theory. The software is suitable for Unix workstations as well as inexpensive personal computers. The model components include reactive solute transport (with diffusion, dispersion, advection and sorption) and single- as well as two-phase flow in the saturated and the vadose zone. The underlying models contain non-standard effects that enable the simulation of a large variety of relevant support strategies for natural attenuation. We present examples for the interaction of the transport of surface-active agents and the water flow or the influence of carriers, which can change predicted residence and travel times of strongly sorbing contaminants by several orders of magnitude.}, author = {Schneid, Eckhard and Prechtel, Alexander and Knabner, Peter}, faupublication = {yes}, journal = {Land Contamination and Reclamation}, keywords = {Carrier facilitation; Finite elements; Numerical flow and transport model; Simulation software; Surfactants}, pages = {357-365}, peerreviewed = {unknown}, title = {{A} comprehensive tool for the simulation of complex reactive transport and flow in soils}, volume = {8}, year = {2000} } @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},
volume = {7},
year = {2007}
}
@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 2006.},
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},
volume = {9},
year = {2005}
}
@article{faucris.119202644,
abstract = {For transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models, the one-particle dispersion (i.e., second central moment of the ensemble average concentration for a point source) is a "memory-free" quantity independent of initial conditions. Nonergodic behavior of large initial plumes, as manifest in deviations of actual solute dispersion from one-particle dispersion, is associated with a "memory term" consisting of correlations between initial positions and displacements of solute molecules. Reliable numerical experiments show that increasing the source dimensions has two opposite effects: it reduces the uncertainty related to the randomness of center of mass, but, at the same time, it yields large memory terms. The memory effects increase with the source dimension and depend on its shape and orientation. Large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one-particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion, whereas for large longitudinal sources, the longitudinal dispersion behaves nonergodically, and the transverse dispersion behaves ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, for instance, calibration of model parameters, forecasting, and identification of the contaminant source. Copyright 2008 by the American Geophysical Union.},
author = {Suciu, Nicolae and Vamo̧s, Cǎlin and Vereecken, Harry and Sabelfeld, Karl and Knabner, Peter},
doi = {10.1029/2007WR006740},
faupublication = {yes},
journal = {Water Resources Research},
peerreviewed = {Yes},
title = {{Memory} effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media},
volume = {44},
year = {2008}
}
@article{faucris.204144387,
author = {Ray, Nadja and Rupp, Andreas and Schulz, Raphael and Knabner, Peter},
doi = {10.1007/s11242-018-1099-x},
faupublication = {yes},
journal = {Transport in Porous Media},
keywords = {Upscaling; Tortuosity; Diffusion in porous media},
pages = {803-824},
peerreviewed = {Yes},
title = {{Old} and {New} {Approaches} {Predicting} the {Diffusion} in {Porous} {Media}},
volume = {124},
year = {2018}
}
@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 parameters.},
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://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=80055066862&origin=inward},
volume = {3},
year = {2011}
}
@article{faucris.204143367,
author = {Rupp, Andreas and Knabner, Peter and Dawson, Clint},
doi = {10.1007/s10596-018-9743-7},
faupublication = {yes},
journal = {Computational Geosciences},
keywords = {Local discontinuous Galerkin method; Instationary Darcy problem; Jump condition; Henry’s Law; Different orders of approximation spaces},
pages = {1149-1159},
peerreviewed = {Yes},
title = {{A} local discontinuous {Galerkin} scheme for {Darcy} flow with internal jumps},
volume = {22},
year = {2018}
}
@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.119193184,
abstract = {We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a "global" nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves "local" nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of water-hydrogen flow involving gas phase appearance and disappearance. © 2012 Springer Science+Business Media B.V.},
author = {Marchand, Estelle and Müller, Torsten and Knabner, Peter},
doi = {10.1007/s10596-012-9279-1},
faupublication = {yes},
journal = {Computational Geosciences},
keywords = {Mixed hybrid finite elements; Nonlinear systems of partial differential equations},
pages = {691-708},
peerreviewed = {Yes},
title = {{Fully} coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. {Part} {II}: {Numerical} scheme and numerical results},
volume = {16},
year = {2012}
}
@article{faucris.122615504,
author = {Suciu, Nicolae and Knabner, Peter},
doi = {10.1029/2008WR007498},
faupublication = {yes},
journal = {Water Resources Research},
peerreviewed = {Yes},
title = {{Comment} on "{Spatial} moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution" by {M}. {Massabó}, {A}. {Bellin}, and {A}. {J}. {Valocchi}},
volume = {45},
year = {2009}
}
@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. © 1995.},
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 = {http://www.sciencedirect.com/science/article/pii/0309170895000054},
volume = {18},
year = {1995}
}
@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 molecules.},
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},
volume = {37},
year = {2008}
}
@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},
peerreviewed = {unknown},
title = {{Drug} release from collagen matrices and transport phenomena in porous media including an evolving microstructure},
venue = {Alicante},
year = {2011}
}
@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 algorithm},
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 = {http://link.springer.com/chapter/10.1007%2F978-3-642-55919-8_4},
volume = {21},
year = {2002}
}
@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 results.},
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},
volume = {8},
year = {1997}
}
@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 reserved.},
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},
volume = {147},
year = {2002}
}
@article{faucris.213035441,
abstract = {We present a numerical framework for efficiently simulating partially miscible two-phase flow with multicomponent reactive transport in porous media using the global implicit approach. The mathematical model consists of coupled and nonlinear partial differential equations, ordinary differential equations, and algebraic equations. Our approach is based on a model-preserving reformulation using the reduction scheme of Krautle and Knabner (Water Resour. Res. 43(3), 2007), Hoffmann et al. (Comput. Geosci. 16(4):1081-1099, 2012) to transform the system. Moreover, a nonlinear, implicitly defined resolution function to reduce its size is employed. By choosing persistent primary variables and using a complementarity approach, mineral reactions and the local appearance and disappearance of the gas phase can be handled without a discontinuous switch of primary variables. In each time step of the Euler-implicit time stepping scheme, the discrete nonlinear systems are solved using the Semismooth Newton method for linearization using the global implicit approach. Thus, we obtain an efficient, robust, and stable simulation method allowing for large time steps and avoiding the potential drawbacks of splitting approaches.},
author = {Brunner, Fabian and Knabner, Peter},
doi = {10.1007/s10596-018-9788-7},
faupublication = {yes},
journal = {Computational Geosciences},
note = {CRIS-Team WoS Importer:2019-03-12},
pages = {127-148},
peerreviewed = {Yes},
title = {{A} global implicit solver for miscible reactive multiphase multicomponent flow in porous media},
volume = {23},
year = {2019}
}
@article{faucris.215929141,
abstract = {In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order epsilon and a periodic heterogeneous structure. The equations inside the layer depend on epsilon and the diffusivity inside the layer on an additional parameter gamma is an element of [-1, 1]. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For epsilon -> 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}},
volume = {13},
year = {2018}
}
@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, LLC.},
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},
volume = {50},
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 dynamics.},
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},
volume = {75},
year = {2015}
}
@article{faucris.107777604,
abstract = {This is the first in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source MATLAB/GNU Octave toolbox. The intention of this ongoing project is to provide a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix/vector operations and is carefully documented; in addition, a direct mapping between discretization terms and code routines is maintained throughout. The present work focuses on a two-dimensional time-dependent diffusion equation with space/time-varying coefficients. The spatial discretization is based on the local discontinuous Galerkin formulation. Approximations of orders zero through four based on orthogonal polynomials have been implemented; more spaces of arbitrary type and order can be easily accommodated by the code structure.},
author = {Frank, Florian and Reuter, Balthasar and Aizinger, Vadym and Knabner, Peter},
doi = {10.1016/j.camwa.2015.04.013},
faupublication = {yes},
journal = {Computers & Mathematics With Applications},
keywords = {MATLAB; GNU Octave; Local discontinuous Galerkin method; Vectorization; Open source},
pages = {11 - 46},
peerreviewed = {Yes},
title = {{FESTUNG}: {A} {MATLAB}/{GNU} {Octave} toolbox for the discontinuous {Galerkin} method. {Part} {I}: {Diffusion} operator},
url = {http://www.sciencedirect.com/science/article/pii/S0898122115001820},
volume = {70},
year = {2015}
}
@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-Verlag.},
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},
volume = {116},
year = {2010}
}
@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. 2009.},
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},
volume = {14},
year = {2010}
}
@incollection{faucris.117982084,
abstract = {Contaminants with very low water solubilities (e.g. polycyclic aromatic hydrocarbons) play an important role in risk assessment of dangerous wastes and development of soil remediation. The mobility of such hydrophobic substances can be strongly affected by the existence of carriers (e.g. dissolved organic carbon), which can adsorb the contaminant and thereby enhance or reduce its velocity. The numerical simulation of the spreading of these contaminants, requires the solution of reactive transport equations for all involved components, coupled by the contaminant’s sorption to the carrier. Our development is based on a model [2], in which all the carrier’s influence on the contaminant transport is contained in an effective adsorption isotherm, depending on the carrier concentration and thereby also on space and time. First we shortly summarize the modelling of reactive transport of a single component (carrier, contaminant, carrier bound contaminant) in a porous medium, then in section 3 we combine the two equations for the contaminant components. The properties of the contaminant’s effective isotherm and its influence on the transport equation are discussed in section 4.},
address = {Berlin, Heidelberg},
author = {Knabner, Peter and Schneid, Eckhard},
booktitle = {Scientific Computing in Chemical Engineering},
doi = {10.1007/978-3-642-80149-5_14},
editor = {Frerich Keil, Wolfgang Mackens, Heinrich Voß, Joachim Werther},
faupublication = {yes},
isbn = {978-3-642-80151-8},
pages = {129-135},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Qualitative} {Properties} of a {Model} for {Carrier} {Facilitated} {Groundwater} {Contaminant} {Transport}},
url = {http://link.springer.com/chapter/10.1007%2F978-3-642-80149-5_14},
year = {1996}
}
@article{faucris.123796244,
abstract = {Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random species concentrations weighted by conserved scalars are formulated as Fokker-Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Itô equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The general FDF approach and the GRW numerical solution are illustrated for a reduced complexity problem consisting of the transport of a single scalar in groundwater. Randomness is induced by the stochastic parameterization of the hydraulic conductivity, characterized by short range correlations and small variance. The objective is to infer the statistics of the random concentration sampled at the plume center of mass, integrated over the transverse dimension of a two-dimensional spatial domain. The PDF/FDF problem can therefore be formulated in a two-dimensional domain as well, a spatial dimension and one in the concentration space. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single-trajectories of diffusion in velocity fields with short-range correlations, owing to their self-averaging property. The mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from the statistical analysis of an ensemble of simulated concentration time series, as well as by classical mixing models. A Gaussian spatial filter applied to a Kraichnan velocity field generator is used to construct coarse-grained simulations (CGS) for FDF problems. The purposes of the CGS simulations are two-fold: first to understand the significance of the FDF approach from a practical point of view and its relation to the PDF approach; second to investigate the limits of the mixing models considered here and the desirable features of the mixing models for groundwater systems.},
author = {Suciu, Nicolae and Schüler, Lennart and Attinger, Sabine and Knabner, Peter},
doi = {10.1016/j.advwatres.2016.02.016},
faupublication = {yes},
journal = {Advances in Water Resources},
keywords = {Global random walk; Groundwater; Mixing; PDF/FDF methods},
pages = {83-98},
peerreviewed = {Yes},
title = {{Towards} a filtered density function approach for reactive transport in groundwater},
volume = {90},
year = {2016}
}
@article{faucris.117782764,
abstract = {Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Péclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM-GRW is tens of times faster than the full MFEM flow and transport simulation. © 2012.},
author = {Suciu, Nicolae and Radu, Adrian Florin and Prechtel, Alexander and Brunner, Fabian and Knabner, Peter},
doi = {10.1016/j.cam.2012.06.027},
faupublication = {yes},
journal = {Journal of Computational and Applied Mathematics},
keywords = {Advection-dominated transport; Porous media; Random walk methods},
pages = {27-37},
peerreviewed = {Yes},
title = {{A} coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity},
volume = {246},
year = {2013}
}
@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.117791344,
abstract = {We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimerlaw while that in the surrounding matrix is governed by Darcy's law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimerlaw is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy's law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimerlaw in the entire domain when the Forchheimer coefficient in the matrix tends toward zero. © 2014 EDP Sciences, SMAI.},
author = {Knabner, Peter and Roberts, Jean E.},
doi = {10.1051/m2an/2014003},
faupublication = {yes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Darcy-Forchheimerflow; Flow in porous media; Fractures; Monotone operators; Regularization; Solvability},
pages = {1451-1472},
peerreviewed = {unknown},
title = {{Mathematical} analysis of a discrete fracture model coupling {Darcy} flow in the matrix with {Darcy}-{Forchheimer} flow in the fracture},
volume = {48},
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 convergence.},
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}},
year = {2006}
}
@article{faucris.123859824,
author = {Van Duijn, C. J. (Hans) and Knabner, Peter},
faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
pages = {329-332},
peerreviewed = {Yes},
title = {{Crystal} dissolution in porous media flow},
url = {https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=33749455829&origin=inward},
volume = {76},
year = {1996}
}
@article{faucris.108689504,
abstract = {This paper presents methods for the time-dependent numerical solution of a two-dimensional model governing combustion an inert porous media. The spatial discretization is based on a mixed finite element method for the flow problem and a cell-centered finite volume scheme for the transport equations. Some results of numerical simulations show the influence of a jump in the porosity of the solid matrix on the localization of the combustion zone.},
author = {Knabner, Peter and De Neef, Michel J. and Summ, Gerhard},
doi = {10.1002/zamm.19990791312},
faupublication = {yes},
journal = {ZAMM - Zeitschrift für angewandte Mathematik und Mechanik},
pages = {45-48},
peerreviewed = {Yes},
title = {{Transient} {Numerical} {Simulation} of {Combustion} in {Inert} {Porous} {Media}},
volume = {79},
year = {1999}
}
@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}},
volume = {54},
year = {2016}
}
@article{faucris.119195164,
author = {Igler, B. A. and Totsche, Kai Uwe and Knabner, Peter},
doi = {10.1016/S0079-1946(98)00016-0},
faupublication = {yes},
journal = {Physics and Chemistry of the Earth},
pages = {215-219},
peerreviewed = {unknown},
title = {{Identification} of {Nonlinear} {Sorption} {Isotherms} by {Soil} {Column} {Breakthrough} {Experiments}},
url = {http://www.sciencedirect.com/science/article/pii/S0079194698000160},
volume = {23},
year = {1998}
}
@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 Union.},
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},
volume = {41},
year = {2005}
}
@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 tool.},
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 = {http://link.springer.com/chapter/10.1007%2F978-3-642-55753-8_12},
year = {2003}
}
@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}
}
@article{faucris.123853664,
abstract = {A drug delivery system, named minirod, containing insoluble non-cross-linked collagen was prepared to investigate the release of model drug compounds. To characterise the complete drug release process properly, a mathematical model was developed. Previously, a mathematical model describing water penetration, matrix swelling and drug release by diffusion from dense collagen matrices has been introduced and tested. However, enzymatic matrix degradation influences the drug release as well. Based on experimental data, a model was developed which describes drug release by collagenolytic matrix degradation based on enzyme diffusion, adsorption and cleavage. Data for swelling, collagen degradation and FITC dextran release from insoluble equine collagen type I minirods were collected. Sorption studies demonstrated a tight sorption of collagenase on collagen surfaces that follows a Freundlich sorption isotherm and results in a degradation constant of 3.8 × 10 mol/l for the minirods. The diffusion coefficients of FITC dextran 20 and 70 (3 × 10 and 2.4 × 10 cm/h) in water were analyzed by fluorescence correlation spectroscopy (FCS). Using these data, the mathematical model was verified by two-dimensional simulations. The numerical results agreed well with the measurements. © 2007 Elsevier B.V. All rights reserved.},
author = {Metzmacher, Iris and Radu, Adrian Florin and Bause, Markus and Knabner, Peter and Friess, Wolfgang},
doi = {10.1016/j.ejpb.2007.02.013},
faupublication = {yes},
journal = {European Journal of Pharmaceutics and Biopharmaceutics},
keywords = {Collagen; Enzymatic adsorption; Enzymatic degradation; FCS; Mathematical model; Numerical simulation},
pages = {349-360},
peerreviewed = {Yes},
title = {{A} model describing the effect of enzymatic degradation on drug release from collagen minirods},
volume = {67},
year = {2007}
}
@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}},
volume = {10},
year = {2017}
}
@article{faucris.123748284,
abstract = {Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdif-fusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.},
author = {Suciu, Nicolae and Attinger, Sabine and Radu, Adrian Florin and Vamo̧s, Cǎlin and Vanderborght, Jan and Vereecken, Harry and Knabner, Peter},
faupublication = {yes},
journal = {The Preprint-Series of the Institute of Applied Mathematics},
keywords = {Porous media; Random fields; Random walk; Transport},
pages = {1-18},
peerreviewed = {No},
title = {{Solute} transport in aquifers with evolving scale heterogeneity},
volume = {346},
year = {2011}
}
@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},
volume = {51},
year = {2017}
}
@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 simulation.},
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 = {http://link.springer.com/chapter/10.1007%2F978-3-642-60155-2_4},
volume = {8},
year = {1999}
}
@article{faucris.106905524,
abstract = {In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one strong solution is shown by applying Schauder's fixed point theorem. Additionally, non-negativity, uniqueness, and global existence or existence up to possible closure of some pores, i.e. up to the limit of degenerating coefficients, is guarantee},
author = {Schulz, Raphael and Ray, Nadja and Frank, Florian and Mahato, Harishankar and Knabner, Peter},
doi = {10.1017/S0956792516000164},
faupublication = {yes},
journal = {European Journal of Applied Mathematics},
keywords = {effective constitutive equations; evolving microstructure; finite element methods; periodic homogenization; strong solutions},
pages = {1-29},
peerreviewed = {Yes},
title = {{Strong} solvability up to clogging of an effective diffusion-precipitation model in an evolving porous medium},
year = {2016}
}
@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) 2008.},
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},
volume = {12},
year = {2009}
}
@article{faucris.123796904,
author = {Knabner, Peter and Tapp, Christoph and Thiele, Kathrin},
faupublication = {yes},
journal = {Acta Mathematica Universitatis Comenianae},
pages = {115-136},
peerreviewed = {unknown},
title = {{Adaptive} {Finite} {Volume} {Discretization} of {Density} {Driven} {Flows} in {Porous} {Media}},
url = {http://www.mso.math.fau.de/fileadmin/am1/users/knabner/publicationen/FinVolDiscDensFlows_AMUniCom_97.pdf},
volume = {67},
year = {1998}
}
@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},
volume = {77},
year = {2019}
}
@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 results.},
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},
volume = {39},
year = {2002}
}
@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 analysis.},
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 = {http://epubs.siam.org/doi/abs/10.1137/S0036142993249024},
volume = {34},
year = {1997}
}
@incollection{faucris.123746744,
abstract = {We propose and study the numerical approximation of an advection-diffusion-reaction model equation by a modified Brezzi–Douglas–Marini mixed finite element method.Nonlinear advection is admitted, arising in complex and coupled flow and transport systems.In contrast to the classical variant of this approach, optimal second-order convergence of the scalar and the vector variable is ensured.No loss of rate of convergence due to the presence of the advection term is observed.},
address = {Berlin Heidelberg},
author = {Bause, Markus and Brunner, Fabian and Knabner, Peter and Radu, Adrian Florin},
booktitle = {Numerical Mathematics and Advanced Applications 2011},
doi = {10.1007/978-3-642-33134-3_27},
editor = {Andrea Cangiani, Ruslan L. Davidchack, Emmanuil Georgoulis, Alexander N. Gorban, Jeremy Levesley, Michael V. Tretyakov},
faupublication = {yes},
isbn = {978-3-642-33133-6},
pages = {247-255},
peerreviewed = {unknown},
publisher = {Springer},
title = {{An} {Improved} {Optimal} {Order} {Mixed} {Finite} {Element} {Method} for {Semilinear} {Transport} {Problems}},
url = {http://link.springer.com/chapter/10.1007%2F978-3-642-33134-3_27},
year = {2012}
}
@article{faucris.217583073,
abstract = {The article concerns the growth of biofilms made by chemotactical bacteria within a saturated porous media. The increase of a biomass on the surface of the solid matrix changes the porosity and impedes the flow through the pores. By formal periodic homogenization we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients given by the evolving microstructure at the pore-scale. Based on the assumption of uniform evolve of the underlying pore geometry and slight self-diffusivity of the bacteria, solvability in a weak sense global in time or at least up to a possible clogging phenomenon is shown in the two-dimensional case. Furthermore,},
author = {Schulz, Raphael and Knabner, Peter},
doi = {10.1137/16M108817X},
faupublication = {yes},
journal = {SIAM Journal on Applied Mathematics},
keywords = {biofilm growth;chemotaxis;porous media;fluid-solid interactions;evolving microstructure;parabolic PDE},
month = {Jan},
pages = {1653-1677},
peerreviewed = {Yes},
title = {{AN} {EFFECTIVE} {MODEL} {FOR} {BIOFILM} {GROWTH} {MADE} {BY} {CHEMOTACTICAL} {BACTERIA} {IN} {EVOLVING} {POROUS} {MEDIA}},
volume = {77},
year = {2017}
}
@article{faucris.119215844,
abstract = {We perform the periodic homogenization (i.e ε → 0) of a non-stationary Stokes-Nernst-Planck-Poisson system using two-scale convergence, where ε is a suitable scale parameter. The objective is to investigate the influence of different boundary conditions and variable choices of scalings in ε 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 electrostatic drift term. As a direct result of the homogenization procedure, various classes of upscaled model equations are obtained. © 2012 Elsevier Inc.},
author = {Ray, Nadja and Muntean, Adrian and Knabner, Peter},
doi = {10.1016/j.jmaa.2012.01.052},
faupublication = {yes},
journal = {Journal of Mathematical Analysis and Applications},
keywords = {Colloidal transport; Homogenization; Porous media; Stokes-Nernst-Planck-Poisson system; Two-scale convergence},
pages = {374-393},
peerreviewed = {Yes},
title = {{Rigorous} homogenization of a {Stokes}-{Nernst}-{Planck}-{Poisson} system},
volume = {390},
year = {2012}
}
@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 microstructure.},
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}},
year = {2002}
}
@article{faucris.123847284,
abstract = {[1] In this article a systematic approach for the efficient computation of the transport and reaction of a multispecies, multireaction system is proposed. The objective of this approach is to reformulate the given system of differential or differential-algebraic equations in such a way that the couplings and the nonlinearities are concentrated in a reduced number of equations (if compared to the original formulation), while some linear equations decouple from the system. The resulting system is handled in the spirit of a global implicit approach ("one step method") avoiding operator splitting techniques. The reduction of the problem size proposed in this article helps to limit the large computational costs of numerical simulations for such problems. The reduction mechanism is a generalization of the method proposed in a previous paper. Now, problems with mixed mobile/immobile species, homogeneous/heterogeneous kinetic/equilibrium reactions are considered, while the previous publication was restricted to problems without heterogeneous equilibrium reactions (such as equilibrium sorption). An application of the reduction mechanism to an example problem is given in order to investigate the reduction of the number of coupled nonlinear equations and to compare it to other methods. Copyright 2007 by the American Geophysical Union.},
author = {Kräutle, Serge and Knabner, Peter},
doi = {10.1029/2005WR004465},
faupublication = {yes},
journal = {Water Resources Research},
peerreviewed = {Yes},
title = {{A} reduction scheme for coupled multicomponent transport-reaction problems in porous media: {Generalization} to problems with heterogeneous equilibrium reactions},
volume = {43},
year = {2007}
}
@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. 2010.},
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},
volume = {14},
year = {2010}
}
@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},
volume = {17},
year = {2013}
}
@incollection{faucris.118046324,
address = {Dresden},
author = {Prechtel, Alexander and Hoffmann, Joachim and Kräutle, Serge and Knabner, Peter},
booktitle = {Modellierung und Prognose von Natural Attenuation-Prozessen im Untergrund},
faupublication = {yes},
pages = {75-90},
peerreviewed = {unknown},
series = {Gemeinsame Mitteilungen des Dresdner Grundwasserforschungszentrum e.V. und seiner Partner},
title = {{Reaktive} {Mehrkomponentenprobleme}: {Sicherung} von {Effizienz} und {Zuverlässigkeit}},
volume = {3},
year = {2006}
}
@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 Mathematics.},
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},
volume = {47},
year = {2010}
}
@incollection{faucris.107107924,
abstract = {"Fluid flow through an unsaturated soil is described by a model equation which contains the hydraulic properties of the considered medium in the form of nonlinear coefficient functions. A common method to determine the hydraulic properties is based on the formulation and solution of an inverse problem for soil column outflow experiments. Here, a formfree approach and a numerical least squares technique in combination with a stability analysis of the inverse problem is used for the determination of the nonlinearities. In accordance with Tikhonov regularization techniques, penalty terms, which can be added to the least squares functional, are derived.''},
author = {Bitterlich, Sandro and Knabner, Peter},
booktitle = {Fluid flow and transport in porous media: mathematical and numerical treatment},
doi = {10.1090/conm/295},
editor = {Zhangxin Chen, Richard E. Ewing},
faupublication = {yes},
pages = {63-74},
peerreviewed = {unknown},
publisher = {American Mathematical Society},
series = {Contemporary Mathematics},
title = {{Adaptive} and formfree identification of nonlinearities in fluid flow from column experiments},
volume = {295},
year = {2002}
}
@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.