An adaptive Newton multigrid method for a model of marine ice sheets - Jouvet, Guillaume; Gräser, Carsten (Journal of Computational Physics, 2013)
In this paper, we consider a model for the time evolution of three-dimensional marine ice sheets. This model combines the Shallow Ice Approximation (SIA) for the ice deformation, the Shallow Shelf Approximation (SSA) for the basal sliding, and the mass conservation principle. At each time step, we solve a scalar p-Laplace minimization-type problem with obstacle (SIA), a vectorial p-Laplace minimization-type problem (SSA) and a transport equation (mass conservation). The two minimization problems are solved using a truncated nonsmooth Newton multigrid method while the transport equation is solved using a vertex-centred finite volume method. Our approach is combined to an heuristic mesh adaptive refinement procedure to face the large gradients of the solution that are expected between the ice sheet and the ice shelf. As applications, we present some simulations of the Marine Ice Sheet Model Intercomparison Project MISMIP (2D and 3D) and validate our results against an analytic solution (2D) and other participant model results (3D). Further numerical results show that the convergence of our Newton multigrid method is insensitive to local refinements making our overall adaptive strategy fully efficient. © 2013 Elsevier Inc.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Adaptive method; Grounding line; Marine ice sheets; Newton multigrid method; P-Laplace

An evolving space framework for Oseen equations on a moving domain - Djurdjevac, Ana; Gräser, Carsten; Herbert, Philip J. (Mathematical Modelling and Numerical Analysis, 2023)
This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.
CRIS-Team Scopus Importer:2023-11-24
Keywords: Evolving domain; Finite difference; Oseen equation

A Variational Approach to Particles in Lipid Membranes - Elliott, Charles M.; Gräser, Carsten; Hobbs, Graham; Kornhuber, Ralf; Wolf, Maren-Wanda (Archive for Rational Mechanics and Analysis, 2016)
A variety of models for the membrane-mediated interaction of particles in lipid membranes, mostly well-established in theoretical physics, is reviewed from a mathematical perspective. We provide mathematically consistent formulations in a variational framework, relate apparently different modelling approaches in terms of successive approximation, and investigate existence and uniqueness. Numerical computations illustrate that the new variational formulations are directly accessible to effective numerical methods.
CRIS-Team Scopus Importer:2022-03-23

Discretization error estimates for penalty formulations of a linearized Canham-Helfrich-type energy - Gräser, Carsten; Kies, Tobias (IMA Journal of Numerical Analysis, 2019)
This article is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of biomembranes with embedded particles. There, the curve conditions result from the imposed particle-membrane coupling. We prove almost-H 5/2 regularity of the solution and then consider two possible penalty formulations. For the combination of these penalty formulations with a Bogner-Fox-Schmit finite element discretization, we prove discretization error estimates that are optimal in view of the solution's reduced regularity. The error estimates are based on a general estimate for linear penalty problems in Hilbert spaces. Finally, we illustrate the theoretical results by numerical computations. An important feature of the presented discretization is that it does not require the particle boundary to be resolved. This is crucial to avoid re-meshing if the presented problem arises as a subproblem in a model where particles are allowed to move or rotate.
CRIS-Team Scopus Importer:2022-03-23
Keywords: biomembrane model; Bogner-Fox-Schmit finite element; discretization error estimate; penalty method

Hierarchical error estimates for the energy functional in obstacle problems - Zou, Qingsong; Veeser, Andreas; Kornhuber, Ralf; Gräser, Carsten (Numerische Mathematik, 2011)
We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the error in the energy minimum is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. We even show that small oscillation implies a related saturation assumption. In addition, we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators. Finally, we illustrate our theoretical results by numerical computations. © 2011 Springer-Verlag.
CRIS-Team Scopus Importer:2022-03-23

Nonsmooth Newton methods for set-valued saddle point problems - Gräser, Carsten; Kornhuber, Ralf (SIAM Journal on Numerical Analysis, 2009)
We present a new class of iterative schemes fo r large scale set-valued saddle point problems as arising, e.g., from optimization problems in the presence of linear and inequality constraints. Our algorithms can be regarded either as nonsmooth Newton-type methods for the nonlinear Schur complement or as Uzawa-type iterations with active set preconditioners. Numerical experiments with a control constrained optimal control problem and a discretized Cahn-Hilliard equation with obstacle potential illustrate the reliability and efficiency of the new approach. © 2009 Societ y for Industrial and Applied Mathematics.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Active set preconditioners; Nonsmooth Newton methods; Set-valued saddle point problems; Uzawa algorithms

Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems - Gräser, Carsten; Kornhuber, Ralf; Sack, Uli (IMA Journal of Numerical Analysis, 2015)
We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete multicomponent Cahn-Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments.
CRIS-Team Scopus Importer:2022-03-23
Keywords: convex minimization; descent methods; finite elements; multigrid methods; phase field models; variational inequalities

Numerical Approximation of Multi-Phase Penrose-Fife Systems - Gräser, Carsten; Kahnt, Max; Kornhuber, Ralf (Computational Methods in Applied Mathematics, 2016)
We consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur-Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Finite Elements; Multi-Phase Field Models; Parabolic Variational Inequality; Partial Differential Inclusion; Phase Transitions; Thermodynamic Consistency

Numerical simulation of coarsening in binary solder alloys - Gräser, Carsten; Kornhuber, Ralf; Sack, Uli (Computational Materials Science, 2014)
Coarsening in solder alloys is a widely accepted indicator for possible failure of joints in electronic devices. Based on the well-established Cahn-Larché model with logarithmic chemical energy density (Dreyer and Müller, 2001) [20], we present a computational framework for the efficient and reliable simulation of coarsening in binary alloys. Main features are adaptive mesh refinement based on hierarchical error estimates, fast and reliable algebraic solution by multigrid and Schur-Newton multigrid methods, and the quantification of the coarsening speed by the temporal growth of mean phase radii. We provide a detailed description and a numerical assessment of the algorithm and its different components, together with a practical application to a eutectic AgCu brazing alloy. © 2014 Elsevier B.V. All rights reserved.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Adaptive finite elements; Cahn-Larché system; Phase separation; Schur-Newton multigrid

Numerical simulation of multiscale fault systems with rate- and state-dependent friction - Gräser, Carsten; Kornhuber, Ralf; Podlesny, Joscha (Computational Geosciences, 2023)
We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich- and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of nonsmooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated nonsmooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.
CRIS-Team Scopus Importer:2023-11-10
Keywords: Mortar methods; Multibody coupling; Nonsmooth multigrid; Rate- and state-dependend friction

On differentiability of the membrane-mediated mechanical interaction energy of discrete continuum membrane particle models - Gräser, Carsten; Kies, Tobias (Interfaces and Free Boundaries, 2021)
We consider a discrete continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membranemediated mechanical particle particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore, we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Biomembranes; Membrane particle systems; Topology optimization

Polyhedral Gauß-Seidel converges - Gräser, Carsten; Sander, O. (Journal of Numerical Mathematics, 2014)
We prove global convergence of an inexact extended polyhedral Gauß-Seidel method for the minimization of strictly convex functionals that are continuously differentiable on each polyhedron of a polyhedral decomposition of their domains of definition. While pure Gauß-Seidel methods are known to be very slow for problems governed by partial differential equations, the presented convergence result also covers multilevel methods that extend the Gauß-Seidel step by coarse level corrections. Our result generalizes the proof of [10] for differentiable functionals on the Gibbs simplex. Example applications are given that require the generality of our approach.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Allen-Cahn equation; discontinuous Galerkin methods; global convergence; multigrid methods; polyhedral Gauß-Seidel

Review article multigrid methods for obstacle problems - Gräser, Carsten; Kornhuber, Ralf (Journal of Computational Mathematics, 2009)
In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Multigrid methods; Variational inequalities

The DUNE framework: Basic concepts and recent developments - Bastian, Peter; Blatt, Markus; Dedner, Andreas; Dreier, Nils-Arne; Engwer, Christian; Fritze, Rene; Gräser, Carsten; Grueninger, Christoph; Kempf, Dominic; Kloefkorn, Robert; Ohlberger, Mario; Sander, Oliver (Computers & Mathematics with Applications, 2021)
This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first DUNE version in 2007 and the main papers describing that state Bastian etal. (2008a, 2008b). This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.
CRIS-Team Scopus Importer:2022-03-23

The dune-subgrid module and some applications - Gräser, Carsten; Sander, Oliver (Computing, 2009)
We present an extension module for the Dune system. This module, called dune-subgrid, allows to mark elements of another Dune hierarchical grid. The set of marked elements can then be accessed as a Dune grid in its own right. dune-subgrid is free software and is available for download (External Dune Modules: www.dune-project.org/downloadext.html ). We describe the functionality and use of dune-subgrid, comment on its implementation, and give two example applications. First, we show how dune-subgrid can be used for micro-FE simulations of trabecular bone. Then we present an algorithm that allows to use exact residuals for the adaptive solution of the spatial problems of time-discretized evolution equations. © 2009 Springer-Verlag.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Dune; Finite elements; Grid interface; Meta grid; Subgrid

Time discretizations of anisotropic Allen-Cahn equations - Gräser, Carsten; Kornhuber, Ralf; Sack, Uli (IMA Journal of Numerical Analysis, 2013)
We consider anisotropic Allen-Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the double-well potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer invariant under rotation of the initial configuration for a fixed orientation of the anisotropy. © The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
CRIS-Team Scopus Importer:2022-03-23
Keywords: multigrid; phase transitions; stability; time discretization

Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis - Gräser, Carsten; Kienle, Daniel; Sander, Oliver (Computational Mechanics, 2023)
We propose the truncated nonsmooth Newton multigrid method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler-Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.
CRIS-Team WoS Importer:2023-06-02

Truncated nonsmooth Newton multigrid methods for block-separable minimization problems - Gräser, Carsten; Sander, Oliver (IMA Journal of Numerical Analysis, 2019)
The Truncated Nonsmooth Newton Multigrid method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere.
CRIS-Team Scopus Importer:2022-03-23
Keywords: Block separable; Convex minimization; Global convergence; Multigrid

Truncated nonsmooth newton multigrid methods for convex minimization problems - Gräser, Carsten; Sack, Uli; Sander, Oliver (Lecture Notes in Computational Science and Engineering, 2009)
We present a new inexact nonsmooth Newton method for the solution of convex minimization problems with piecewise smooth, pointwise nonlinearities. The algorithm consists of a nonlinear smoothing step on the fine level and a linear coarse correction. Suitable postprocessing guarantees global convergence even in the case of a single multigrid step for each linear subproblem. Numerical examples show that the overall efficiency is comparable to multigrid for similar linear problems. © 2009 Springer-Verlag Berlin Heidelberg.
CRIS-Team Scopus Importer:2022-03-23

Well-posedness of an evaporation model for a spherical droplet exposed to an air flow - Bänsch, Eberhard; Doß, Martin; Gräser, Carsten; Ray, Nadja (Interfaces and Free Boundaries, 2025)
In this paper, we address the well-posedness of an evaporation model for a spherical liquid droplet taking into account the convective impact of an air flow in the ambient gas phase. From a mathematical perspective, we are dealing with a coupled ODE–PDE system for the droplet radius, the temperature distribution, and the vapor concentration. The nonlinear coupling arises from the evaporation rate modeled by the Hertz–Knudsen equation. Under physically meaningful assumptions, we prove existence and uniqueness of a weak solution until the droplet has evaporated completely. Numerical simulations are performed to illustrate how different air flows affect the evaporation process.
CRIS-Team Scopus Importer:2025-03-14
Keywords: coupled ODE–PDE system; direct numerical simulation; Hertz–Knudsen equation; nonlinear boundary condition; single droplet evaporation