Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

Authors:
Jessica Bosch;Martin Stoll;Peter Benner
Affiliations:
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany;Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany;Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany and Mathematics in Industry and Technol ...
Venue:
Journal of Computational Physics
Year:
2014

Citing 15
Cited 0

GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

SIAM Journal on Scientific and Statistical Computing
BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems

SIAM Journal on Scientific and Statistical Computing
Monotone multigrid methods for elliptic variational inequalities I

Numerische Mathematik
Finite Element Approximation of the Cahn--Hilliard Equation with Degenerate Mobility

SIAM Journal on Numerical Analysis
A Note on Preconditioning for Indefinite Linear Systems

SIAM Journal on Scientific Computing
Semismooth Newton Methods for Operator Equations in Function Spaces

SIAM Journal on Optimization
The Primal-Dual Active Set Strategy as a Semismooth Newton Method

SIAM Journal on Optimization
Iterative Methods for Sparse Linear Systems

Iterative Methods for Sparse Linear Systems
Conservative multigrid methods for Cahn-Hilliard fluids

Journal of Computational Physics
A multigrid finite element solver for the Cahn-Hilliard equation

Journal of Computational Physics
An Introduction to Algebraic Multigrid

Computing in Science and Engineering
deal.II—A general-purpose object-oriented finite element library

ACM Transactions on Mathematical Software (TOMS)
An adaptive finite-element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem

Optimization Methods & Software - Advances in Shape and Topology Optimization: Theory, Numerics and New Applications Areas
Preconditioning for Allen-Cahn variational inequalities with non-local constraints

Journal of Computational Physics
Inpainting of Binary Images Using the Cahn–Hilliard Equation

IEEE Transactions on Image Processing

Quantified Score

Hi-index	31.45

Visualization

Abstract

We consider the efficient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an effective Schur complement approximation. Numerical results illustrate the competitiveness of this approach.