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

We consider the e�cient 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 e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach

Fast Solvers for Cahn-Hilliard Inpainting

We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential

Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems

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

Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations

We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete vector-valued Cahn-Hilliard equations with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems as 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

Computational methods for Cahn-Hilliard variational inequalities

We consider the non-standard fourth order parabolic Cahn-Hilliard variational inequality with constant as well as non-constant diffusional mobility. We propose a primal-dual active set method as solution technique for the discrete variational inequality given by a (semi-)implicit Euler discretization in time and linear finite elements in space. We show local convergence of the method by reinterpretation as a semi-smooth Newton method. The discrete saddle point system arising in each iteration step is handled by either a Gauss-Seidel type method, the application of a multi-frontal direct solver or a preconditioned conjugate gradient method applied to the Schur complement. Finally we show the efficiency of the method and the preconditioning with several numerical simulations

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach