Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Abstract

Abstract. The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.Mathematics Subject Classification. 35K55, 35K85, 90C33, 49N90, 80A22, 82C26, 65M60.

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

“…Otherwise (2.1) may not be solvable. Furthermore we have shown in [3] using the equivalence of the PDAS-algorithm to a semismooth Newton method:…”

“…However, the appropriate scaling of the Lagrange multiplier µ by 1 ε , or respectively the choice of the parameter c is essential to avoid oscillatory behaviour due to bilateral constraints (see [3,4]). …”

“…denotes the L 2 -inner product. The mass-conserving H −1 -inner product yields in the obstacle case the fourth order Cahn-Hilliard variational inequality: together with |u| ≤ 1 a.e.. For these formulations it can be shown that under Neumann boundary conditions for w and initial conditions for u a unique solution (u, w) of (1.5)-(1.6) exists where u is H 2 -regular in space, see [7,3]. Using the H 2 -regularity the formulation (1.4) and (1.5), (1.6) can be restated in the complementary form 10) and homogeneous Neumann boundary conditions for u and w together with an initial phase distribution u(0) = u 0 .…”

“…We also remark that the non-local mean value constraints appear since the Cahn-Hilliard evolution variational inequality conserves mass. Identifying the Lagrange multiplier w for (1.16) with v up to a constant we obtain the reduced KKT-system (1.17)-(1.19), see [3] for details:…”

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

“…Then the multipliers may still exist but are only measures. This effect is also known for obstacle problems, see [29], and is discussed in more detail for Cahn-Hilliard problems in [8]. Therefore, the pointwise definition of the active sets A k i is not possible.…”