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

Abstract

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

A Relaxation Approach to Vector-Valued Allen–Cahn MPEC Problems

“…The rest of the paper is organized as follows. In section 2 we analyze the vector-valued Allen-Cahn inequality as the lower level problem; the existence of a solution to the inequality is proven by a penalization technique, see for similar results in [4]. Furthermore the complementarity formulation for the Allen-Cahn inequality is given.…”

“…Following [4] the problem (ACVI) can be reformulated with the help of the slack variable (Lagrange multiplier of the lower level problem) ξ corresponding to the inequality constraint y ≥ 0, which results in the following complementarity-problem (CCP):…”

“…We will give here a sketch of the main steps of the proof. For detailed calculations we refer to a similar proof in [4]. 1.…”

“…For every δ ∈ (0, 1] one can show the unique solvability of (2.3) by classical theory of parabolic partial differential equations and then pass to the limit. Following [4] we reformulate (2.3) by using Ψ δ y (y δ ) = 1 δΦ (y δ ) − y δ and defining ξ δ := − 1 δΦ (y δ ). Hence, we have…”

“…Furthermore, the convex and closed subset V hN Σ is weakly closed and we obtain that y ∈ V hN Σ . For proving y ≥ 0 we refer the reader to [4,9]. We get moreover as δ 0…”