# Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries

## TL;DR

This paper develops an energy stable numerical scheme for Allen-Cahn type models of phase boundary motion, ensuring energy decay at discrete levels and demonstrating effectiveness through theoretical proofs and numerical tests.

## Contribution

It introduces a conforming Galerkin spatial discretization and an implicit time-stepping scheme that guarantee energy decay and well-posedness for Allen-Cahn type problems.

## Key findings

- Proven energy decay for semi-discrete and fully discrete schemes
- Numerical tests confirm theoretical energy decay and interface behavior
- Scheme effectively captures asymptotic interface velocities

## Abstract

We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al.