Title: Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries,
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.