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We study the systematic numerical approximation of a class of Allen-Cahn typeproblems modeling the motion of phase interfaces. The common feature of thesemodels is an underlying gradient flow structure which gives rise to a decay ofan associated energy functional along solution trajectories. We first study thediscretization in space by a conforming Galerkin approximation of a variationalprinciple which characterizes smooth solutions of the problem. Well-posednessof the resulting semi-discretization is established and the energy decay alongdiscrete solution trajectories is proven. A problem adapted implicittime-stepping scheme is then proposed and we establish its well-posed and decayof the free energy for the fully discrete scheme. Some details about thenumerical realization by finite elements are discussed, in particular theiterative solution of the nonlinear problems arising in every time-step. Thetheoretical results are illustrated by numerical tests which also providefurther evidence for asymptotic expansions of the interface velocities derivedby Alber et al.

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