Stability Analysis and Best Approximation Error Estimates of Discontinuous Time-Stepping Schemes for the Allen-Cahn Equation

Abstract

Fully-discrete approximations of the Allen-Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove best approximation a-priori error estimates, with constants depending polynomially ypon $(1/\epsilon )$ by circumventing Grönwall Lemma arguments. We also prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. The key feature of our approach is an appropriate duality argument, combined with a boot-strap technique.