STABILITY ANALYSIS AND BEST APPROXIMATION ERROR ESTIMATES OF DISCONTINUOUS TIME-STEPPING SCHEMES FOR THE ALLEN-CAHN EQUATION

摘要

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 that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/epsilon) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.