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

TL;DR

This paper develops high-order discontinuous Galerkin time-stepping schemes for the Allen-Cahn equation, providing unconditional stability and optimal error estimates without Grönwall Lemma reliance.

Contribution

It introduces a novel duality and bootstrap approach to establish stability and error bounds for arbitrary order schemes in Allen-Cahn equation approximation.

Findings

Unconditionally stable schemes under minimal regularity.

Best approximation error estimates with polynomial dependence on 1/ε.

Applicable to arbitrary order discontinuous Galerkin methods.

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 Gr\"onwall 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.