Analysis of interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow

Abstract: This paper develops and analyzes two fully discrete interior penalty discontinuous Galerkin (IP-DG) methods for the Allen-Cahn equation, which is a nonlinear singular perturbation of the heat equation and originally arises from phase transition of binary alloys in materials science, and its sharp interface limit (the mean curvature flow) as the perturbation parameter tends to zero. Both fully implicit and energy-splitting time-stepping schemes are proposed. The primary goal of the paper is to derive sharp error bounds which depend on the reciprocal of the perturbation parameter $\epsilon$ (also called "interaction length") only in some lower polynomial order, instead of exponential order, for the proposed IP-DG methods. The derivation is based on a refinement of the nonstandard error analysis technique first introduced in [12]. The centerpiece of this new technique is to establish a spectrum estimate result in totally discontinuous DG finite element spaces with a help of a similar spectrum estimate result in the conforming finite element spaces which was established in [12]. As a nontrivial application of the sharp error estimates, they are used to establish convergence and the rates of convergence of the zero level sets of the fully discrete IP-DG solutions to the classical and generalized mean curvature flow. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete IP-DG methods.