Numerical analysis of a second-order IPDGFE method for the Allen–Cahn equation and the curvature-driven geometric flow

Abstract

The paper focuses on proposing and analyzing a nonlinear interior penalty discontinuous Galerkin finite element (IPDGFE) method for the Allen–Cahn equation, which is a reaction–diffusion model with a nonlinear singular perturbation arising from the phase separation process. We firstly present a fully discrete IPDGFE formulation based on the modified Crank–Nicolson scheme and a mid-point approximation of the potential term f(u). We then derive the energy-stability and the second-order-in-time error estimates for the proposed IPDGFE method under some regularity assumptions on the initial function u0. There are two key works in our paper. One is to establish a second-order-in-time and energy-stable IPDGFE scheme. The other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions um and um+1 in the nonlinear term, instead of using the standard Gronwall inequality technique, so we obtain that all our error bounds depend on the reciprocal of the perturbation parameter ϵ only in some lower polynomial order, instead of exponential order. As a nontrivial byproduct of our paper, we also analyze the convergence of the zero-level sets of fully discrete IPDGFE solutions to the curvature-driven geometric flow. Finally, numerical experiments are provided to demonstrate the good performance of our presented IPDGFE method, including the time and space error estimates of the discrete solutions, discrete energy-stability, and the convergence of numerical interfaces governed by the curvature-driven geometric flow in the classical motion and generalized motion.