Adaptive discontinuous Galerkin finite element methods for the Allen-Cahn equation on polygonal meshes

In this paper, we develop a polygonal mesh adaptation algorithm for a fully implicit scheme based on discontinuous Galerkin (DG) finite element methods in space and backward Euler method in time to solve the Allen-Cahn equation. The mathematical framework and a procedure for solving this nonlinear equation are given. We extend the DG discretization with a polygonal mesh adaptation method to save computation time and capture the thin interfaces more accurately. A criterion based on the local value of the phase field function gradient is used to select the target element for refinement and coarsening, and then a 4-node polygonal mesh refinement strategy is adopted by connecting the midpoint of each edge to the barycenter of the target element. Using numerical tests, including motion by the mean curvature, curvature-driven flow, the Allen-Cahn equation with a logarithmic free energy, the Allen-Cahn equation with advection, and the application for image segmentation, we verify the accuracy, efficiency, and capabilities of the adaptive DG on polygonal meshes and confirm the decreasing property of the discrete energy.