An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons (Communications in Computational Physics, 2020)
In this paper we consider the numerical solution of the Allen-Cahn typediffuse interface model in a polygonal domain. The intersection of the interface withthe re-entrant corners of the polygon causes strong corner singularities in the solution.To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method withexponential mesh refinement in the vicinity of the singular corners, that is based on($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized byemploying a first-order, semi-implicit time stepping scheme with the Invariant EnergyQuadratization approach in time, which is an unconditionally energy stable method.It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatialelements, where $h$ and $τ$ are the mesh and time steps, respectively. The numericalresults obtained support the analysis made.