An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons
Year: 2020
Author: Emine Celiker, Ping Lin
Communications in Computational Physics, Vol. 28 (2020), Iss. 4 : pp. 1536–1560
Abstract
In this paper we consider the numerical solution of the Allen-Cahn type diffuse interface model in a polygonal domain. The intersection of the interface with the 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 with exponential 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 by employing a first-order, semi-implicit time stepping scheme with the Invariant Energy Quadratization 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$h^{−1}$) spatial elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical results obtained support the analysis made.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2020-0036
Communications in Computational Physics, Vol. 28 (2020), Iss. 4 : pp. 1536–1560
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Allen-Cahn equation non-convex polygon mesh refinement corner singularities finite element method invariant energy quadratization error estimation.