@Article{CiCP-28-4,
author = {Emine, Celiker and Ping, Lin},
title = {An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons},
journal = {Communications in Computational Physics},
year = {2020},
volume = {28},
number = {4},
pages = {1536--1560},
abstract = {<p style="text-align: justify;">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 <span style="text-align: justify;">$\mathcal{O}$</span>($h^{−2}$ln$h^{−1}$) spatial
elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical
results obtained support the analysis made.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2020-0036},
url = {https://global-sci.com/article/79730/an-efficient-finite-element-method-with-exponential-mesh-refinement-for-the-solution-of-the-allen-cahn-equation-in-non-convex-polygons}
}