Volume 28, Issue 4
An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

Emine CelikerPing Lin

Commun. Comput. Phys., 28 (2020), pp. 1536-1560.

Published online: 2020-08

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  • 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.

  • Keywords

Allen-Cahn equation, non-convex polygon, mesh refinement, corner singularities, finite element method, invariant energy quadratization, error estimation.

  • AMS Subject Headings

65M50, 65M60, 65M15, 65Z05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1536, author = {Celiker , Emine and Lin , Ping}, 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 = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0036}, url = {http://global-sci.org/intro/article_detail/cicp/18110.html} }
TY - JOUR T1 - An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons AU - Celiker , Emine AU - Lin , Ping JO - Communications in Computational Physics VL - 4 SP - 1536 EP - 1560 PY - 2020 DA - 2020/08 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2020-0036 UR - https://global-sci.org/intro/article_detail/cicp/18110.html KW - Allen-Cahn equation, non-convex polygon, mesh refinement, corner singularities, finite element method, invariant energy quadratization, error estimation. AB -

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.

Emine Celiker & Ping Lin. (2020). 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. 28 (4). 1536-1560. doi:10.4208/cicp.OA-2020-0036
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