Finite Element Method on locally refined composite meshes for Dirichlet fractional Laplacian

IF 3.1 3区计算机科学 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Science Pub Date : 2024-08-30 DOI:10.1016/j.jocs.2024.102433

Jun Zhou, Hongbin Chen

引用次数: 0

Abstract

It is known that the solution of the Dirichlet fractional Laplacian in a bounded domain exhibits singular behavior near the boundary. Consequently, numerical discretizations on quasi-uniform meshes lead to low accuracy and nonphysical solutions. We adopt a finite element discretization on locally refined composite meshes, which consist in a combination of graded meshes near the singularity and uniform meshes where the solution is smooth. We also provide a reference strategy on parameter selection of locally refined composite meshes. Numerical tests confirm that finite element method on locally refined composite meshes has higher accuracy than uniform meshes, but the computational cost is less than that of graded meshes. Our method is applied to discrete the fractional-in-space Allen–Cahn equation and the fractional Burgers equation with Dirichlet fractional Laplacian, some new observations are discovered from our numerical results.

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针对 Dirichlet 分数拉普拉斯的局部细化复合网格有限元法

众所周知，有界域中的 Dirichlet 分数拉普拉斯解在边界附近表现出奇异行为。因此，在准均匀网格上进行数值离散会导致低精度和非物理解。我们在局部细化的复合网格上采用有限元离散化方法，这种方法由奇异点附近的分级网格和求解平滑的均匀网格组合而成。我们还提供了局部细化复合网格参数选择的参考策略。数值试验证实，局部细化复合网格上的有限元方法比均匀网格具有更高的精度，但计算成本却低于分级网格。我们的方法被应用于离散分式空间 Allen-Cahn 方程和具有 Dirichlet 分式拉普拉斯的分式 Burgers 方程，从数值结果中发现了一些新的观察结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Science COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS-COMPUTER SCIENCE, THEORY & METHODS

CiteScore

5.50

自引率

3.00%

发文量

227

审稿时长

41 days

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