Optimal Quadrilateral Finite Elements on Polygonal Domains

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• 作者：Hengguang Li ; Qinghui Zhang
• 关键词：Corner singularity ; Finite element ; Graded quadrilateral mesh ; Error analysis
• 刊名：Journal of Scientific Computing
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：70
• 期：1
• 页码：60-84
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer US
• ISSN：1573-7691
• 卷排序：70

文摘

We propose three quadrilateral mesh refinement algorithms to improve the convergence of the finite element method approximating the singular solutions of elliptic equations, which are due to the non-smoothness of the domain. These algorithms result in graded meshes consisting of convex and shape-regular quadrilaterals. With analysis in weighted spaces, we provide the selection criteria for the grading parameter, such that the optimal convergence rate can be recovered for the associated finite element approximation. Various numerical tests verify the theory. In addition to the bi-k elements, we also investigate the serendipity elements on the graded quadrilateral meshes in the numerical experiments.