Abstract
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.
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H. Li was partially supported by the NSF Grants DMS-1158839 and DMS-1418853, and by the Wayne State University Grants Plus Program. Q. Zhang was partially supported by the Natural Science Foundation of China Grants 11001282 and 11471343, and by Guangdong Provincial Natural Science Foundation of China under Grant 2015A030306016.
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Li, H., Zhang, Q. Optimal Quadrilateral Finite Elements on Polygonal Domains. J Sci Comput 70, 60–84 (2017). https://doi.org/10.1007/s10915-016-0242-5
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DOI: https://doi.org/10.1007/s10915-016-0242-5