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Volume 34, Issue 1
A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Yanlong Zhang

DOI: 10.4208/cicp.OA-2022-0307

Commun. Comput. Phys., 34 (2023), pp. 116-131.

Published online: 2023-08

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  • Abstract

Based on the idea of serendipity element, we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonal meshes in this article. The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates, and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space. Moreover, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to finite volume element scheme, and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. Finally, under certain geometric assumption conditions, the optimal $H^1$ error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained, and verified by numerical experiments.

  • Keywords

Quadratic serendipity polygonal finite volume element method, arbitrary convex polygonal meshes, Wachspress coordinate, unified dual partitions, optimal $H^1$ error estimate.

  • AMS Subject Headings

65N08, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-116, author = {Zhang , Yanlong}, title = {A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {1}, pages = {116--131}, abstract = {

Based on the idea of serendipity element, we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonal meshes in this article. The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates, and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space. Moreover, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to finite volume element scheme, and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. Finally, under certain geometric assumption conditions, the optimal $H^1$ error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained, and verified by numerical experiments.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0307}, url = {http://global-sci.org/intro/article_detail/cicp/21882.html} }
TY - JOUR T1 - A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes AU - Zhang , Yanlong JO - Communications in Computational Physics VL - 1 SP - 116 EP - 131 PY - 2023 DA - 2023/08 SN - 34 DO - http://doi.org/10.4208/cicp.OA-2022-0307 UR - https://global-sci.org/intro/article_detail/cicp/21882.html KW - Quadratic serendipity polygonal finite volume element method, arbitrary convex polygonal meshes, Wachspress coordinate, unified dual partitions, optimal $H^1$ error estimate. AB -

Based on the idea of serendipity element, we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonal meshes in this article. The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates, and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space. Moreover, we construct a family of unified dual partitions for arbitrary convex polygonal meshes, which is crucial to finite volume element scheme, and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom. Finally, under certain geometric assumption conditions, the optimal $H^1$ error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained, and verified by numerical experiments.

Yanlong Zhang. (2023). A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes. Communications in Computational Physics. 34 (1). 116-131. doi:10.4208/cicp.OA-2022-0307
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