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Volume 33, Issue 5
New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method

Xiang Wang, Yuqing Zhang & Zhimin Zhang

DOI: 10.4208/cicp.OA-2022-0295

Commun. Comput. Phys., 33 (2023), pp. 1332-1356.

Published online: 2023-06

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  • Abstract

New superconvergent structures are proposed and analyzed for the finite volume element (FVE) method over tensorial meshes in general dimension $d$ (for $d≥2$); we call these orthogonal superconvergent structures. In this framework, one has the freedom to choose the superconvergent points of tensorial $k$-order FVE schemes (for $k≥3$). This flexibility contrasts with the superconvergent points (such as Gauss points and Lobatto points) for current FE schemes and FVE schemes, which are fixed. The orthogonality condition and the modified M-decomposition (MMD) technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes. Numerical experiments are provided to validate our theoretical results.

  • Keywords

Superconvergence, finite volume, orthogonality condition, tensorial mesh, rectangular mesh.

  • AMS Subject Headings

65N12, 65N08, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-33-1332, author = {Wang , XiangZhang , Yuqing and Zhang , Zhimin}, title = {New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method}, journal = {Communications in Computational Physics}, year = {2023}, volume = {33}, number = {5}, pages = {1332--1356}, abstract = {

New superconvergent structures are proposed and analyzed for the finite volume element (FVE) method over tensorial meshes in general dimension $d$ (for $d≥2$); we call these orthogonal superconvergent structures. In this framework, one has the freedom to choose the superconvergent points of tensorial $k$-order FVE schemes (for $k≥3$). This flexibility contrasts with the superconvergent points (such as Gauss points and Lobatto points) for current FE schemes and FVE schemes, which are fixed. The orthogonality condition and the modified M-decomposition (MMD) technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes. Numerical experiments are provided to validate our theoretical results.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0295}, url = {http://global-sci.org/intro/article_detail/cicp/21763.html} }
TY - JOUR T1 - New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method AU - Wang , Xiang AU - Zhang , Yuqing AU - Zhang , Zhimin JO - Communications in Computational Physics VL - 5 SP - 1332 EP - 1356 PY - 2023 DA - 2023/06 SN - 33 DO - http://doi.org/10.4208/cicp.OA-2022-0295 UR - https://global-sci.org/intro/article_detail/cicp/21763.html KW - Superconvergence, finite volume, orthogonality condition, tensorial mesh, rectangular mesh. AB -

New superconvergent structures are proposed and analyzed for the finite volume element (FVE) method over tensorial meshes in general dimension $d$ (for $d≥2$); we call these orthogonal superconvergent structures. In this framework, one has the freedom to choose the superconvergent points of tensorial $k$-order FVE schemes (for $k≥3$). This flexibility contrasts with the superconvergent points (such as Gauss points and Lobatto points) for current FE schemes and FVE schemes, which are fixed. The orthogonality condition and the modified M-decomposition (MMD) technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes. Numerical experiments are provided to validate our theoretical results.

Xiang Wang, Yuqing Zhang & Zhimin Zhang. (2023). New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method. Communications in Computational Physics. 33 (5). 1332-1356. doi:10.4208/cicp.OA-2022-0295
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