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Volume 14, Issue 1
On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate

Xiang Wang & Yuqing Zhang

DOI: 10.4208/nmtma.OA-2020-0027

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 47-70.

Published online: 2020-10

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  • Abstract

We provide a general construction method for a finite volume element (FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable domain in $k$-dimension.
In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element, which open up more possibilities of the FVE schemes to be applied to some complex problems, such as the convection-dominated problems. It worth mentioning that, the construction can be extended to the quadrilateral meshes in 2D. The stability and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.

  • Keywords

Finite volume, $L^2$ estimate, sufficient and necessary condition, orthogonal condition.

  • AMS Subject Headings

65N12, 65N08, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-47, author = {Wang , Xiang and Zhang , Yuqing}, title = {On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2020}, volume = {14}, number = {1}, pages = {47--70}, abstract = {

We provide a general construction method for a finite volume element (FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable domain in $k$-dimension.
In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element, which open up more possibilities of the FVE schemes to be applied to some complex problems, such as the convection-dominated problems. It worth mentioning that, the construction can be extended to the quadrilateral meshes in 2D. The stability and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0027}, url = {http://global-sci.org/intro/article_detail/nmtma/18327.html} }
TY - JOUR T1 - On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate AU - Wang , Xiang AU - Zhang , Yuqing JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 47 EP - 70 PY - 2020 DA - 2020/10 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0027 UR - https://global-sci.org/intro/article_detail/nmtma/18327.html KW - Finite volume, $L^2$ estimate, sufficient and necessary condition, orthogonal condition. AB -

We provide a general construction method for a finite volume element (FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable domain in $k$-dimension.
In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element, which open up more possibilities of the FVE schemes to be applied to some complex problems, such as the convection-dominated problems. It worth mentioning that, the construction can be extended to the quadrilateral meshes in 2D. The stability and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.

Xiang Wang & Yuqing Zhang. (2020). On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate. Numerical Mathematics: Theory, Methods and Applications. 14 (1). 47-70. doi:10.4208/nmtma.OA-2020-0027
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