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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2503.01420 |
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| _version_ | 1866909571145007104 |
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| author | Huang, Weizhang Wang, Xiang Zhang, Xinyuan |
| author_facet | Huang, Weizhang Wang, Xiang Zhang, Xinyuan |
| contents | A two-layer dual strategy is proposed in this work to construct a new family of high-order finite volume element (FVE-2L) schemes that can avoid main common drawbacks of the existing high-order finite volume element (FVE) schemes. The existing high-order FVE schemes are complicated to construct since the number of the dual elements in each primary element used in their construction increases with a rate $O((k+1)^2)$, where $k$ is the order of the scheme. Moreover, all $k$th-order FVE schemes require a higher regularity $H^{k+2}$ than the approximation theory for the $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L schemes has a much simpler construction since they have a fixed number (four) of dual elements in each primary element. They also reduce the regularity requirement for the $L^2$ theory to $H^{k+1}$ and preserve the local conservation law on all dual elements of the second dual layer for both flux and equation forms. Their stability and $H^1$ and $L^2$ convergence are proved. Numerical results are presented to illustrate the convergence and conservation properties of the FVE-2L schemes. Moreover, the condition number of the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown to have the same growth rate as those for the existing FVE and finite element schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New finite volume element schemes based on a two-layer dual strategy Huang, Weizhang Wang, Xiang Zhang, Xinyuan Numerical Analysis 65N08, 65N12, 65N30 A two-layer dual strategy is proposed in this work to construct a new family of high-order finite volume element (FVE-2L) schemes that can avoid main common drawbacks of the existing high-order finite volume element (FVE) schemes. The existing high-order FVE schemes are complicated to construct since the number of the dual elements in each primary element used in their construction increases with a rate $O((k+1)^2)$, where $k$ is the order of the scheme. Moreover, all $k$th-order FVE schemes require a higher regularity $H^{k+2}$ than the approximation theory for the $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L schemes has a much simpler construction since they have a fixed number (four) of dual elements in each primary element. They also reduce the regularity requirement for the $L^2$ theory to $H^{k+1}$ and preserve the local conservation law on all dual elements of the second dual layer for both flux and equation forms. Their stability and $H^1$ and $L^2$ convergence are proved. Numerical results are presented to illustrate the convergence and conservation properties of the FVE-2L schemes. Moreover, the condition number of the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown to have the same growth rate as those for the existing FVE and finite element schemes. |
| title | New finite volume element schemes based on a two-layer dual strategy |
| topic | Numerical Analysis 65N08, 65N12, 65N30 |
| url | https://arxiv.org/abs/2503.01420 |