A hybrid high-order method for the biharmonic problem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911563929092096 |
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| author | Liang, Yizhou Tran, Ngoc Tien |
| author_facet | Liang, Yizhou Tran, Ngoc Tien |
| contents | This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16608 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A hybrid high-order method for the biharmonic problem Liang, Yizhou Tran, Ngoc Tien Numerical Analysis 65N30, 65N25, 65N15 This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions. |
| title | A hybrid high-order method for the biharmonic problem |
| topic | Numerical Analysis 65N30, 65N25, 65N15 |
| url | https://arxiv.org/abs/2504.16608 |