The grad-div conforming virtual element method for the quad-div problem in three dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910014775492608 |
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| author | Dong, Xiaojing Han, Yibing Huang, Yunqing |
| author_facet | Dong, Xiaojing Han, Yibing Huang, Yunqing |
| contents | We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The grad-div conforming virtual element method for the quad-div problem in three dimensions Dong, Xiaojing Han, Yibing Huang, Yunqing Numerical Analysis 65N30, 65N15 We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results. |
| title | The grad-div conforming virtual element method for the quad-div problem in three dimensions |
| topic | Numerical Analysis 65N30, 65N15 |
| url | https://arxiv.org/abs/2410.18375 |