A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915312753967104 |
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| author | Barker, Mary Cao, Shuhao Stern, Ari |
| author_facet | Barker, Mary Cao, Shuhao Stern, Ari |
| contents | We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_10567 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian Barker, Mary Cao, Shuhao Stern, Ari Numerical Analysis 65N30, 65N15, 35Q60 We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities. |
| title | A nonconforming primal hybrid finite element method for the two-dimensional vector Laplacian |
| topic | Numerical Analysis 65N30, 65N15, 35Q60 |
| url | https://arxiv.org/abs/2206.10567 |