Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908886528688128 |
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| author | Zhu, Chengbin Christiansen, Snorre H. Hu, Kaibo Hirani, Anil N. |
| author_facet | Zhu, Chengbin Christiansen, Snorre H. Hu, Kaibo Hirani, Anil N. |
| contents | We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions Zhu, Chengbin Christiansen, Snorre H. Hu, Kaibo Hirani, Anil N. Numerical Analysis 65N30 We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincaré inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes. |
| title | Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions |
| topic | Numerical Analysis 65N30 |
| url | https://arxiv.org/abs/2505.08966 |