Finite element spaces of double forms
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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_ | 1866912389378605056 |
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| author | Berchenko-Kogan, Yakov Gawlik, Evan S. |
| author_facet | Berchenko-Kogan, Yakov Gawlik, Evan S. |
| contents | The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or $(p,q)$-forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of $(p,q)$-forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17243 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite element spaces of double forms Berchenko-Kogan, Yakov Gawlik, Evan S. Numerical Analysis Differential Geometry 65N30, 15A69, 58A10, 53A45 The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or $(p,q)$-forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of $(p,q)$-forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor. |
| title | Finite element spaces of double forms |
| topic | Numerical Analysis Differential Geometry 65N30, 15A69, 58A10, 53A45 |
| url | https://arxiv.org/abs/2505.17243 |