Multisymplecticity in finite element exterior calculus
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913866240229376 |
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| author | Stern, Ari Zampa, Enrico |
| author_facet | Stern, Ari Zampa, Enrico |
| contents | We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold-Falk-Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder-Weyl "grad-div" form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_03657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Multisymplecticity in finite element exterior calculus Stern, Ari Zampa, Enrico Numerical Analysis 65N30 (Primary) 70S05 (Secondary) We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold-Falk-Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder-Weyl "grad-div" form. |
| title | Multisymplecticity in finite element exterior calculus |
| topic | Numerical Analysis 65N30 (Primary) 70S05 (Secondary) |
| url | https://arxiv.org/abs/2312.03657 |