{"title":"有限元外部微积分中的多折射性","authors":"Ari Stern, Enrico Zampa","doi":"arxiv-2312.03657","DOIUrl":null,"url":null,"abstract":"We consider the application of finite element exterior calculus (FEEC)\nmethods to a class of canonical Hamiltonian PDE systems involving differential\nforms. Solutions to these systems satisfy a local multisymplectic conservation\nlaw, which generalizes the more familiar symplectic conservation law for\nHamiltonian systems of ODEs, and which is connected with physically-important\nreciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We\ncharacterize hybrid FEEC methods whose numerical traces satisfy a version of\nthe multisymplectic conservation law, and we apply this characterization to\nseveral specific classes of FEEC methods, including conforming\nArnold-Falk-Winther-type methods and various hybridizable discontinuous\nGalerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming\nmethods are shown, in general, to be multisymplectic in a stronger sense than\nthe conforming FEEC methods. This substantially generalizes previous work of\nMcLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more\nrestricted class of canonical Hamiltonian PDEs in the de Donder-Weyl \"grad-div\"\nform.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multisymplecticity in finite element exterior calculus\",\"authors\":\"Ari Stern, Enrico Zampa\",\"doi\":\"arxiv-2312.03657\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the application of finite element exterior calculus (FEEC)\\nmethods to a class of canonical Hamiltonian PDE systems involving differential\\nforms. Solutions to these systems satisfy a local multisymplectic conservation\\nlaw, which generalizes the more familiar symplectic conservation law for\\nHamiltonian systems of ODEs, and which is connected with physically-important\\nreciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We\\ncharacterize hybrid FEEC methods whose numerical traces satisfy a version of\\nthe multisymplectic conservation law, and we apply this characterization to\\nseveral specific classes of FEEC methods, including conforming\\nArnold-Falk-Winther-type methods and various hybridizable discontinuous\\nGalerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming\\nmethods are shown, in general, to be multisymplectic in a stronger sense than\\nthe conforming FEEC methods. This substantially generalizes previous work of\\nMcLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more\\nrestricted class of canonical Hamiltonian PDEs in the de Donder-Weyl \\\"grad-div\\\"\\nform.\",\"PeriodicalId\":501061,\"journal\":{\"name\":\"arXiv - CS - Numerical Analysis\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2312.03657\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2312.03657","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Multisymplecticity in finite element exterior calculus
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.