{
  "id": "2312.03657",
  "version": "v1",
  "published": "2023-12-06T18:27:21.000Z",
  "updated": "2023-12-06T18:27:21.000Z",
  "title": "Multisymplecticity in finite element exterior calculus",
  "authors": [
    "Ari Stern",
    "Enrico Zampa"
  ],
  "comment": "34 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-06T18:27:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65N30",
      "37K06"
    ],
    "keywords": [
      "finite element exterior calculus",
      "local multisymplectic conservation law",
      "familiar symplectic conservation law",
      "multisymplecticity",
      "characterize hybrid feec methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}