An exterior calculus framework for polytopal methods

  • Francesco Bonaldi

    Université de Perpignan, Perpignan, France
  • Daniele Antonio Di Pietro

    Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Montpellier, France
  • Jérôme Droniou

    Institut Montpelliérain Alexander Grothendieck, CNRS, Montpellier, France; Monash University, Melbourne, Australia
  • Kaibo Hu

    University of Edinburgh, Edinburgh, UK
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Abstract

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the discrete de Rham and the virtual element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

Cite this article

Francesco Bonaldi, Daniele Antonio Di Pietro, Jérôme Droniou, Kaibo Hu, An exterior calculus framework for polytopal methods. J. Eur. Math. Soc. (2025), published online first

DOI 10.4171/JEMS/1602