An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

Daniele A. Di Pietro, Jérôme Droniou

Research output: Contribution to journalArticleResearchpeer-review

16 Citations (Scopus)

Abstract

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of L 2-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

Original languageEnglish
Pages (from-to)85-164
Number of pages80
JournalFoundations of Computational Mathematics
Volume23
Issue number1
DOIs
Publication statusPublished - Feb 2023

Keywords

  • Arbitrary order
  • Compatible discretizations
  • Discrete de Rham complex
  • Polyhedral methods

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