Geometric decompositions and local bases for spaces of finite element differential forms

Douglas N. Arnold, Richard S. Falk, Ragnar Winther

Research output: Contribution to journalArticlepeer-review

36 Scopus citations

Abstract

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nédélec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.

Original languageEnglish (US)
Pages (from-to)1660-1672
Number of pages13
JournalComputer Methods in Applied Mechanics and Engineering
Volume198
Issue number21-26
DOIs
StatePublished - May 1 2009

Bibliographical note

Funding Information:
The work of the first author was supported in part by NSF Grant DMS-0713568. The work of the second author was supported in part by NSF Grant DMS06-09755. The work of the third author was supported by the Norwegian Research Council.

Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

Keywords

  • Berstein bases
  • Finite element exterior calculus
  • finite element bases

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