A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Alan Demlow, Anil N. Hirani

Research output: Contribution to journalArticlepeer-review

Abstract

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.

Original languageEnglish (US)
Pages (from-to)1337-1371
Number of pages35
JournalFoundations of Computational Mathematics
Volume14
Issue number6
DOIs
StatePublished - Nov 1 2014

Keywords

  • A posteriori error estimates
  • Adaptivity
  • Exterior calculus
  • Finite element methods

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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