Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Finite element exterior calculus: from Hodge theory to numerical stability
HTML articles powered by AMS MathViewer

by Douglas N. Arnold, Richard S. Falk and Ragnar Winther PDF
Bull. Amer. Math. Soc. 47 (2010), 281-354 Request permission


This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge-theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 65N30, 58A14
  • Retrieve articles in all journals with MSC (2000): 65N30, 58A14
Additional Information
  • Douglas N. Arnold
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 27240
  • Email:
  • Richard S. Falk
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • Email:
  • Ragnar Winther
  • Affiliation: Centre of Mathematics for Applications and Department of Informatics, University of Oslo, 0316 Oslo, Norway
  • MR Author ID: 183665
  • Email:
  • Received by editor(s): June 23, 2009
  • Received by editor(s) in revised form: August 12, 2009
  • Published electronically: January 25, 2010
  • Additional Notes: 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 DMS-0609755.
    The work of the third author was supported by the Norwegian Research Council
  • © Copyright 2010 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 47 (2010), 281-354
  • MSC (2000): Primary 65N30, 58A14
  • DOI:
  • MathSciNet review: 2594630