Title: Finite element exterior calculus: from Hodge theory to numerical stability
Authors: Douglas N. Arnold, Richard S. Falk, and Ragnar Winther
Source: Bull. Amer. Math. Soc. 47 (2010) 281-353
Status: Published
Abstract: 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 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.
Keywords: finite element exterior calculus, exterior calculus, de Rham
cohomology, Hodge theory, Hodge Laplacian, mixed finite elements
Subj. class.: 65N30, 58A14
URL: http://www.math.rutgers.edu/~falk/papers/bulletin-rev2.pdf