Finite element exterior calculus: From hodge theory to numerical stability

By Douglas N. Arnold, Richard S. Falk, Ragnar Winther. 2009

671 citations recorded by the source corpus.

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 the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, 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 to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/681b5347129176f3bae73b3c7ec577df93cf6b6a

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Citations in the corpus

Hodge Laplacians on graphs (2015, background): “” Unlike physical problems arising from areas such as continuum mechanics or electromagnetics, where the differentiable Hodge–de Rham theory has been applied with great efficacy for both modeling and computations [3, 30, 45, 53, 65, 66, 70], those arising from data analytic applications are likely t”
A review of unified a posteriori finite element error control (2012, background): “With appropriate differential operators, these spaces form an exact sequence [5,6] which is shown in the next commuting diagram together with corresponding conforming discrete spaces.”
Finite element approximation of eigenvalue problems (2010, methodology): “A deep understanding of the subject which leads to the formalism of the finite element exterior calculus is presented in the following works: Arnold (2002) and Arnold et al. (2006a, 2006b, 2010).”
Mathematical analysis of variational isogeometric methods* (2014, methodology): “…Picard (1984), the commutative diagram (5.23) and the stability of the projectors in the L2-norm from Corollary 5.12, we can apply Theorem 3.19 in Arnold et al. (2010) to show the convergence of the associated vector Hodge–Laplacian eigenvalue problem:∫ Ω φhψh dx− ∫ Ω rEh · gradψh dx = 0 for all…”
Mimetic finite difference method (2014, methodology): “Finite element techniques have been recently recasted in the framework of the Whitney forms and formalized in the finite element exterior calculus, cf. [15] [16].”
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows (2015, background): “During the past 30 years, the construction of de Rham sub-complexes consisting of finite element spaces has been an invaluable tool to develop stable finite element pairs for problems in porous media flow, electromagnetics, and linear elasticity [2, 3, 48, 16].”
Finite Element Methods for Maxwell's Equations (2019, background): “The recognition by Bossavit [11] (see also [31]) that this, and a similar equality for H(curl; Ω) and H(div; Ω) interpolation, constitute a discrete de Rham complex motivated in part the Finite Element Exterior Calculus of Arnold, Falk and Winther [3, 2].”