Finite element exterior calculus: from Hodge theory to numerical stability

TL;DR

This paper develops a unified framework connecting topology, geometry, and finite element methods to analyze the stability and convergence of discretized differential equations related to Hodge theory.

Contribution

It introduces an abstract Hilbert space framework for stable discretization of differential complexes, applying it to finite element methods for the Hodge Laplacian and elasticity equations.

Findings

Stable discretization requires subcomplexes and bounded cochain projections.

Constructed finite element differential forms form subcomplexes of the de Rham complex.

Framework ensures convergence of finite element approximations for Hodge Laplacian.

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 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.