Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

Abstract

A recent paper of Arnold, Falk, and Winther (Bull. Am. Math. Soc. 47:281-354, 2010) showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk (Lecture Notes in Math., vol. 1357, pp. 142-155, 1988) analyzed a class of nodal finite elements for the Laplace-Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis (SIAM J. Numer. Anal. 47:805-827, 2009) to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge-de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk's and Demlow's a priori estimates for 2-and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results gener- Communicated by 264 Found Comput Math (2012) 12:263-293 alize these earlier estimates dramatically, extending them from nodal finite elements for Laplace-Beltrami to mixed finite elements for the Hodge Laplacian, and from 2-and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.

On the linearization of Regge calculus

“…Also highly relevant are [8] and [15]. Numerical methods based on differential forms have been studied on manifolds [39] and applied to general relativity in a simplified setting [53]. Regge elements have been rediscovered as a tool for solving equations of elasticity [28].…”

A discrete exterior approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems

“…A major challenge from the numerical analysis standpoint is to offer a careful study of the convergence properties of discrete exterior calculus. In future, in the context of [1], [17], it would be interesting to study structurepreserving discretization of port-Hamiltonian systems in the framework of Hilbert complexes.…”

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

“…Additionally, we aim to analyze variational crimes in this semilinear setting, extending our earlier analysis of the linear case in Holst and Stern [24].…”

“…Another recent development in this area has been the analysis of "variational crimes" on Hilbert complexes (Holst and Stern [24]). By analogy with Strang's lemmas for variational crimes on Hilbert spaces, this work extended the estimates of Arnold et al [4] to problems where certain conditions on the discretization have been violated.…”

“…Our main motivation for extending the estimates of Arnold et al [4] and of Holst and Stern [24], from linear to semilinear problems, is to enable the use of finite element exterior calculus for nonlinear problems on hypersurfaces, allowing for a complete analysis of the additional errors due to nonlinearity, as well as those due to surface approximation.…”