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A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed thata large class of mixed finite element methods can be formulated naturally onHilbert complexes, where using a Galerkin-like approach, one solves avariational problem on a finite-dimensional subcomplex. In a seeminglyunrelated research direction, Dziuk [Lect Notes in Math, vol 1357 (1988)]analyzed a class of nodal finite elements for the Laplace-Beltrami equation onsmooth 2-surfaces approximated by a piecewise-linear triangulation; Demlowlater extended this analysis [SIAM J Numer Anal, 47 (2009)] to 3-surfaces, aswell as to higher-order surface approximation. In this article, we bring theselines of research together, first developing a framework for the analysis ofvariational crimes in abstract Hilbert complexes, and then applying thisabstract framework to the setting of finite element exterior calculus onhypersurfaces. Our framework extends the work of Arnold, Falk, and Winther toproblems that violate their subcomplex assumption, allowing for the extensionof finite element exterior calculus to approximate domains, most notably theHodge-de Rham complex on approximate manifolds. As an application of thelatter, we recover Dziuk's and Demlow's a priori estimates for 2- and3-surfaces, demonstrating that surface finite element methods can be analyzedcompletely within this abstract framework. Moreover, our results generalizethese earlier estimates dramatically, extending them from nodal finite elementsfor Laplace-Beltrami to mixed finite elements for the Hodge Laplacian, and from2- and 3-dimensional hypersurfaces to those of arbitrary dimension. Bydeveloping this analytical framework using a combination of general tools fromdifferential geometry and functional analysis, we are led to a more geometricanalysis of surface finite element methods, whereby the main results becomemore transparent.

Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces