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

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

Content Provider	Springer Nature Link
Author	Holst, Michael Stern, Ari
Copyright Year	2012
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 generalize 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.
Ending Page	293
Page Count	31
Starting Page	263
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	3
Volume Number	12
Language	English
Publisher	Springer-Verlag
Publisher Date	2012-04-05
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Economics general Variational crimes Surface finite elements Linear and Multilinear Algebras, Matrix Theory Finite element exterior calculus de Rham theory Numerical Analysis Isoparametric finite elements Computer Science Math Applications in Computer Science Applications of Mathematics Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods Mixed finite elements
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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