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Foundations of Computational Mathematics

, Volume 12, Issue 3, pp 263–293 | Cite as

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

  • Michael Holst
  • Ari Stern
Article

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.

Keywords

Finite element exterior calculus Variational crimes Mixed finite elements Surface finite elements Isoparametric finite elements 

Mathematics Subject Classification

65N30 58A12 

Notes

Acknowledgements

We are grateful to the editor and anonymous referees for their valuable comments and suggestions. Their diligence and attention to detail during the review process was truly extraordinary, and the final paper has benefited greatly from their efforts. We also thank Paul Leopardi for catching a typographical error in one of the equations shortly before this article went to press. Finally, we wish to express our appreciation to Douglas Arnold, Snorre Christiansen, Alan Demlow, Richard Falk, and Ragnar Winther for reading earlier versions of the manuscript so carefully, and for providing helpful feedback.

M.H. was supported in part by NSF DMS/CM Awards 0715146 and 0915220, NSF MRI Award 0821816, NSF PHY/PFC Award 0822283, and by DOD/DTRA Award HDTRA-09-1-0036.

A.S. was supported in part by NSF DMS/CM Award 0715146 and by NSF PHY/PFC Award 0822283, as well as by NIH, HHMI, CTBP, and NBCR.

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Copyright information

© SFoCM 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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