Skip to main content
Log in

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

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abraham, J.E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, 1978).

    MATH  Google Scholar 

  2. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155 (2006). doi:10.1017/S0962492906210018.

    Article  MathSciNet  MATH  Google Scholar 

  3. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull., New Ser., Am. Math. Soc. 47(2), 281–354 (2010). doi:10.1090/S0273-0979-10-01278-4.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. Aubin, Nonlinear Analysis on Manifolds. Monge–Ampère Equations, Grundlehren der Mathematischen Wissenschaften, vol. 252 (Springer, New York, 1982).

    Book  MATH  Google Scholar 

  5. D. Boffi, F. Brezzi, L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comput. 69(229), 121–140 (2000). doi:10.1090/S0025-5718-99-01072-8.

    MathSciNet  MATH  Google Scholar 

  6. A. Bossavit, Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, IEE Proc. A, Sci. Meas. Technol. 135(8), 493–500 (1988).

    Google Scholar 

  7. D. Braess, Finite elements. Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd edn. (Cambridge University Press, Cambridge, 2007). Translated from the German by L.L. Schumaker.

    Book  MATH  Google Scholar 

  8. J. Brüning, M. Lesch, Hilbert complexes, J. Funct. Anal. 108(1), 88–132 (1992). doi:10.1016/0022-1236(92)90147-B.

    Article  MathSciNet  MATH  Google Scholar 

  9. J.M. Cascon, C. Kreuzer, R.H. Nochetto, K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46(5), 2524–2550 (2008). doi:10.1137/07069047X.

    Article  MathSciNet  MATH  Google Scholar 

  10. S.H. Christiansen, Résolution des équations intégrales pour la diffraction d’ondes acoustiques et électromagnétiques: Stabilisation d’algorithmes itératifs et aspects de l’analyse numérique. Ph.D. thesis, École Polytechnique, 2002. http://tel.archives-ouvertes.fr/tel-00004520/.

  11. S.H. Christiansen, R. Winther, Smoothed projections in finite element exterior calculus, Math. Comput. 77(262), 813–829 (2008). doi:10.1090/S0025-5718-07-02081-9.

    MathSciNet  MATH  Google Scholar 

  12. K. Deckelnick, G. Dziuk, Convergence of a finite element method for non-parametric mean curvature flow, Numer. Math. 72(2), 197–222 (1995). doi:10.1007/s002110050166.

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Deckelnick, G. Dziuk, Numerical approximation of mean curvature flow of graphs and level sets, in Mathematical Aspects of Evolving Interfaces, Funchal, 2000, Lecture Notes in Math., vol. 1812, pp. 53–87 (Springer, Berlin, 2003).

    Chapter  Google Scholar 

  14. K. Deckelnick, G. Dziuk, C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer. 14, 139–232 (2005). doi:10.1017/S0962492904000224.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal. 47(2), 805–827 (2009). doi:10.1137/070708135.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Demlow, G. Dziuk, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal. 45(1), 421–442 (2007) (electronic). doi:10.1137/050642873.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations, Lecture Notes in Math., vol. 1357, pp. 142–155 (Springer, Berlin, 1988). doi:10.1007/BFb0082865.

    Chapter  Google Scholar 

  18. G. Dziuk, An algorithm for evolutionary surfaces, Numer. Math. 58(6), 603–611 (1991). doi:10.1007/BF01385643.

    MathSciNet  MATH  Google Scholar 

  19. G. Dziuk, C.M. Elliott, Finite elements on evolving surfaces, IMA J. Numer. Anal. 27(2), 262–292 (2007). doi:10.1093/imanum/drl023.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Dziuk, J.E. Hutchinson, Finite element approximations to surfaces of prescribed variable mean curvature, Numer. Math. 102(4), 611–648 (2006). doi:10.1007/s00211-005-0649-7.

    Article  MathSciNet  MATH  Google Scholar 

  21. P.W. Gross, P.R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach. Mathematical Sciences Research Institute Publications, vol. 48 (Cambridge University Press, Cambridge, 2004).

    Book  MATH  Google Scholar 

  22. M. Holst, Adaptive numerical treatment of elliptic systems on manifolds, Adv. Comput. Math. 15(1–4), 139–191 (2001). doi:10.1023/A:1014246117321.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Holst, G. Tsogtgerel, Y. Zhu, Local convergence of adaptive methods for nonlinear partial differential equations. arXiv:1001.1382 [math.NA] (2010)

  24. S. Lang, Introduction to Differentiable Manifolds, 2nd edn. Universitext (Springer, New York, 2002).

    MATH  Google Scholar 

  25. J.M. Lee, Riemannian Manifolds. Graduate Texts in Mathematics, vol. 176 (Springer, New York, 1997).

    MATH  Google Scholar 

  26. P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18(5), 707–737 (2008). doi:10.1142/S0218202508002838.

    Article  MathSciNet  MATH  Google Scholar 

  27. J.C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in ℝ3, Comput. Methods Appl. Mech. Eng. 8(1), 61–80 (1976).

    Article  MATH  Google Scholar 

  28. J.C. Nédélec, Mixed finite elements in ℝ3, Numer. Math. 35(3), 315–341 (1980). doi:10.1007/BF01396415.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.C. Nédélec, A new family of mixed finite elements in ℝ3, Numer. Math. 50(1), 57–81 (1986). doi:10.1007/BF01389668.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Stern, L p change of variables inequalities on manifolds. Math. Inequal. Appl. (2012, in press). arXiv:1004.0401 [math.DG].

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ari Stern.

Additional information

Communicated by Douglas Arnold.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holst, M., Stern, A. Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces. Found Comput Math 12, 263–293 (2012). https://doi.org/10.1007/s10208-012-9119-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-012-9119-7

Keywords

Mathematics Subject Classification

Navigation