Finite element exterior calculus: from Hodge theory to numerical stability

Arnold, Douglas; Falk, Richard; Winther, Ragnar

doi:10.1090/S0273-0979-10-01278-4

Finite element exterior calculus: from Hodge theory to numerical stability

Authors: Douglas N. Arnold, Richard S. Falk and Ragnar Winther
Journal: Bull. Amer. Math. Soc. 47 (2010), 281-354
MSC (2000): Primary 65N30, 58A14
DOI: https://doi.org/10.1090/S0273-0979-10-01278-4
Published electronically: January 25, 2010
MathSciNet review: 2594630
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Abstract: This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge-theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

1. Scot Adams and Bernardo Cockburn, A mixed finite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005), no. 3, 515–521. MR 2221175, https://doi.org/10.1007/s10915-004-4807-3
2. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, https://doi.org/10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
3. Douglas N. Arnold, Gerard Awanou, and Ragnar Winther, Finite elements for symmetric tensors in three dimensions, Math. Comp. 77 (2008), no. 263, 1229–1251. MR 2398766, https://doi.org/10.1090/S0025-5718-08-02071-1
4. Douglas N. Arnold, Franco Brezzi, and Jim Douglas Jr., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347–367. MR 840802, https://doi.org/10.1007/BF03167064
5. Douglas N. Arnold, Jim Douglas Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. MR 761879, https://doi.org/10.1007/BF01379659
6. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. I. The de Rham complex, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 24–46. MR 2249344, https://doi.org/10.1007/0-387-38034-5
7. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. II. The elasticity complex, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 47–67. MR 2249345, https://doi.org/10.1007/0-387-38034-5_3
8. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, https://doi.org/10.1017/S0962492906210018
9. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007), no. 260, 1699–1723. MR 2336264, https://doi.org/10.1090/S0025-5718-07-01998-9
10. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 21-26, 1660–1672. MR 2517938, https://doi.org/10.1016/j.cma.2008.12.017
11. Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), no. 3, 401–419. MR 1930384, https://doi.org/10.1007/s002110100348
12. V. I. Arnol′d, Mathematical methods of classical mechanics, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR 0690288
13. Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322–333. MR 0288971, https://doi.org/10.1007/BF02165003
14. I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
15. Garth A. Baker, Combinatorial Laplacians and Sullivan-Whitney forms, Differential geometry (College Park, Md., 1981/1982) Progr. Math., vol. 32, Birkhäuser, Boston, Mass., 1983, pp. 1–33. MR 702525
16. Pavel B. Bochev and James M. Hyman, Principles of mimetic discretizations of differential operators, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 89–119. MR 2249347, https://doi.org/10.1007/0-387-38034-5_5
17. D. Boffi, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett. 14 (2001), no. 1, 33–38. MR 1793699, https://doi.org/10.1016/S0893-9659(00)00108-7
18. Daniele Boffi, Compatible discretizations for eigenvalue problems, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 121–142. MR 2249348, https://doi.org/10.1007/0-387-38034-5_6
19. Daniele Boffi, Approximation of eigenvalues in mixed form, discrete compactness property, and application to ℎ𝑝 mixed finite elements, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3672–3681. MR 2339993, https://doi.org/10.1016/j.cma.2006.10.024
20. Daniele Boffi, Franco Brezzi, and Lucia Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), no. 229, 121–140. MR 1642801, https://doi.org/10.1090/S0025-5718-99-01072-8
21. Alain Bossavit, Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, IEE Trans. Mag. 135, Part A (1988), 493-500.
22. Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
23. J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 0263214, https://doi.org/10.1137/0707006
24. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with loose French summary). MR 0365287
25. Franco Brezzi and Klaus-Jürgen Bathe, A discourse on the stability conditions for mixed finite element formulations, Comput. Methods Appl. Mech. Engrg. 82 (1990), no. 1-3, 27–57. Reliability in computational mechanics (Austin, TX, 1989). MR 1077650, https://doi.org/10.1016/0045-7825(90)90157-H
26. Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, https://doi.org/10.1007/BF01389710
27. Franco Brezzi, Konstantin Lipnikov, and Mikhail Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43 (2005), no. 5, 1872–1896. MR 2192322, https://doi.org/10.1137/040613950
28. J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88–132. MR 1174159, https://doi.org/10.1016/0022-1236(92)90147-B
29. J. G. Charney, R. Fjörtoft, and J. von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2 (1950), 237–254. MR 0042799, https://doi.org/10.3402/tellusa.v2i4.8607
30. Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, https://doi.org/10.2307/1971113
31. Snorre H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension, Numer. Math. 107 (2007), no. 1, 87–106. MR 2317829, https://doi.org/10.1007/s00211-007-0081-2
32. Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, https://doi.org/10.1090/S0025-5718-07-02081-9
33. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
34. Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
35. Martin Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991), no. 2, 527–541. MR 1112332, https://doi.org/10.1016/0022-247X(91)90104-8
36. R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), no. 1, 32–74 (German). MR 1512478, https://doi.org/10.1007/BF01448839
37. L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz, de Rham diagram for ℎ𝑝 finite element spaces, Comput. Math. Appl. 39 (2000), no. 7-8, 29–38. MR 1746160, https://doi.org/10.1016/S0898-1221(00)00062-6
38. Mathieu Desbrun, Anil N. Hirani, Melvin Leok, and Jerrold E. Marsden, Discrete exterior calculus, 2005, available from arXiv.org/math.DG/0508341.
39. Jozef Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms, Amer. J. Math. 98 (1976), no. 1, 79–104. MR 0407872, https://doi.org/10.2307/2373615
40. J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1–52 (1977). MR 0488179
41. Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39–52. MR 771029, https://doi.org/10.1090/S0025-5718-1985-0771029-9
42. Patrick Dular and Christophe Geuzaine, GetDP: A general environment for the treatment of discrete problems, http://geuz.org/getdp/.
43. Michael Eastwood, A complex from linear elasticity, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), 2000, pp. 23–29. MR 1758075
44. Anders Logg et al., The FEniCS project, http://www.fenics.org.
45. Daniel White et al., EMSolve: Unstructured grid computational electromagnetics using mixed finite element methods, https://www-eng.llnl.gov/emsolve/emsolve_home.html.
46. Wolfgang Bangerth et al., deal.II: A finite element differential equations analysis library, http://www.dealii.org.
47. Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
48. R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249–277 (English, with French summary). MR 592753
49. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
50. Matthew Fisher, Peter Schröder, Mathieu Desbrun, and Hugues Hoppe, Design of tangent vector fields, SIGGRAPH '07: ACM SIGGRAPH 2007 Papers (New York), ACM, 2007, paper 56.
51. Badouin M. Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method, Stress Analysis (O. C. Zienkiewicz and G. S. Holister, eds.), Wiley, New York, 1965, pp. 145-197.
52. Matthew P. Gaffney, The harmonic operator for exterior differential forms, Proc. Nat. Acad. Sci. U. S. A. 37 (1951), 48–50. MR 0048138
53. N. V. Glotko, On the complex of Sobolev spaces associated with an abstract Hilbert complex, Sibirsk. Mat. Zh. 44 (2003), no. 5, 992–1014 (Russian, with Russian summary); English transl., Siberian Math. J. 44 (2003), no. 5, 774–792. MR 2019553, https://doi.org/10.1023/A:1025924417135
54. Steven J. Gortler, Craig Gotsman, and Dylan Thurston, Discrete one-forms on meshes and applications to 3D mesh parameterization, Comput. Aided Geom. Design 23 (2006), no. 2, 83–112. MR 2189438, https://doi.org/10.1016/j.cagd.2005.05.002
55. M. Gromov and M. A. Shubin, Near-cohomology of Hilbert complexes and topology of non-simply connected manifolds, Astérisque 210 (1992), 9–10, 283–294. Méthodes semi-classiques, Vol. 2 (Nantes, 1991). MR 1221363
56. Xianfeng David Gu and Shing-Tung Yau, Computational conformal geometry, Advanced Lectures in Mathematics (ALM), vol. 3, International Press, Somerville, MA; Higher Education Press, Beijing, 2008. With 1 CD-ROM (Windows, Macintosh and Linux). MR 2439718
57. R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, https://doi.org/10.1090/S0025-5718-99-01166-7
58. -, Higher order Whitney forms, Geometrical Methods in Computational Electromagnetics (F. Teixeira, ed.), PIER, vol. 32, EMW Publishing, Cambridge, MA, 2001, pp. 271-299.
59. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, https://doi.org/10.1017/S0962492902000041
60. Lars Hörmander, The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR 2304165
61. inuTech GmbH, Diffpack: Expert tools for expert problems, http://www.diffpack.com.
62. Klaus Jänich, Vector analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. Translated from the second German (1993) edition by Leslie Kay. MR 1811820
63. C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103–116. MR 0483904, https://doi.org/10.1007/BF01403910
64. Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
65. Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
66. P. Robert Kotiuga, Hodge decompositions and computational electromagnetics, Ph.D. thesis in Electrical Engineering, McGill University, 1984.
67. Serge Lang, Differential and Riemannian manifolds, 3rd ed., Graduate Texts in Mathematics, vol. 160, Springer-Verlag, New York, 1995. MR 1335233
68. Jean-Louis Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970
69. Anders Logg and Kent-Andre Mardal, Finite element exterior calculus, http://www.fenics.org/wiki/Finite_Element_Exterior_Calculus, 2009.
70. Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447
71. Werner Müller, Analytic torsion and 𝑅-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233–305. MR 498252, https://doi.org/10.1016/0001-8708(78)90116-0
72. J.-C. Nédélec, Mixed finite elements in 𝑅³, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, https://doi.org/10.1007/BF01396415
73. J.-C. Nédélec, A new family of mixed finite elements in 𝑅³, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, https://doi.org/10.1007/BF01389668
74. R. A. Nicolaides and K. A. Trapp, Covolume discretization of differential forms, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 161–171. MR 2249350, https://doi.org/10.1007/0-387-38034-5_8
75. R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), no. 2, 151–164. MR 753428, https://doi.org/10.1007/BF01161700
76. P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
77. Yves Renard and Julien Pommier, Getfem++, http://home.gna.org/getfem/.
78. Joachim Schöberl, NGSolve - D finite element solver, http://www.hpfem.jku.at/ngsolve/.
79. Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, https://doi.org/10.1090/S0025-5718-07-02030-3
80. Rolf Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. Math. 48 (1986), no. 4, 447–462. MR 834332, https://doi.org/10.1007/BF01389651
81. Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, https://doi.org/10.1007/BF01397550
82. R. Stenberg, Two low-order mixed methods for the elasticity problem, The mathematics of finite elements and applications, VI (Uxbridge, 1987) Academic Press, London, 1988, pp. 271–280. MR 956898
83. D. Sullivan, Differential forms and the topology of manifolds, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 37–49. MR 0370611
84. Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 0646078
85. Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148
86. John von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021–1099. MR 0024235, https://doi.org/10.1090/S0002-9904-1947-08909-6
87. Ke Wang, Weiwei, Yiying Tong, Desbrun Mathieu, and Peter Schröder, Edge subdivision schemes and the construction of smooth vector fields, ACM Trans. on Graphics 25 (2006), 1041-1048.
88. Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
89. Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. MR 1971217, https://doi.org/10.1007/s002110100308
90. Kōsaku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR 1336382