Finite element exterior calculus: From hodge theory to numerical stability

Bulletin of the American Mathematical Society

Volumn 47, Issue 2, 2010, Pages 281-354

  Arnold, Douglas N ^a   Falk, Richard S ^b   Winther, Ragnar ^c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b RUTGERS UNIVERSITY   (United States)

^c UNIVERSITY OF OSLO   (Norway)

Author keywords

De Rham cohomology; Exterior calculus; Finite element exterior calculus; Hodge laplacian; Hodge theory; Mixed finite elements

Indexed keywords

EID: 77956502215     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-10-01278-4     Document Type: Article

References (90)

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.
2. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), 823-864.
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.
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), 347-367.
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), 1-22.
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.
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.
8. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1-155.
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.
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), 1660-1672.
11. Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), 401-419.
12. Vladimir I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
13. Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322-333.
14. Ivo Babuška and John Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991, pp. 641-787.
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.
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.
17. Daniele Boffi, A note on the de Rham complex and a discrete compactness property, Appl. Math. Lett. 14 (2001), 33-38.
18. Daniele Boffi. Compatible discretizations for eigenvalue problems, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 121-142.
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.
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), 121-140.
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, New York, 1982.
23. James H. Bramble and Stephen 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.
24. Franco 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), 129-151.
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), 27-57.
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), 217-235.
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 (electronic).
28. Jochen Brüning and Matthias Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88-132.
29. Jules G. Charney, Ragnar Fjörtoft, and John von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2 (1950), 237-254.
30. Jeffrey Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259-322.
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.
32. Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813-829.
33. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam, 1978.
34. Philippe Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, RAIRO Analyse Numérique 9 (1975), 77-84.
35. Martin Costabel, A coercive bilinear form for Maxwell's equations, J. Math. Anal. Appl. 157 (1991), 527-541.
36. Richard Courant, Kurt Friedrichs, and Hans Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928), no. 1, 32-74.
37. Leszek Demkowicz, Peter Monk, Leon Vardapetyan, and Waldemar Rachowicz, de Rham diagram for hp finite element spaces, Comput. Math. Appl. 39 (2000), 29-38.
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.
40. Jozef Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1-52.
41. Jim Douglas, Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), 39-52.
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, Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 23-29.
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.
48. Richard S. Falk and John E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), 249-277.
49. Herbert Federer, Geometric Measure Theory, Vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, 1969.
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. USA 37 (1951), 48-50.
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.
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.
55. M. Gromov and M. A. Shubin, Near-cohomology of Hilbert complexes and topology of nonsimply connected manifolds, Astérisque (1992), no. 210, 9-10, 283-294, Méthodes semiclassiques, Vol. 2 (Nantes, 1991).
56. Xianfeng David Gu and Shing-Tung Yau, Computational conformal geometry, Advanced Lectures in Mathematics (ALM), vol. 3, International Press, Somerville, MA, 2008, With 1 CDROM (Windows, Macintosh and Linux).
57. Ralf Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), 1325-1346.
58. Ralf Hiptmair, Higher order Whitney forms, Geometrical Methods in Computational Electromagnetics (F. Teixeira, ed.), PIER, vol. 32, EMW Publishing, Cambridge, MA, 2001, pp. 271-299.
59. Ralf Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, vol. 11, Cambridge University Press, Cambridge, 2002, pp. 237-339.
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.
61. inuTech GmbH, Diffpack: Expert tools for expert problems, http://www.diffpack.com.
62. Klaus Jänich, Vector Analysis, Undergraduate Texts in Mathematics, Springer, New York, 2001, Translated from the second German (1993) edition by Leslie Kay
63. Claes Johnson and Bertram Mercier, Some equilibrium finite element methods for twodimensional elasticity problems, Numer. Math. 30 (1978), 103-116.
64. Tosio Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1995, Reprint of the 1980 edition
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.
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, Graduate Texts in Mathematics, vol. 160, Springer, New York, 1995.
68. Jean-Louis Loday, Cyclic Homology, Grundlehren der Mathematischen Wissenschaften, vol. 301, Springer, Berlin, 1992.
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
71. Werner Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233-305.
72. 3, Numer. Math. 35 (1980), 315-341.
73. 3, Numer. Math. 50 (1986), 57-81.
74. Roy A. Nicolaides and Kathryn A. Trapp, Covolume discretization of differential forms, Compatible Spatial Discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 161-171.
75. Rainer Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), 151-164.
76. Pierre-Arnaud Raviart and Jean-Marie 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), Vol. 606 of Lecture Notes in Mathematics, Springer, Berlin, 1977, pp. 292-315.
77. Yves Renard and Julien Pommier, Getfem++, http://home.gna.org/getfem/.
78. Joachim Schöberl, NGSolve - 3D 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.
80. Rolf Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. Math. 48 (1986), 447-462.
81. Rolf Stenberg. A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), 513-538.
82. Rolf 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.
83. Dennis Sullivan, Differential forms and the topology of manifolds, Manifolds-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), Univ. Tokyo Press, Tokyo, 1975, pp. 37-49.
84. Dennis Sullivan. Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269-331 (1978).
85. Michael E. Taylor, Partial Differential Equations. I: Basic Theory, Applied Mathematical Sciences, vol. 115, Springer, New York, 1996.
86. John von Neumann and Herman H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021-1099.
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, NJ, 1957
89. Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195-202.
90. Kōsaku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the sixth (1980) edition