×

Finite element exterior calculus: From Hodge theory to numerical stability. (English) Zbl 1207.65134

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 the authors consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are the key tools for exploring the well-posedness of the continuous problem. The discretization methods they consider are finite element methods, in which a variational or weak formulation of the partial differential equation problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace.
An abstract Hilbert space framework is developed for analysing 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.
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, the authors 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. The Koszul complex is used to construct two families of finite element differential forms. The authors show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each they 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.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
58A14 Hodge theory in global analysis
14F40 de Rham cohomology and algebraic geometry
58A12 de Rham theory in global analysis
58A15 Exterior differential systems (Cartan theory)
58J10 Differential complexes
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[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. · Zbl 1125.74382 · doi: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). , https://doi.org/10.1002/(SICI)1099-1476(199806)21:93.0.CO;2-B · Zbl 0914.35094
[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. · Zbl 1285.74013
[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. · Zbl 0633.73074 · doi: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. · Zbl 0558.73066 · doi: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. · Zbl 1119.65398 · doi: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. · Zbl 1119.65399 · doi: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. · Zbl 1185.65204 · doi: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. · Zbl 1118.74046
[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. · Zbl 1227.65091 · doi: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. · Zbl 1090.74051 · doi:10.1007/s002110100348
[12] V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. V. I. Arnol\(^{\prime}\)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.
[13] Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/1971), 322 – 333. · Zbl 0214.42001 · doi: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. · Zbl 0875.65087
[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. · Zbl 1110.65103 · doi: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. · Zbl 0983.65125 · doi: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. · Zbl 1110.65104 · doi:10.1007/0-387-38034-5_6
[19] Daniele Boffi, Approximation of eigenvalues in mixed form, discrete compactness property, and application to \?\? mixed finite elements, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3672 – 3681. · Zbl 1173.65349 · doi: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. · Zbl 0938.65126
[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. · Zbl 0496.55001
[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. · Zbl 0201.07803 · doi: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). · Zbl 0338.90047
[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). · Zbl 0736.73062 · doi: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. · Zbl 0599.65072 · doi: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. · Zbl 1108.65102 · doi:10.1137/040613950
[28] J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88 – 132. · Zbl 0826.46065 · doi: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.
[30] Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259 – 322. , https://doi.org/10.2307/1971113 Werner Müller, Analytic torsion and \?-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233 – 305. · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0
[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. · Zbl 1127.65085 · doi: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. · Zbl 1140.65081
[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. · Zbl 0383.65058
[34] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77 – 84 (English, with Loose French summary). · Zbl 0368.65008
[35] Martin Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991), no. 2, 527 – 541. · Zbl 0738.35095 · doi: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). · JFM 54.0486.01 · doi:10.1007/BF01448839
[37] L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz, de Rham diagram for \?\? finite element spaces, Comput. Math. Appl. 39 (2000), no. 7-8, 29 – 38. · Zbl 0955.65084 · doi: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. · Zbl 1080.39021
[39] Jozef Dodziuk, Finite-difference approach to the Hodge theory of harmonic forms, Amer. J. Math. 98 (1976), no. 1, 79 – 104. · Zbl 0324.58001 · doi: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). · Zbl 0435.58004
[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. · Zbl 0624.65109
[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. · Zbl 0965.58029
[44] Anders Logg et al., The FEniCS project, http://www.fenics.org. · Zbl 1158.74048
[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. · Zbl 1006.65105
[47] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002
[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). · Zbl 0467.65062
[49] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[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 , 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. · Zbl 0042.10205
[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. · Zbl 1037.46029 · doi: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. · Zbl 1088.65015 · doi: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).
[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). · Zbl 1144.65008
[57] R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325 – 1346. · Zbl 0938.65132
[58] -, Higher order Whitney forms, Geometrical Methods in Computational Electromagnetics , 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. · Zbl 1123.78320 · doi: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. · Zbl 1115.35005
[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. · Zbl 0964.58002
[63] C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103 – 116. · Zbl 0427.73072 · doi:10.1007/BF01403910
[64] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009
[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. · Zbl 0698.65067
[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. · Zbl 0824.58003
[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. · Zbl 0780.18009
[69] Anders Logg and Kent-Andre Mardal, Finite element exterior calculus, http://www.fenics.org/wiki/Finite_Element_Exterior_Calculus, 2009. · Zbl 1247.65105
[70] Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. · Zbl 1024.78009
[71] Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259 – 322. , https://doi.org/10.2307/1971113 Werner Müller, Analytic torsion and \?-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), no. 3, 233 – 305. · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0
[72] J.-C. Nédélec, Mixed finite elements in \?³, Numer. Math. 35 (1980), no. 3, 315 – 341. · Zbl 0419.65069 · doi:10.1007/BF01396415
[73] J.-C. Nédélec, A new family of mixed finite elements in \?³, Numer. Math. 50 (1986), no. 1, 57 – 81. · Zbl 0625.65107 · doi: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. · Zbl 1110.65024 · doi: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. · Zbl 0527.58038 · doi: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.
[77] Yves Renard and Julien Pommier, Getfem++, http://home.gna.org/getfem/.
[78] Joachim Schöberl, NGSolve - \( 3\)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. · Zbl 1136.78016
[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. · Zbl 0563.65072 · doi:10.1007/BF01389651
[81] Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513 – 538. · Zbl 0632.73063 · doi: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.
[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.
[84] Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269 – 331 (1978). · Zbl 0374.57002
[85] Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. · Zbl 0869.35001
[86] John von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021 – 1099. · Zbl 0031.31402
[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. · Zbl 0083.28204
[89] Jinchao Xu and Ludmil Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195 – 202. · Zbl 1028.65115 · doi:10.1007/s002110100308
[90] Kōsaku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.