On high order finite element spaces of differential forms

Christiansen, Snorre; Rapetti, Francesca

doi:10.1090/mcom/2995

On high order finite element spaces of differential forms
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by Snorre H. Christiansen and Francesca Rapetti PDF

Math. Comp. 85 (2016), 517-548 Request permission

Abstract:

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nédelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms.

References

Mark Ainsworth and Joe Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes, Internat. J. Numer. Methods Engrg. 58 (2003), no. 14, 2103–2130. MR 2022172, DOI 10.1002/nme.847
Douglas N. Arnold, Differential complexes and numerical stability, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 137–157. MR 1989182
Douglas N. Arnold, Spaces of finite element differential forms, Analysis and numerics of partial differential equations, Springer INdAM Ser., vol. 4, Springer, Milan, 2013, pp. 117–140. MR 3051399, DOI 10.1007/978-88-470-2592-9_{9}
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, DOI 10.1017/S0962492906210018
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, DOI 10.1016/j.cma.2008.12.017
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. MR 2594630, DOI 10.1090/S0273-0979-10-01278-4
Marcel Berger, Geometry. I, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882541, DOI 10.1007/978-3-540-93815-6
Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1–120. MR 2652780, DOI 10.1017/S0962492910000012
A. Bossavit, Mixed finite elements and the complex of Whitney forms, The mathematics of finite elements and applications, VI (Uxbridge, 1987) Academic Press, London, 1988, pp. 137–144. MR 956893
Alain Bossavit, Computational electromagnetism, Electromagnetism, Academic Press, Inc., San Diego, CA, 1998. Variational formulations, complementarity, edge elements. MR 1488417
A. Bossavit, Generating Whitney forms of polynomial degree one and higher, IEEE Trans. Magn. 38 (2002), no. 2, 341–344.
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Annalisa Buffa and Snorre H. Christiansen, A dual finite element complex on the barycentric refinement, Math. Comp. 76 (2007), no. 260, 1743–1769. MR 2336266, DOI 10.1090/S0025-5718-07-01965-5
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, DOI 10.1007/s00211-007-0081-2
Snorre H. Christiansen, A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739–757. MR 2413036, DOI 10.1142/S021820250800284X
Snorre H. Christiansen, Foundations of finite element methods for wave equations of Maxwell type, Applied wave mathematics, Springer, Berlin, 2009, pp. 335–393. MR 2553283, DOI 10.1007/978-3-642-00585-5_{1}7
Snorre H. Christiansen, Éléments finis mixtes minimaux sur les polyèdres, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 217–221 (French, with English and French summaries). MR 2600081, DOI 10.1016/j.crma.2010.01.017
Snorre Harald Christiansen, Upwinding in finite element systems of differential forms, Foundations of computational mathematics, Budapest 2011, London Math. Soc. Lecture Note Ser., vol. 403, Cambridge Univ. Press, Cambridge, 2013, pp. 45–71. MR 3137633
Snorre H. Christiansen, Hans Z. Munthe-Kaas, and Brynjulf Owren, Topics in structure-preserving discretization, Acta Numer. 20 (2011), 1–119. MR 2805152, DOI 10.1017/S096249291100002X
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. MR 2373181, DOI 10.1090/S0025-5718-07-02081-9
Snorre H. Christiansen and Ragnar Winther, On variational eigenvalue approximation of semidefinite operators, IMA J. Numer. Anal. 33 (2013), no. 1, 164–189. MR 3020954, DOI 10.1093/imanum/drs002
Martin Costabel and Alan McIntosh, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265 (2010), no. 2, 297–320. MR 2609313, DOI 10.1007/s00209-009-0517-8
L. Demkowicz and I. Babuška, $p$ interpolation error estimates for edge finite elements of variable order in two dimensions, SIAM J. Numer. Anal. 41 (2003), no. 4, 1195–1208. MR 2034876, DOI 10.1137/S0036142901387932
L. Demkowicz and A. Buffa, $H^1$, $H(\textrm {curl})$ and $H(\textrm {div})$-conforming projection-based interpolation in three dimensions. Quasi-optimal $p$-interpolation estimates, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 2-5, 267–296. MR 2105164, DOI 10.1016/j.cma.2004.07.007
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 488179
J. Gopalakrishnan, L. E. García-Castillo, and L. F. Demkowicz, Nédélec spaces in affine coordinates, Comput. Math. Appl. 49 (2005), no. 7-8, 1285–1294. MR 2141266, DOI 10.1016/j.camwa.2004.02.012
R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, DOI 10.1090/S0025-5718-99-01166-7
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
Robert C. Kirby, Low-complexity finite element algorithms for the de Rham complex on simplices, SIAM J. Sci. Comput. 36 (2014), no. 2, A846–A868. MR 3198602, DOI 10.1137/130927693
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Francesca Rapetti, Weights computation for simplicial Whitney forms of degree one, C. R. Math. Acad. Sci. Paris 341 (2005), no. 8, 519–523 (English, with English and French summaries). MR 2180821, DOI 10.1016/j.crma.2005.09.005
Francesca Rapetti and Alain Bossavit, Whitney forms of higher degree, SIAM J. Numer. Anal. 47 (2009), no. 3, 2369–2386. MR 2519607, DOI 10.1137/070705489
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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 523–639. MR 1115239
Joachim Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp. 77 (2008), no. 262, 633–649. MR 2373173, DOI 10.1090/S0025-5718-07-02030-3
Joachim Schöberl and Sabine Zaglmayr, High order Nédélec elements with local complete sequence properties, COMPEL 24 (2005), no. 2, 374–384. MR 2169504, DOI 10.1108/03321640510586015
André Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119–145 (French). MR 50280, DOI 10.1007/BF02564296
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148