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Local bounded cochain projections

Authors: Richard S. Falk and Ragnar Winther
Journal: Math. Comp. 83 (2014), 2631-2656
MSC (2010): Primary 65N30
Published electronically: March 11, 2014
MathSciNet review: 3246803
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Abstract: We construct projections from $ H \Lambda ^k(\Omega )$, the space of differential $ k$ forms on $ \Omega $ which belong to $ L^2(\Omega )$ and whose exterior derivative also belongs to $ L^2(\Omega )$, to finite dimensional subspaces of $ H \Lambda ^k(\Omega )$ consisting of piecewise polynomial differential forms defined on a simplicial mesh of $ \Omega $. Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the $ H \Lambda ^k(\Omega )$ norm independent of the mesh size $ h$. Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clément interpolant. A double complex structure is introduced as a key tool to carry out the construction.

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  • [1] 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 (2007j:58002),
  • [2] 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 (2010b:58002),
  • [3] 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 (2011f:58005),
  • [4] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York, 1982. MR 658304 (83i:57016)
  • [5] 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 (51 #1540)
  • [6] 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 (92d:65187)
  • [7] 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 (87g:65133),
  • [8] W. Cao and L. Demkowicz, Optimal error estimate of a projection based interpolation for the $ p$-version approximation in three dimensions, Comput. Math. Appl. 50 (2005), no. 3-4, 359-366. MR 2165425 (2006d:65121),
  • [9] 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 (2008c:65318),
  • [10] Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813-829. MR 2373181 (2009a:65310),
  • [11] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique 9 (1975), no. R-2, 77-84 (English, with Loose French summary). MR 0400739 (53 #4569)
  • [12] Daniele Boffi, Franco Brezzi, Leszek F. Demkowicz, Ricardo G. Durán, Richard S. Falk, and Michel Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics, vol. 1939, Springer-Verlag, Berlin, 2008. Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26-July 1, 2006; Edited by Boffi and Lucia Gastaldi. MR 2459075 (2010h:65219)
  • [13] 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 (2004m:65191),
  • [14] L. Demkowicz and A. Buffa, $ H^1$, $ H({\rm curl})$ and $ H({\rm 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 (2005j:65139),
  • [15] L. Demkowicz and J. Kurtz, Projection-based interpolation and automatic $ hp$-adaptivity for finite element discretizations of elliptic and Maxwell problems, ESAIM Proceedings. Vol. 21 (2007) [Journées d'Analyse Fonctionnelle et Numérique en l'honneur de Michel Crouzeix], ESAIM Proc., vol. 21, EDP Sci., Les Ulis, 2007, pp. 1-15. MR 2404049 (2009c:65300),
  • [16] Alan Demlow, Localized pointwise error estimates for mixed finite element methods, Math. Comp. 73 (2004), no. 248, 1623-1653 (electronic). MR 2059729 (2005e:65184),
  • [17] A. Demlow and A. N. Hirani, A posteriori error estimates for finite element exterior calculus: The de Rham complex, arXiv:1203.0803 [math.NA] (2012).
  • [18] J.-C. Nédélec, Mixed finite elements in $ {\bf R}^{3}$, Numer. Math. 35 (1980), no. 3, 315-341. MR 592160 (81k:65125),
  • [19] J.-C. Nédélec, A new family of mixed finite elements in $ {\bf R}^3$, Numer. Math. 50 (1986), no. 1, 57-81. MR 864305 (88e:65145),
  • [20] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292-315. Lecture Notes in Math., Vol. 606. MR 0483555 (58 #3547)
  • [21] J. Schöberl, A multilevel decomposition result in H(curl), in Multigrid, Multilevel and Multiscale Methods, EMG 2005 CD, Eds: P. Wesseling, C.W. Oosterlee, P. Hemker, ISBN 90-9020969-7
  • [22] Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148 (19,309c)

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Additional Information

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, 0316 Oslo, Norway

Keywords: Cochain projections, finite element exterior calculus
Received by editor(s): November 22, 2012
Received by editor(s) in revised form: April 15, 2013
Published electronically: March 11, 2014
Additional Notes: The work of the first author was supported in part by NSF grant DMS-0910540.
The work of the second author was supported by the Norwegian Research Council.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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