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Foundations of Computational Mathematics
Published in:

20-09-2022

The Bubble Transform and the de Rham Complex

Authors: Richard S. Falk, Ragnar Winther

Published in: Foundations of Computational Mathematics | Issue 1/2024

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Abstract

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh \({{\mathscr {T}}}\) of the domain \(\varOmega \), we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in \(L^2\) and also on the appropriate subspace consisting of k-forms with exterior derivatives in \(L^2\).
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Literature
1.
Douglas N. Arnold, Finite Element Exterior Calculus, CBMS-NSF Regional Conf. Ser. in Appl. Math. 93, SIAM, Philadelphia, 2018. xi+120 pp.
2.
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), 1–155.ADSMathSciNetCrossRef
3.
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.ADSMathSciNetCrossRef
4.
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.MathSciNetCrossRef
5.
Douglas N. Arnold and Johnny Guzmán, Local\(L^2\)-bounded commuting projections in FEEC, ESAIM Math. Model. Numer. Anal. 55 (2021), no. 5, 2169–2184.MathSciNetCrossRef
6.
Ivo Babuška and Manil Suri, The optimal convergence rate of the\(p\)-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776.ADSMathSciNetCrossRef
7.
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.
8.
Daniele Boffi, Martin Costabel, Monique Dauge, Leszek Demkowicz, and Ralf Hiptmair, Discrete compactness for the\(p\)-version of discrete differential forms, SIAM J. Numer. Anal. 49 (2011), no. 1, 135–158.MathSciNetCrossRef
9.
Phillipe G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, Vol. 4. North-Holland Publishing Co., Amsterdam-New York-Oxford, (1978), xix+530 pp.,
10.
Snorre H. Christiansen and Ragnar Winther, Smoothed projections in finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829.MathSciNet
11.
Leszek Demkowicz and Ivo 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.MathSciNetCrossRef
12.
Leszek Demkowicz and Annalisa 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.
13.
Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl, Polynomial extension operators. I, SIAM J. Numer. Anal. 46 (2008), no. 6, 3006–3031.MathSciNetCrossRef
14.
Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl, Polynomial extension operators. II, SIAM J. Numer. Anal. 47 (2009), no. 5, 3293–3324.MathSciNetCrossRef
15.
Leszek Demkowicz, Jayadeep Gopalakrishnan, and Joachim Schöberl, Polynomial extension operators. Part III, Math. Comp. 81 (2012), no. 279, 1289–1326.MathSciNetCrossRef
16.
Alexandre Ern, Thirupathi Gudi, Iain Smears, and Martin Vohralík, Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div), IMA J. Numer. Anal., 42, (2022), no, 2, 1023–1049.MathSciNetCrossRef
17.
Richard S. Falk and Ragnar Winther, Local bounded cochain projections, Math. Comp. 83, (2014), no. 290, 2631–2656.MathSciNetCrossRef
18.
Richard S. Falk and Ragnar Winther, Double complexes and local cochain projections Numer. Methods Partial Differential Equations 31 (2015), no. 2, 541–551.MathSciNetCrossRef
19.
Richard S. Falk and Ragnar Winther, The Bubble Transform: A New Tool for Analysis of Finite Element Methods, Found. Comput. Math., 16 (2016), no. 1, 297–328MathSciNetCrossRef
20.
Jens M. Melenk and C. Rojik, On commuting p-version projection-based interpolation on tetrahedra, Math. Comp. 89 (2020), 45–87.
21.
Rafael Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the\(h\)-\(p\)version of the finite element method in three dimensions, SIAM J. Numer. Anal. 34 (1997), no. 1, 282–314.MathSciNetCrossRef
22.
Joachim Schöberl, A multilevel decomposition result in H(curl). In: Proceedings from the 8th European Multigrid, Multilevel, and Multiscale Conference, edited by P.H.P. Wesseling and C.W. Oosterlee (2005).
23.
Joachim Schöberl, Jens M. Melenk, Clemens Pechstein, and Sabine Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements, 28(1) (2008), IMA J. Numer. Anal. 28 (2008), no. 1, 1–24.
Metadata
Title
The Bubble Transform and the de Rham Complex
Authors
Richard S. Falk
Ragnar Winther
Publication date
20-09-2022
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 1/2024
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-022-09589-1

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