Smoothed projections over weakly Lipschitz domains

Licht, Martin

doi:10.1090/mcom/3329

Smoothed projections over weakly Lipschitz domains
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by Martin W. Licht HTML | PDF

Math. Comp. 88 (2019), 179-210 Request permission

Abstract:

We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $L^p$ de Rham complexes over weakly Lipschitz domains onto finite element de Rham complexes. The projections satisfy uniform bounds for finite element spaces with bounded polynomial degree over shape-regular families of triangulations. Thus we extend the theory of finite element differential forms to polyhedral domains that are weakly Lipschitz but not strongly Lipschitz. As new mathematical tools, we use the collar theorem in the Lipschitz category, and we show that the degrees of freedom in finite element exterior calculus are flat chains in the sense of geometric measure theory.

References

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
Andreas Axelsson and Alan McIntosh, Hodge decompositions on weakly Lipschitz domains, Advances in analysis and geometry, Trends Math., Birkhäuser, Basel, 2004, pp. 3–29. MR 2077077
Sebastian Bauer, Dirk Pauly, and Michael Schomburg, The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions, SIAM J. Math. Anal. 48 (2016), no. 4, 2912–2943. MR 3542004, DOI 10.1137/16M1065951
Kevin Brewster and Marius Mitrea, Boundary value problems in weighted Sobolev spaces on Lipschitz manifolds, Mem. Differ. Equ. Math. Phys. 60 (2013), 15–55 (English, with English and Georgian summaries). MR 3288169
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, 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
Alan Demlow and Anil N. Hirani, A posteriori error estimates for finite element exterior calculus: the de Rham complex, Found. Comput. Math. 14 (2014), no. 6, 1337–1371. MR 3273681, DOI 10.1007/s10208-014-9203-2
Alexandre Ern and Jean-Luc Guermond, Mollification in strongly Lipschitz domains with application to continuous and discrete de Rham complexes, Comput. Methods Appl. Math. 16 (2016), no. 1, 51–75. MR 3441095, DOI 10.1515/cmam-2015-0034
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Riccardo Ghiloni, On the complexity of collaring theorem in the Lipschitz category, Raag preprint no. 315, available at http://www.maths.manchester.ac.uk/raag/index.php?preprint=0315, 2010.
V. Gol’dshtein, I. Mitrea, and M. Mitrea, Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds, J. Math. Sci. (N.Y.) 172 (2011), no. 3, 347–400. Problems in mathematical analysis. No. 52. MR 2839867, DOI 10.1007/s10958-010-0200-y
V. M. Gol’dshtein, V. I. Kuz’minov, and I. A. Shvedov, Differential forms on Lipschitz manifolds, Sib. Math. J. 23 (1982), no. 2, 151–161.
Vladimir Gol′dshtein and Marc Troyanov, Sobolev inequalities for differential forms and $L_{q,p}$-cohomology, J. Geom. Anal. 16 (2006), no. 4, 597–631. MR 2271946, DOI 10.1007/BF02922133
J. Gopalakrishnan and M. Oh, Commuting smoothed projectors in weighted norms with an application to axisymmetric Maxwell equations, J. Sci. Comput. 51 (2012), no. 2, 394–420. MR 2902212, DOI 10.1007/s10915-011-9513-3
Jay Gopalakrishnan and Weifeng Qiu, Partial expansion of a Lipschitz domain and some applications, Front. Math. China 7 (2012), no. 2, 249–272. MR 2897704, DOI 10.1007/s11464-012-0189-2
P. Grisvard, Boundary Value Problems in Non-Smooth Domains, Lecture Notes, vol. 19, Department of Mathematics, University of Maryland, MD, 1980.
Steve Hofmann, Marius Mitrea, and Michael Taylor, Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains, J. Geom. Anal. 17 (2007), no. 4, 593–647. MR 2365661, DOI 10.1007/BF02937431
John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), no. 1, 85–122. MR 515647, DOI 10.5186/aasfm.1977.0315
Dorina Mitrea and Marius Mitrea, Finite energy solutions of Maxwell’s equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal. 190 (2002), no. 2, 339–417. MR 1899489, DOI 10.1006/jfan.2001.3870
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. Schöberl, A multilevel decomposition result in $H(\operatorname {curl})$, Proceedings of the 8th European Multigrid Conference, EMG, 2005.
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
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148