Smoothed projections and mixed boundary conditions
Author:
Martin W. Licht
Journal:
Math. Comp. 88 (2019), 607-635
MSC (2010):
Primary 65N30; Secondary 58A12
DOI:
https://doi.org/10.1090/mcom/3330
Published electronically:
April 10, 2018
MathSciNet review:
3882278
Full-text PDF
View in AMS MathViewer
Abstract | References | Similar Articles | Additional Information
Abstract: Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions.
- Ana Alonso Rodríguez and Mirco Raffetto, Unique solvability for electromagnetic boundary value problems in the presence of partly lossy inhomogeneous anisotropic media and mixed boundary conditions, Math. Models Methods Appl. Sci. 13 (2003), no. 4, 597–611. MR 1976304, DOI https://doi.org/10.1142/S0218202503002672
- Douglas N. Arnold and Gerard Awanou, Finite element differential forms on cubical meshes, Math. Comp. 83 (2014), no. 288, 1551–1570. MR 3194121, DOI https://doi.org/10.1090/S0025-5718-2013-02783-4
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1137/16M1065951
- Francesca Bonizzoni, Annalisa Buffa, and Fabio Nobile, Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term, IMA J. Numer. Anal. 34 (2014), no. 4, 1328–1360. MR 3269428, DOI https://doi.org/10.1093/imanum/drt041
- Alain Bossavit, Computational electromagnetism, Electromagnetism, Academic Press, Inc., San Diego, CA, 1998. Variational formulations, complementarity, edge elements. MR 1488417
- Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235
- J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), no. 1, 88–132. MR 1174159, DOI https://doi.org/10.1016/0022-1236%2892%2990147-B
- A. Buffa, M. Costabel, and D. Sheen, On traces for ${\bf H}({\bf curl},\Omega )$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845–867. MR 1944792, DOI https://doi.org/10.1016/S0022-247X%2802%2900455-9
- Annalisa Buffa, Trace theorems on non-smooth boundaries for functional spaces related to Maxwell equations: an overview, Computational electromagnetics (Kiel, 2001) Lect. Notes Comput. Sci. Eng., vol. 28, Springer, Berlin, 2003, pp. 23–34. MR 1986130, DOI https://doi.org/10.1007/978-3-642-55745-3_3
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1090/S0025-5718-07-02081-9
- 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 https://doi.org/10.1007/s00209-009-0517-8
- Martin Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991), no. 2, 527–541. MR 1112332, DOI https://doi.org/10.1016/0022-247X%2891%2990104-8
- Leszek Demkowicz, Computing with $hp$-adaptive finite elements. Vol. 1, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. One and two dimensional elliptic and Maxwell problems; With 1 CD-ROM (UNIX). MR 2267112
- 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 https://doi.org/10.1007/s10208-014-9203-2
- P. Doktor, On the density of smooth functions in certain subspaces of Sobolev space, Comment. Math. Univ. Carolinae 14 (1973), 609–622. MR 336317
- Pavel Doktor and Alexander Ženíšek, The density of infinitely differentiable functions in Sobolev spaces with mixed boundary conditions, Appl. Math. 51 (2006), no. 5, 517–547. MR 2261637, DOI https://doi.org/10.1007/s10492-006-0019-5
- 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 https://doi.org/10.1515/cmam-2015-0034
- Richard S. Falk and Ragnar Winther, Local bounded cochain projections, Math. Comp. 83 (2014), no. 290, 2631–2656. MR 3246803, DOI https://doi.org/10.1090/S0025-5718-2014-02827-5
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- Paolo Fernandes and Gianni Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions, Math. Models Methods Appl. Sci. 7 (1997), no. 7, 957–991. MR 1479578, DOI https://doi.org/10.1142/S0218202597000487
- 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 https://doi.org/10.1007/s10958-010-0200-y
- V. M. Gol’dshtein, V. I. Kuz’minov, and I, A. Shvedov, Differential forms on Lipschitz manifolds, Siberian Mathematical Journal 23 (1982), no. 2, 151–161.
- 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 https://doi.org/10.1007/s10915-011-9513-3
- J. Gopalakrishnan and W. Qiu, Partial expansion of a Lipschitz domain and some applications, Front. Math. China (2011), 1–24.
- R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI https://doi.org/10.1017/S0962492902000041
- Tünde Jakab, Irina Mitrea, and Marius Mitrea, On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains, Indiana Univ. Math. J. 58 (2009), no. 5, 2043–2071. MR 2583491, DOI https://doi.org/10.1512/iumj.2009.58.3678
- F. Jochmann, A compactness result for vector fields with divergence and curl in $L^q(\Omega )$ involving mixed boundary conditions, Appl. Anal. 66 (1997), no. 1-2, 189–203. MR 1612136, DOI https://doi.org/10.1080/00036819708840581
- Frank Jochmann, Regularity of weak solutions of Maxwell’s equations with mixed boundary-conditions, Math. Methods Appl. Sci. 22 (1999), no. 14, 1255–1274. MR 1710708, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819990925%2922%3A14%3C1255%3A%3AAID-MMA83%3E3.0.CO%3B2-N
- Rainer Kreß, Ein kombiniertes Dirichlet-Neumannsches Randwertproblem bei harmonischen Vektorfeldern, Arch. Rational Mech. Anal. 42 (1971), 40–49 (German). MR 350033, DOI https://doi.org/10.1007/BF00282316
- P. Kuhn, Die Maxwellgleichung mit wechselnden Randbedingungen, Shaker, 1999.
- John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
- M. W. Licht, Smoothed projections over weakly Lipschitz domains, Math. of Comp. (2018). To appear in Math. Comp.
- M. W. Licht, Discrete distributional differential forms and their homology theory, Found. Comput. Math. (2016).
- 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 https://doi.org/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 https://doi.org/10.1006/jfan.2001.3870
- Dorina Mitrea, Marius Mitrea, and Mei-Chi Shaw, Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions, Indiana Univ. Math. J. 57 (2008), no. 5, 2061–2095. MR 2463962, DOI https://doi.org/10.1512/iumj.2008.57.3338
- Marius Mitrea, Dirichlet integrals and Gaffney-Friedrichs inequalities in convex domains, Forum Math. 13 (2001), no. 4, 531–567. MR 1830246, DOI https://doi.org/10.1515/form.2001.021
- Marius Mitrea, Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J. 125 (2004), no. 3, 467–547. MR 2166752, DOI https://doi.org/10.1215/S0012-7094-04-12322-1
- Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447
- J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer Science & Business Media, 2011.
- R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), no. 2, 151–164. MR 753428, DOI https://doi.org/10.1007/BF01161700
- J. Schöberl, A Multilevel Decomposition Result in ${H(\rm curl)}$, 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 https://doi.org/10.1090/S0025-5718-07-02030-3
- Chad Scott, $L^p$ theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995), no. 6, 2075–2096. MR 1297538, DOI https://doi.org/10.1090/S0002-9947-1995-1297538-7
- Ch. Weber, A local compactness theorem for Maxwell’s equations, Math. Methods Appl. Sci. 2 (1980), no. 1, 12–25. MR 561375, DOI https://doi.org/10.1002/mma.1670020103
- N. Weck, Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries, J. Math. Anal. Appl. 46 (1974), 410–437. MR 343771, DOI https://doi.org/10.1016/0022-247X%2874%2990250-9
- N. Weck, Traces of differential forms on Lipschitz boundaries, Analysis 24 (2004), no. 1-4, 147–170.
- Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
- Karl J. Witsch, A remark on a compactness result in electromagnetic theory, Math. Methods Appl. Sci. 16 (1993), no. 2, 123–129. MR 1200159, DOI https://doi.org/10.1002/mma.1670160205
Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 58A12
Retrieve articles in all journals with MSC (2010): 65N30, 58A12
Additional Information
Martin W. Licht
Affiliation:
Department of Mathematics, University of California San Diego, 9500 Gilman Drive MC0112, La Jolla, California 92093-0112
MR Author ID:
1225084
Email:
mlicht@ucsd.edu
Keywords:
Finite element exterior calculus,
Hodge Laplace equation,
smoothed projection,
partial boundary conditions,
mixed boundary conditions
Received by editor(s):
November 6, 2016
Received by editor(s) in revised form:
May 22, 2017, and January 1, 2017
Published electronically:
April 10, 2018
Additional Notes:
This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS
Article copyright:
© Copyright 2018
American Mathematical Society