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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 16
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Research Article

The elasticity complex: compact embeddings and regular decompositions

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Pages 4393-4421 | Received 06 Jul 2020, Accepted 30 Sep 2021, Published online: 16 Sep 2022

References

  • Pauly D, Zulehner W. The divDiv-complex and applications to biharmonic equations. Appl Anal. 2020;99(9):1579–1630.
  • Pauly D. On the Maxwell constants in 3D. Math Meth Appl Sci. 2017;40(2):435–447.
  • Pauly D. A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A0∗- A1-lemma in Hilbert spaces. Analysis (Berlin). 2019;39:33–58.
  • Pauly D. On the Maxwell and Friedrichs/Poincare constants in ND. Math Z. 2019;293(3):957–987.
  • Pauly D. Solution theory, variational formulations, and functional a posteriori error estimates for general first order systems with applications to electro-magneto-statics and more. Numer Funct Anal Optim. 2020;41(1):16–112.
  • Pauly D, Zulehner W. The elasticity complex: compact embeddings and regular decompositions. Preprint 2020 Aug. Available from: arXiv:2001.11007v4. e-prints.
  • Weck N. Maxwell's boundary value problems on Riemannian manifolds with nonsmooth boundaries. J Math Anal Appl. 1974;46:410–437.
  • Weber C. A local compactness theorem for Maxwell's equations. Math Meth Appl Sci. 1980;2:12–25.
  • Picard R. An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math Z. 1984;187:151–164.
  • Bauer S, Pauly D, Schomburg M. The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions. SIAM J Math Anal. 2016;48(4):2912–2943.
  • Bauer S, Pauly D, Schomburg M. Weck's selection theorem: the Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions. Preprint 2018. Available from: https://arxiv.org/abs/1809.01192 .
  • Bauer S, Pauly D, Schomburg M. Weck's selection theorem: the Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions. Maxwell's Equations: Analysis and Numerics. De Gruyter; 2019. p. 77–104. (Radon Series on Computational and Applied Mathematics).
  • Amrouche C, Ciarlet PG, Gratie L, et al. On Saint Venant's compatibility conditions and Poincaré's lemma. C R Math Acad Sci Paris. 2006;342(11):887–891.
  • Amrouche C, Ciarlet PG, Gratie L, et al. On the characterizations of matrix fields as linearized strain tensor fields. J Math Pures Appl (9). 2006;86(2):116–132.
  • Arnold DN. Finite element exterior calculus. Vol. 93, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); 2018.
  • Arnold DN, Awanou G, Winther R. Finite elements for symmetric tensors in three dimensions. Math Comput. 2008;77(263):1229–1251.
  • Arnold DN, Falk RS, Winther R. Differential complexes and stability of finite element methods. II: the elasticity complex. Compatible spatial discretizations. Papers presented at IMA hot topics workshop: compatible spatial discretizations for partial differential equations, Minneapolis, MN, USA, May 11–15, 2004. New York, NY: Springer; 2006. p. 47–67.
  • Arnold DN, Falk RS, Winther R. Finite element exterior calculus, homological techniques, and applications. Acta Numer. 2006;15:1–155.
  • Arnold DN, Falk RS, Winther R. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math Comput. 2007;76(260):1699–1723.
  • Arnold DN, Falk RS, Winther R. Finite element exterior calculus: from Hodge theory to numerical stability. Bull Am Math Soc New Ser. 2010;47(2):281–354.
  • Arnold DN, Winther R. Mixed finite elements for elasticity. Numer Math. 2002;92(3):401–419.
  • Ciarlet PG, Ciarlet P, Geymonat G, et al. Characterization of the kernel of the operator CURL  CURL. C R Math Acad Sci Paris. 2007;344:305–308.
  • Eastwood M. Variations on the de Rham complex. Notices Am Math Soc. 1999;46(11):1368–1376.
  • Eastwood M. A complex from linear elasticity. The Proceedings of the 19th Winter School “Geometry and Physics”, Srní, Czech Republic, January 9–15, 1999; Palermo: Circolo Matematico di Palermo; 2000. p. 23–29.
  • Geymonat G, Krasucki F. Beltrami's solutions of general equilibrium equations in continuum mechanics. C R Math Acad Sci Paris. 2006;342(5):359–363.
  • Geymonat G, Krasucki F. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Commun Pure Appl Anal. 2009;8(1):295–309.
  • Pauly D, Zulehner W. The elasticity complex: compact embeddings and regular decompositions. Preprint 2020 Jan. Available from: arXiv:2001.11007v1. e-prints.
  • Arnold DN, Hu K. Complexes from complexes. Preprint 2020. Available from: https://arxiv.org/abs/2005.12437v1.
  • Costabel M, McIntosh A. On Bogovskii a nd regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math Z. 2010;265(2):297–320.
  • Pauly D, Waurick M. The index of some mixed order Dirac-type operators and generalised Dirichlet–Neumann tensor fields. Preprint 2020. Available from: https://arxiv.org/abs/2005.07996, submitted.
  • Picard R. On the boundary value problems of electro- and magnetostatics. Proc Roy Soc Edinburgh Sect A. 1982;92:165–174.

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