Analysis of weakly symmetric mixed finite elements for elasticity

Lederer, Philip; Stenberg, Rolf

doi:10.1090/mcom/3865

Analysis of weakly symmetric mixed finite elements for elasticity
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by Philip L. Lederer and Rolf Stenberg

Math. Comp. 93 (2024), 523-550

DOI: https://doi.org/10.1090/mcom/3865

Published electronically: June 14, 2023

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Abstract:

We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.

References

M. Amara and J. M. Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979), no. 4, 367–383. MR 553347, DOI 10.1007/BF01399320
Thomas Apel, Anisotropic finite elements: local estimates and applications, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1999. MR 1716824
D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32 (English, with French summary). MR 813687, DOI 10.1051/m2an/1985190100071
Douglas N. Arnold, Gerard Awanou, and Ragnar Winther, Finite elements for symmetric tensors in three dimensions, Math. Comp. 77 (2008), no. 263, 1229–1251. MR 2398766, DOI 10.1090/S0025-5718-08-02071-1
Douglas N. Arnold, Franco Brezzi, and Jim Douglas Jr., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math. 1 (1984), no. 2, 347–367. MR 840802, DOI 10.1007/BF03167064
Douglas N. Arnold, Jim Douglas Jr., and Chaitan P. Gupta, A family of higher order mixed finite element methods for plane elasticity, Numer. Math. 45 (1984), no. 1, 1–22. MR 761879, DOI 10.1007/BF01379659
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. II. The elasticity complex, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 47–67. MR 2249345, DOI 10.1007/0-387-38034-5_{3}
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, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007), no. 260, 1699–1723. MR 2336264, DOI 10.1090/S0025-5718-07-01998-9
Ivo Babuška and Manil Suri, Locking effects in the finite element approximation of elasticity problems, Numer. Math. 62 (1992), no. 4, 439–463. MR 1174468, DOI 10.1007/BF01396238
Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
Susanne C. Brenner, Korn’s inequalities for piecewise $H^1$ vector fields, Math. Comp. 73 (2004), no. 247, 1067–1087. MR 2047078, DOI 10.1090/S0025-5718-03-01579-5
C. Carstensen, G. Dolzmann, S. A. Funken, and D. S. Helm, Locking-free adaptive mixed finite element methods in linear elasticity, Comput. Methods Appl. Mech. Engrg. 190 (2000), no. 13-14, 1701–1718. MR 1807476, DOI 10.1016/S0045-7825(00)00185-7
Carsten Carstensen and Georg Dolzmann, A posteriori error estimates for mixed FEM in elasticity, Numer. Math. 81 (1998), no. 2, 187–209. MR 1657768, DOI 10.1007/s002110050389
Bernardo Cockburn, Jayadeep Gopalakrishnan, and Johnny Guzmán, A new elasticity element made for enforcing weak stress symmetry, Math. Comp. 79 (2010), no. 271, 1331–1349. MR 2629995, DOI 10.1090/S0025-5718-10-02343-4
Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
B. M. Fraeijs de Veubeke, Discretization of rotational equilibrium in the finite element method, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin-New York, 1977, pp. 87–112. MR 669405
B. M. Fraeijs de Veubeke and A. Millard, Discretization of stress fields in the finite element method, J. Franklin Inst. 302 (1976), no. 5-6, 389–412. Basis of the finite element method. MR 455773, DOI 10.1016/0016-0032(76)90032-6
B. Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, pages 145–197 of Stress Analysis, Edited by O. C. Zienkiewicz and G. S. Holister, Published by John Wiley & Sons, 1965, Internat. J. Numer. Methods Engrg. 52 (2001), no. 3, 287–342. Edited by O. C. Zienkiewicz and G. S. Holister and with introductory remarks by Zienkiewicz. MR 3618552, DOI 10.1002/nme.339
B. M. Fraejis de Veubeke, Stress function approach, Proceedings of the World Congress on Finite Element Methods in Structural Mechanics, vol. 1, Dorset, England, October 12–17, 1975, pp. J.1–J.51.
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal. 32 (2012), no. 1, 352–372. MR 2875255, DOI 10.1093/imanum/drq047
Johnny Guzmán and Michael Neilan, Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions, Numer. Math. 126 (2014), no. 1, 153–171. MR 3149075, DOI 10.1007/s00211-013-0557-1
Antti Hannukainen, Rolf Stenberg, and Martin Vohralík, A unified framework for a posteriori error estimation for the Stokes problem, Numer. Math. 122 (2012), no. 4, 725–769. MR 2995179, DOI 10.1007/s00211-012-0472-x
C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103–116. MR 483904, DOI 10.1007/BF01403910
Kwang-Yeon Kim, Guaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry, SIAM J. Numer. Anal. 49 (2011), no. 6, 2364–2385. MR 2854600, DOI 10.1137/110823031
Philip L. Lederer and Rolf Stenberg, Energy norm analysis of exactly symmetric mixed finite elements for linear elasticity, Math. Comp. 92 (2023), no. 340, 583–605. MR 4524103, DOI 10.1090/mcom/3784
Marco Lonsing and Rüdiger Verfürth, A posteriori error estimators for mixed finite element methods in linear elasticity, Numer. Math. 97 (2004), no. 4, 757–778. MR 2127931, DOI 10.1007/s00211-004-0519-8
Jindřich Nečas and Ivan Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: an introduction, Studies in Applied Mechanics, vol. 3, Elsevier Scientific Publishing Co., Amsterdam-New York, 1980. MR 600655
W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
J. Schöberl, NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci. 1 (1997), no. 1, 41–52.
L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 111–143 (English, with French summary). MR 813691, DOI 10.1051/m2an/1985190101111
Rolf Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513–538. MR 954768, DOI 10.1007/BF01397550
Rolf Stenberg, Some new families of finite elements for the Stokes equations, Numer. Math. 56 (1990), no. 8, 827–838. MR 1035181, DOI 10.1007/BF01405291
Rolf Stenberg, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 1, 151–167 (English, with French summary). MR 1086845, DOI 10.1051/m2an/1991250101511
Barna Szabó and Ivo Babuška, Finite element analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. MR 1164869
Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. MR 3059294, DOI 10.1093/acprof:oso/9780199679423.001.0001