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Finite elements for symmetric tensors in three dimensions

Authors: Douglas N. Arnold, Gerard Awanou and Ragnar Winther
Journal: Math. Comp. 77 (2008), 1229-1251
MSC (2000): Primary 65N30; Secondary 74S05
Published electronically: January 29, 2008
MathSciNet review: 2398766
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Abstract: We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when standard discontinuous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

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  • 1. Scot Adams and Bernardo Cockburn, A mixed finite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005), no. 3, 515-521. MR 2221175 (2006m:65251)
  • 2. M. Amara and J. M. Thomas, Equilibrium finite elements for the linear elastic problem, Numer. Math. 33 (1979), no. 4, 367-383. MR 553347 (81b:65096)
  • 3. Douglas N. Arnold, Differential complexes and numerical stability, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, pp. 137-157. MR 1989182 (2004h:65115)
  • 4. Douglas N. Arnold and Gerard Awanou, Rectangular mixed finite elements for elasticity, Math. Models Methods Appl. Sci. 15 (2005), no. 9, 1417-1429. MR 2166210 (2006f:65112)
  • 5. 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 (87h:65189)
  • 6. 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 (86a:65112)
  • 7. Douglas N. Arnold and Richard S. Falk, A new mixed formulation for elasticity, Numer. Math. 53 (1988), no. 1-2, 13-30. MR 946367 (89f:73020)
  • 8. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007), 1699-1723. MR 2336264
  • 9. -, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1-155. MR 2269741
  • 10. Douglas N. Arnold and Ragnar Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), no. 3, 401-419. MR 1930384 (2003i:65103)
  • 11. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129-151. MR 0365287 (51:1540)
  • 12. Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 13. Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér., RAIRO Analyse Numérique 9 (1975), no. R-2, 77-84. MR 0400739 (53:4569)
  • 14. R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249-277. MR 592753 (82j:65076)
  • 15. Badouin M. Fraejis de Veubeke, Displacement and equilibrium models in the finite element method, Stress analysis (New York) (O.C Zienkiewics and G.S. Holister, eds.), Wiley, 1965, pp. 145-197.
  • 16. C. Johnson and B. Mercier, Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. Math. 30 (1978), no. 1, 103-116. MR 0483904 (58:3856)
  • 17. E. Stein and R. Rolfes, Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity, Comput. Methods Appl. Mech. Engrg. 84 (1990), no. 1, 77-95. MR 1082821 (91i:73045)
  • 18. R. Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. Math. 48 (1986), no. 4, 447-462. MR 834332 (87i:73062)
  • 19. -, A family of mixed finite elements for the elasticity problem, Numer. Math. 53 (1988), no. 5, 513-538. MR 954768 (89h:65192)
  • 20. -, Two low-order mixed methods for the elasticity problem, The mathematics of finite elements and applications, VI (Uxbridge, 1987), Academic Press, London, 1988, pp. 271-280. MR 956898 (89j:73074)
  • 21. V.B. Watwood Jr. and B.J. Hartz, An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems, Internat. J. Solids Structures 4 (1968), 857-873.

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Additional Information

Douglas N. Arnold
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455

Gerard Awanou
Affiliation: Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois 60115

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Received by editor(s): January 17, 2007
Received by editor(s) in revised form: May 8, 2007, and December 4, 2007
Published electronically: January 29, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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