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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
DOI: https://doi.org/10.1090/S0025-5718-08-02071-1
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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Additional Information

Douglas N. Arnold
Affiliation: Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
Email: arnold@ima.umn.edu

Gerard Awanou
Affiliation: Department of Mathematical Sciences, Northern Illinois University, Dekalb, Illinois 60115
Email: awanou@math.niu.edu

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email: ragnar.winther@cma.uio.no

DOI: https://doi.org/10.1090/S0025-5718-08-02071-1
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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