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Mixed finite element methods for linear elasticity with weakly imposed symmetry

Authors: Douglas N. Arnold, Richard S. Falk and Ragnar Winther
Journal: Math. Comp. 76 (2007), 1699-1723
MSC (2000): Primary 65N30; Secondary 74S05
Published electronically: May 9, 2007
MathSciNet review: 2336264
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Abstract: In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

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Additional Information

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

Richard S. Falk
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019

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

Keywords: Mixed method, finite element, elasticity
Received by editor(s): October 31, 2005
Received by editor(s) in revised form: September 11, 2006
Published electronically: May 9, 2007
Additional Notes: The work of the first author was supported in part by NSF grant DMS-0411388
The work of the second author was supported in part by NSF grant DMS03-08347
The work of the third author was supported by the Norwegian Research Council
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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