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A new elasticity element made for enforcing weak stress symmetry

Authors: Bernardo Cockburn, Jayadeep Gopalakrishnan and Johnny Guzmán
Journal: Math. Comp. 79 (2010), 1331-1349
MSC (2000): Primary 65M60, 65N30, 35L65
Published electronically: March 12, 2010
MathSciNet review: 2629995
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We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer $ k$, the row-wise divergence of the element space spans the set of polynomials of total degree $ k$. The degrees of freedom are suited to achieve continuity of the normal stresses. What makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress. This is achieved using certain ``bubble matrices'', which are special divergence-free matrix-valued polynomials. We prove that the approximation error is of order $ k+1$ in both the displacement and the stress, and that a postprocessed displacement approximation converging at order $ k+2$ can be computed element by element. We also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries.

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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Jayadeep Gopalakrishnan
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611–8105

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Keywords: Finite element, elasticity, weakly imposed symmetry, mixed method
Received by editor(s): February 23, 2009
Received by editor(s) in revised form: July 31, 2009
Published electronically: March 12, 2010
Additional Notes: The first author was supported in part by the National Science Foundation (grant DMS-0712955) and by the University of Minnesota Supercomputing Institute
The second author was supported in part by the National Science Foundation (grants DMS-0713833 and SCREMS-0619080)
The third author was partially supported by the National Science Foundation grant DMS-0914596
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.