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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We introduce a new mixed method for linear elasticity. The novelty is a simplicial element for the approximate stress. For every positive integer , the row-wise divergence of the element space spans the set of polynomials of total degree . 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 in both the displacement and the stress, and that a postprocessed displacement approximation converging at order 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

Email:
cockburn@math.umn.edu

**Jayadeep Gopalakrishnan**

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

Email:
jayg@ufl.edu

**Johnny Guzmán**

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

Email:
johnny_guzman@brown.edu

DOI:
https://doi.org/10.1090/S0025-5718-10-02343-4

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.