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).
Elasticity, Mathematical physics, Functional analysis, Approximation theory
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.
Bernardo Cockburn, Jayadeep Gopalakrishnan and Johnny Guzmán. A New Elasticity Element Made for Enforcing Weak Stress Symmetry. (2010). Mathematics of Computation, 79(271), 1331-1349.