This work was supported by the National Science Foundation under grants DMS-1014817 and DMS- 0914596.
IMA Journal of Numerical Analysis
Finite element method, Boundary value problems, Elasticity
We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in Cockburn et al. (2010) A new elasticity element made for enforcing weak stress symmetry. Math. Comput., 79, 1331–1349 . In this sequel we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements is of one degree less than the first method. The analysis, while similar to the first, requires a few adjustments as the new Fortin projector may not preserve weak symmetry, but we are able to prove optimal convergence for all the variables. Finally, we present a sufficient condition wherein a mixed method with weakly imposed stress symmetry in fact yields an exactly symmetric stress tensor approximation.
Gopalakrishnan, Jay and Guzmán, Johnny, "A Second Elasticity Element Using the Matrix Bubble" (2011). Mathematics and Statistics Faculty Publications and Presentations. 46.