A Second Elasticity Element Using the Matrix Bubble

We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in [Math. Comp., 79 (2010), 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 are of one degree less than the first method. The analysis, while similar to the first, requires a few new 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.