From the de Rham Sequence to Mixed Elasticity

Saturday, May 15, 2004 - 1:30pm - 2:30pm
Keller 3-180
Ragnar Winther (University of Oslo)
There have been many attempts during the past four decades to construct stable mixed finite elements for the Hellinger-Reissner formulation of linear elasticity, i.e., the stress-displacement formulation. Unfortunately, these efforts have not been as successful as expected. There is now renewed interest in this topic due to applications of mixed models in areas such as viscoelasticity, where the stress-strain relation may be nonlocal, and, as a consequence, a pure displacement model is excluded. In this talk, we first explain why the condition of symmetry of the stress tensor makes it difficult to construct stable, low order, mixed finite elements for elasticity. By introducing a proper commuting diagram, we establish a connection between the standard de Rham sequence and a corresponding elasticity sequence. Utilizing discrete versions of this connection, we are then able to construct new stable elements in two and three space dimensions, which satisfy either the usual symmetry condition or a weak version of this condition.

This represents joint work with Douglas N. Arnold, Univ. of Minnesota and Richard S. Falk, Rutgers University.