Reduced symmetry elements in linear elasticity

Abstract

In continuum mechanics problems, we have to work in most cases with symmetric tensors, symmetry expressing the conservation of angular momentum. Discretization of symmetric tensors is however difficult and a classical solution is to employ some form of reduced symmetry. We present two ways of introducing elements with reduced symmetry. The first one is based on Stokes problems, and in the two-dimensional case allows to recover practically all interesting elements on the market. This however is (definitely) not true in three dimensions. On the other hand the second approach (based on a very nice property of several interpolation operators) works for three-dimensional problems as well, and allows, in particular, to prove the convergence of the Arnold-Falk-Winther element with simple and standard arguments, without the use of the Berstein-Gelfand-Gelfand resolution.