Discontinuous Galerkin methods for incompressible elastic materials

In this paper, we introduce and analyze a local discontinuous Galerkin method for linear elasticity. A simple post-processing is introduced which takes advantage of the special structure of the method. It allows us to construct an approximation to the displacement which is H(div)-conforming and to enforce the equation that links the pressure to the divergence of the displacement strongly inside each element. As a consequence, when the material is exactly incompressible, the displacement is also exactly incompressible. This is achieved without having to deal with the almost impossible task of constructing finite dimensional subspaces of incompressible displacements. We provide an error analysis of the method that holds uniformly with respect to the Poisson ratio. In particular, we show that the displacement converges in L² with order k + 1 when polynomials of degree k > 0 are used. We also display numerical experiments confirming that the theoretical orders of convergence are actually achieved and that they do not deteriorate when the material becomes incompressible.