A note on the local discontinuous Galerkin method for linear problems in elasticity

Abstract

In this paper we present a mixed local discontinuous Galerkin formulation for linear elasticity problems in the plane with Dirichlet boundary conditions. The approach follows previous dual-mixed methods and introduces the stress and strain tensors, and the rotation, as auxiliary unknowns. Next, we use suitable lifting operators to eliminate part of the unknowns of the corresponding discrete system, and obtain an equivalent variational formulation. We discuss about the unique solvability of the discrete scheme and the main difficulty that arises to derive the a-priori error estimates. Finally, we propose a computable a-posteriori error estimate and include some numerical examples, which show the expected rates of convergence for the error (with respect to a suitable meshdependent norm), as well as the good behaviour of the adaptivity algorithm to recover the optimal rates of convergence, results that are not covered yet by the theory.