Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions

Abstract

We propose a new tool, which we call |${\boldsymbol{M}}$|-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an |${\boldsymbol{M}}$|-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element post-processing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit |${\boldsymbol{M}}$|-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.