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

TL;DR

This paper introduces $M$-decompositions as a new tool to develop superconvergent HDG methods for linear elasticity, achieving optimal convergence and superconvergence on unstructured meshes with symmetric stresses.

Contribution

It presents the concept of $M$-decompositions, enabling the design of locking-free, superconvergent HDG methods with symmetric stresses on polygonal meshes, including explicit construction of approximation spaces.

Findings

Achieves optimal convergence of approximate stress.

Demonstrates superconvergence of displacement postprocessing.

Validates theoretical results with numerical experiments.

Abstract

We propose a new tool, which we call $M$-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 $M$-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.