Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by $M$-decompositions
Provides a systematic framework for constructing locking-free, superconvergent HDG methods for linear elasticity, benefiting computational mechanics researchers.
The authors introduce M-decompositions as a new tool to design superconvergent HDG methods for linear elasticity with symmetric approximate stresses, achieving optimal stress convergence and superconvergent displacement postprocessing on unstructured polygonal meshes. Numerical results on triangular meshes validate the theory.
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.