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

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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.

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Cockburn, B., & Fu, G. (2018). Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA Journal of Numerical Analysis, 38(2), 566–604.

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