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

Abstract

We propose a new tool, which we call M M 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 M 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 M M-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

“…This postprocess technique is inspired by the work of Stenberg on mixed finite elements and allows to retrieve the uniqueness of the postprocessed solution, but the superconvergence is lost for low‐order approximations. Alternatively, in recent years, the extremely elegant, but rather complicated, framework of the M ‐decomposition has been extensively studied to devise the proper discrete spaces to obtain optimal convergence and superconvergence of the postprocessed solution.…”

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

“…34,37 To remedy this issue, several techniques have been proposed in the context of hybrid discretization techniques. 25,41,55,56,117 The HDG-Voigt formulation introduced in 57, 118 exploits Voigt notation for second-order tensors, see, 105 to strongly enforce symmetry by storing solely m sd :=n sd (n sd −1)/2 non-redundant off-diagonal components of the stress tensor in a vector form σ v , namely,…”

New analytical tools for HDG in elasticity, with applications to elastodynamics

“…One way to achieve this is to construct the projection directly on the physical element, instead of first constructing the projection on the reference element and then using a push-forward operator (this is what we did in this paper). This alternative approach is feasible since the M-decomposition can be applied on general polyhedral elements (see [5,4]). (B) HDG+ for elliptic diffusion.…”

“…The second and the third approaches are all based on strong symmetric stress formulations, where the conservation of angular momentum is automatically preserved. The second approach [4] uses M-decomposition [5] to enrich the approximation space for stress by adding some basis functions. The approach recovers optimal convergence and also provides an associated tailor projection as an useful tool for error analysis.…”

“…(3) build connections between M-decompositions [4,5] and HDG+ methods. To be more specific, we present a semi-discrete HDG+ method for transient elastic waves that has a uniform-in-time optimal convergence.…”