An HDG method for linear elasticity with strong symmetric stresses

TL;DR

This paper introduces a novel HDG method for linear elasticity that employs a strong symmetric stress formulation, achieving optimal convergence rates on polyhedral meshes with efficient implementation and robust error analysis.

Contribution

The paper develops a new HDG method with a strong symmetric stress formulation, featuring a unique numerical trace approach and proven optimal convergence for stresses and displacements.

Findings

Optimal convergence for stresses and displacements on polyhedral meshes.

Efficient implementation with only the displacement trace globally coupled.

Robust error bounds in the nearly incompressible case.

Abstract

This paper presents a new hybridizable discontinuous Galerkin (HDG) method for linear elasticity on general polyhedral meshes, based on a strong symmetric stress formulation. The key feature of this new HDG method is the use of a special form of the numerical trace of the stresses, which makes the error analysis different from the projection-based error analyzes used for most other HDG methods. For arbitrary polyhedral elements, we approximate the stress by using polynomials of degree k>=1 and the displacement by using polynomials of degree k+1. In contrast, to approximate the numerical trace of the displacement on the faces, we use polynomials of degree k only. This allows for a very efficient implementation of the method, since the numerical trace of the displacement is the only globally-coupled unknown, but does not degrade the convergence properties of the method. Indeed, we prove optimal orders of convergence for both the stresses and displacements on the elements. In the almost incompressible case, we show the error of the stress is also optimal in the standard L2-norm. These optimal results are possible thanks to a special superconvergence property of the numerical traces of the displacement, and thanks to the use of a crucial elementwise Korn's inequality. Several numerical results are presented to support our theoretical findings in the end.