An HDG method for linear elasticity with strong symmetric stresses

Abstract

Abstract. This paper presents a new hybridizable discontinuous Galerkin (HDG) method forlinear elasticity, on tetrahedral meshes, based on a strong symmetric stress formulation. The keyfeature 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 otherHDG methods. On each element, we approximate the stress by using polynomials of degree k ≥ 1and the displacement by using polynomials of degree k+1. In contrast, to approximate the numericaltrace of the displacement on the faces, we use polynomials of degree k only. This allows for a veryefficient implementation of the method, since the numerical trace of the displacement is the onlyglobally-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.These optimal results are possible thanks to a special superconvergence property of the numericaltraces of the displacement, and thanks to the use of a crucial elementwise Korn’s inequality.Key words. hybridizable; discontinuous Galerkin; superconvergence; linear elasticity.AMS subject classifications. 65N30, 65L12, 35L15