An HDG method for linear elasticity with strong symmetric stresses

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 projectionbased 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 L 2 −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.2000 Mathematics Subject Classification. 65N30, 65L12. ], non-conforming methods using symmetric stress elements are introduced. But, methods in [3,7,8,30,37,38] use low order finite element spaces only (most of them are restricted to rectangular or cubical meshes except [3,7]). In [26], a family of simplicial elements (one for each k ≥ 1) are developed in both two and three dimensions. (The degrees of freedom of P k+1 (S, K) were studied in [26] and then used to design the projection operator Π (div,S) in [27]). However, the convergence rate of stress is suboptimal. The first HDG method for linear and nonlinear elasticity was introduced in [34,35]; see also the related HDG method proposed in [39]. These methods also use simplexes and polynomial approximations of degree k in all variables. For general polyhedral elements, this method was recently analyzed in [23] where it was shown that the method converges optimally in the displacement with order k + 1, but with the suboptimal order of k + 1/2 for the pressure and the stress. For k = 1, these orders of convergence were numerically shown to be sharp for triangular elements. In this paper, we prove that by enriching the local stress space to be polynomials of degree no more than k + 1, and by using a modified numerical trace, we are able to obtain optimal order of convergence for all unknowns. In addition, this analysis is valid for general polyhedral meshes. To the best of our knowledge, this is so far the only result which has optimal accuracy with general polyhedral triangulations for linear elasticity problems.Like many hybrid methods, our HDG method provides approximation to stress and displacement in each elem...

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

“…As a concequence, we obtained optimal order of convergence for all unknowns and superconvergence for the velocity without postprocessing. In addition, similar as in [25,26], the analysis in this paper is valid for general polygonal meshes.…”

“…The work can be seen as a continuation of our previous work on HDG methods for linear problems, see [25,26]. Comparing with the original HDG method for Navier-Stokes equation [6,23], our method uses an enriched polynomial space for the velocity in each element, a modified numerical flux and a modified HDG formulation.…”

“…He numerically showed that the methods provide optimal order of convergence for all unknowns without analysis. In [25], we gave rigorous analysis for this approach for linear elasticity problems. Optimal order of convergence for all unknowns is obtained for both equations.…”

“…In [24], Oikawa analyzed a HDG method for diffusion problem which uses the same polynomial spaces as in [20], with a different choice of the numerical flux, he proved the optimality of the method for all unknowns. Since the polynomial order of the numerical trace of these HDG methods [20,25,24] is one less then that of the approximation space of the primary variable, from the point of view of degrees of freedom of the globally coupled unknown, they obtain superconvergence for the primary variable without postprocessing. In addition, all these methods work on general polyhedral meshes.…”

An HDG Method for Convection Diffusion Equation

“…Thus, if V (K ) denotes a space of scalar-valued functions defined on K , the corresponding space of vector-valued functions is V (K ) := [V (K )] d . Our HDG method, which is a generalization of the HDG methods for the diffusion problem [12] and for the linear elasticity [13], seeks an approximation (u h , q h , u h ) to the exact solution (u, q, u| E h ) in the finite dimensional space…”