Analysis of an HDG method for linear elasticity

Abstract

SUMMARYWe present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058-1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewisepolynomial approximations of degree k > 0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k C 1=2, and the displacement converges with order k C 1. We also provide numerical results showing that the orders of convergence are actually sharp.

A hybridizable discontinuous Galerkin formulation for non-linear elasticity

“…However, it does occur for k = 3, 4. These results are in full agreement with those recently obtained for linear elasticity in [26]. This confirms the close relation between the method proposed in [28] and in [2].…”

“…Near the incompressibility limit, the orders of convergence of the stresses were shown to degrade to 1.5 for k = 1, and to 2.75 for k = 2; no superconvergence of the displacement took place for k = 2. Recently, the first error analysis for this HDG method was carried out in [26]. For general polyhedral elements, the orders of convergence of k + 1 for the displacement and k + 1/2 for the stress were proven; in addition, the method was shown to be free of volumetric locking.…”

A face‐centred finite volume method for second‐order elliptic problems

“…In a series of papers by Cockburn and coworkers, the optimal rate of convergence of HDG has been proved and numerically verified for a wide class of problems. More precisely, for Poisson and Stokes equations using constant degree of approximation, both the primal variables ( u in the Poisson equation and velocity u and pressure p in Stokes equation) and the dual variables representing the fluxes ( q =− ∇ u in Poisson equation and in Stokes equation) converge with first‐order accuracy.…”

A hybridizable discontinuous Galerkin phase‐field model for brittle fracture with adaptive refinement

“…For this formulation, u converges with order p +1 in L 2 norm and J with order p +1/2, if an approximation of degree p is used . In a more recent formulation, Sevilla et al report convergence of order p +1 for J based on numerical experiments.…”