We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a "trial-to-test" operator T , which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T . In practical computations, T is approximated using polynomials of some degree r > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r ≥ p + N , where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.2000 Mathematics Subject Classification. 65N30, 65L12.

Abstract. We provide a projection-based analysis of a large class of finite element methods for second order elliptic problems. It includes the hybridized version of the main mixed and hybridizable discontinuous Galerkin methods. The main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. Sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable are obtained. New mixed and hybridizable discontinuous Galerkin methods with these properties are devised which are defined on squares, cubes and prisms.

We present the first a priori error analysis of the hybridizable discontinuous Galerkin method for the approximation of the Navier-Stokes equations proposed in J. Comput. Phys. vol. 230 (2011), pp. 1147-1170. The method is defined on conforming meshes made of simplexes and provides piecewise polynomial approximations of fixed degree k to each of the components of the velocity gradient, velocity and pressure. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L 2 -norm of the error in each of the above-mentioned variables converges with the optimal order of k + 1 for k ≥ 0. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, H(div)-conforming and divergence-free, which converges with order k + 2 for k ≥ 1. In addition, we show that these results only depend on the inverse of the stabilization parameter of the jump of the normal component of the velocity. Thus, if we superpenalize those jumps, these converegence results do hold by assuming that the pressure lies in H 1 (Ω) only. Moreover, by letting such stabilization parameters go to infinity, we obtain new H(div)-conforming methods with the above-mentioned convergence properties.

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...