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Summary We 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 piecewise-polynomial 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 + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k⩾0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k⩾2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper we present an overview of some new hybridization techniques for linear secondorder elliptic problems. We begin by introducing the hybridization technique through a simple one dimensional example. We then introduce a new point of view with which previously unsuspected applications of hybridization have become possible. Presentation of these applications is the main objective of this review. One such application is in comparing and establishing connections between mixed methods. Next we show how hybridization makes possible the construction of high order variable degree mixed methods. We develop a new error analysis for mixed methods making essential use of hybridization resulting in new error estimates for our new as well as old methods. Finally, we show that via hybridization one can solve the long standing research problem of computing numerical approximations to Stokes flow that are exactly divergence free. We show this for a discontinuous Galerkin method and then for a classical mixed method. (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

International audience; This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuousGalerkin methods and describe and discuss their main features. Since the methods use completely discontinuous approximations, they produce mass matrices that are block-diagonal. This renders the methods highly parallelizable when applied tohyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handleirregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements.Theyare thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued bythe computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high-orderaccurate. Even more, when applied to non-linear hyperbolic problems, the discontinuous Galerkin methods are able to capturehighly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on the exposition of theideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a varietyof problems.

In this paper, we review the development of local discontinuous Galerkin methods for elliptic problems. We explain the derivation of these methods and present the corresponding error estimates; we also mention how to couple them with standard conforming finite element methods. Numerical examples are displayed which confirm the theoretical results and show that the coupling works very well. Copyright © 2001 John Wiley & Sons, Ltd.