An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems

By Paul E. Castillo, Bernardo Cockburn, Ilaria Perugia, Dominik Schötzau. SIAM Journal on Numerical Analysis, 2000

558 citations recorded by the source corpus.

In this paper, we present the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L2-norm of the gradient and the L2-norm of the potential are of order k and k+1/2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h-1 are taken, the order of convergence of the potential increases to k+1. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.

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Citations in the corpus

Analysis of an HDG method for linear elasticity (2015, methodology): “back in 2000 [30] to analyze general DG methods for second-order elliptic equations.”
A comparison of mortar and Nitsche techniques for linear elasticity (2004, methodology): “By using special lifting operators, as is standard for discontinuous Galerkin approaches (see, e.g., [1, 11 ]) the surface integral can be replaced by a volume integral.”
An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations (2009, unknown): “In the diffusive limit c ¼ 0 , expressions (20) become This case has been originally studied in [5]; see also [7,10,11].”
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems (2001, background): “In recent years there has been renewed interest in this class of techniques, stimulated by the computational convenience of DGFEMs due to a high degree of locality, the need to approximate advectiondominated diffusion problems without excessive numerical stabilization, the necessity to accommodate h”
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods (2016, methodology): “It is actually the particular case of discontinuous Galerkin methods proposed in [7] for which we have, at each interior face F P F i h such that F “ BT` X BT ́, letting τ ̆ F :“ τBT ̆ |F , p uh “ˆ τ` F τ ́ F ` τ ` F ̇ uh ` ˆ τ ́ F τ ́ F ` τ ` F ̇ uh ` ˆ 1 τ` F ` τ ́ F ̇ pqh ̈n ` ` qh ̈n ́q, p q”
Devising HDG methods for Stokes flow: An overview (2014, methodology): “Instead, we use simple, standard L(2) projections, as was done for the original analysis of DG methods for diffusion problems on general meshes [22].”
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems (2007, unknown): “(40) The above estimates require that the stabilization parameter C11 in (29) is taken to be of order O(h), where h is the characteristic mesh size (see also [7]).”
Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples (2008, methodology): “This analysis studied the effect of stabilization on the performance for the Babuška-Zlámal method [4], the IP method [23], the LDG method [13] and NIPG methods [30].”
A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS (2009, methodology): “Key words: Singularly perturbed problems, Local discontinuous Galerkin method, Numerical fluxes, Uniform superconvergence.”
A locally conservative LDG method for the incompressible Navier-Stokes equations (2004, methodology): “As will be shown later, the role of the parameter κ is to ensure the stability of the method; see also [6].”
Local Discontinuous Galerkin Methods for the Stokes System (2002, methodology): “Finally, let us point out that the technique we use in our analysis is an extension of that used in [8] for the Laplacian.”