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Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous…
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The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
TLDR
It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case and in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations.
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Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
TLDR
The theoretical and algorithmic aspects of the Runge–Kutta discontinuous Galerkin methods are reviewed and several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations are shown.
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TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework
Construction et analyse d'une classe de methodes a elements finis de Galerkin discontinues a variation totale bornee pour la resolution des lois de conservation. Etude de la convergence. Resultats…
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The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multi-D systems
TLDR
The algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters are discussed, both in the triangular and the rectangular element cases.
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Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
TLDR
A unifying framework for hybridization of finite element methods for second order elliptic problems is introduced, thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom.
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TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems
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The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case
TLDR
The two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods are studied, which can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate.
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An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
TLDR
This paper presents the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem and shows that, for stabilization parameters of order one, the L2-norm of the gradient and the L1- norm of the potential are of order k and k+1/2, respectively.
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Discontinuous Galerkin methods
TLDR
The exposition of the ideas behind the devising of these methods as well as on the mechanisms that allow them to perform so well in such a variety of problems are concentrated on.
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