Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

By Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, L. Donatella Marini. SIAM Journal on Numerical Analysis, 2001

3,541 citations recorded by the source corpus.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/f9015222467701cfde75b48f65bf9bc71be28326

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Citations in the corpus

THE DIRECT DISCONTINUOUS GALERKIN (DDG) METHOD FOR DIFFUSION WITH INTERFACE CORRECTIONS (2010, methodology): “[2] have set the existing DG methods into a unified framework with a systematic analysis of these methods via linear elliptic problems.”
Discontinuous Galerkin methods for reactive transport in porous media (2003, methodology): “A unified analysis of DG for elliptic problems was carried out by Arnold, Brezzi, Cockburn and Marini [11].”
Symmetric Interior Penalty DG Methods for the CompressibleNavier-Stokes Equations I: Method Formulation (2005, methodology): “Finite element methods, discontinuous Galerkin methods, compressible Navier– Stokes equations”
A discontinuous Galerkin finite element model for river bed evolution under shallow flows (2007, methodology): “For the diffusive term in the bed evolution equation, we used the primal formulation of [3, 4, 5].”
The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements (2016, neutral): “On the other hand, the non-symmetric Nitsche method leads to unsymmetric system matrices, convergence is currently only proved for basis functions of degree p ≥ 2, and convergence rates of the L(2) error are suboptimal [3, 4, 46, 58, 59].”
Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations (2002, background): “The aim of this paper is to present a critical comparison between two of the previously mentioned methods, namely that introduced by the authors, see e.g. References [2; 3] and the method of Baumann and Oden described in References [5; 6].”
A stabilized mixed discontinuous Galerkin method for Darcy flow (2006, positive): “In particular, it is noted that the method presented herein is ‘‘adjoint consistent’’ (see [3]), guaranteeing, under appropriate regularity assumptions, optimal L2-convergence rate for pressure.”
Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method (2010, background): “, [2]) and is seen as an important ingredient in the formulation.”
Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples (2008, methodology): “As comprehensively discussed in [3], discontinuous Galerkin methods for elliptic problems (see, e.”
Navier-Stokes/Darcy Coupling: Modeling, Analysis, and Numerical Approximation (2009, methodology): “Other approaches are based on the Discontinuous Galerkin (DG) methods (see [11, 47]) which are attractive for porous media flow due to their high order convergence property, local conservation of mass, flexibility with respect of meshing and hp-adaptive refinement, and their robustness with respect ”
Discontinuous Galerkin Methods for Friedrichs’ Systems (2006, positive): “[2], where it is shown that it is possible to cast many DG methods for the Poisson equation with homogeneous Dirichlet boundary conditions into a single framework amenable to a unified error analysis.”
A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics (2010, methodology): “The form Ah associated with the Laplacian is chosen as the standard interior penalty form [3,4]:”
Approximation of the Velocity by Coupling Discontinuous Galerkin and Mixed Finite Element Methods for Flow Problems (2002, methodology): “Moreover, these authors pointed out that their results also follow from the unified analysis of DG methods by Arnold et al. [ 7 ] as their method fits in the general framework proposed therein.”