The Development of Discontinuous Galerkin Methods

By Bernardo Cockburn, G. Em Karniadakis, Chi-Wang Shu. 2000

988 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/485272fdfe34dc9522663fe3a293a3f0ab8307b1

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In Computational Mechanics

Citations in the corpus

A hybridizable discontinuous Galerkin method for linear elasticity (2009, methodology): “Subsequently, the method was extended to non-linear hyperbolic systems by Cockburn and Shu and later to many other problems arising in fluid dynamics; see [1] for a review and [6] for a history of the development of the method until 1999.”
A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS (2009, methodology): “A more detail review about the LDG method was given in [7, 15].”
High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter (2004, methodology): “Currently, there is a pronounced trend towards the use of discontinuous Galerkin methods [7],[8] and fluctuation splitting / residual distribution schemes [6],[9] which are applicable to unstructured meshes and share some characteristics of both finite element and finite volume techniques.”
A Parameter-Free Generalized Moment Limiter for High- Order Methods on Unstructured Grids (2009, methodology): “Examples of such methods include the discontinuous Galerkin (DG) method [4, 5, 30], the multi-domain staggered- grid method [16, 17], the spectral volume method [41, 42], the spectral difference (SD) method [22, 34].”
Local discontinuous Galerkin methods for nonlinear Schrödinger equations (2005, methodology): “For a detailed description of the method as well as its implementation and applications, we refer the readers to the lecture notes [7], the survey paper [9], other papers in that Springer volume, and the review paper [14].”
Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations (2002, neutral): “See also [16] for a more thorough discussion on the history of DG methods.”
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 ”
Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters (2005, neutral): “DISCONTINUOUS GALERKIN METHOD USING WENO LIMITERS 909 For these reasons, this scheme is widely used in applications; see, for example the survey paper [6], and other papers in that volume.”
hp‐Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity (2002, positive): “It has been shown through numerical experiments in Reference [5] (cf. also Reference [6]) that ED is typically an order of magnitude smaller than EP.”
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations (2004, methodology): “We refer to the lecture notes [7], the survey paper [9] and other papers in that Springer volume, and the review paper [13] for details and history of the discontinuous Galerkin method.”
Optimal Error Estimates for the hp-Version Interior Penalty Discontinuous Galerkin Finite Element Method (2005, background): “Justifications for the renewed interest in these methods, which date back to the 1970s and the early 1980s [6, 10, 1], can be found in the very attractive properties they exhibit, such as great flexibility in mesh design (irregular grids are perfectly admissible), the freedom of choosing the element”