Discontinuous Galerkin methods

By Bernardo Cockburn. 2003

913 citations recorded by the source corpus.

This paper is a short essay on discontinuous Galerkin methods intended for a very wide audience. We present the discontinuous Galerkin 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 to hyperbolic problems. Another consequence of the use of discontinuous approximations is that these methods can easily handle irregular meshes with hanging nodes and approximations that have polynomials of different degrees in different elements. They are thus ideal for use with adaptive algorithms. Moreover, the methods are locally conservative (a property highly valued by the computational fluid dynamics community) and, in spite of providing discontinuous approximations, stable, and high‐order accurate. Even more, when applied to non‐linear hyperbolic problems, the discontinuous Galerkin methods are able to capture highly complex solutions presenting discontinuities with high resolution. In this paper, we concentrate on 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.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/45517b38f4bc9a0d6cf22e46b5deef8a2c19916c

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Citations in the corpus

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation, parallel implementation and scalability properties (2008, neutral): “This approach to the stabilization of the DG formulation is similar to the so-called interior penalty method [21] in which plays the role of a penalty parameter, but without the symmetrization term.”
hp-Version discontinuous Galerkin methods for advection-diffusion-reaction problems on polytopic meshes (2016, background): “first appeared around the same time [12, 13]; for reviews of some of the main developments in the subject, we refer to the monographs [27,28] and the articles [8, 24].”
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes (2008, background): “in [14,24], for triangles in 2D and tetrahedrons in 3D.”
A finite volume discontinuous Galerkin scheme¶for nonlinear convection–diffusion problems (2002, methodology): “Another possibility is to use the DG method (see [7] or [ 9 ] (and the references there) for a general survey).”
A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications (2006, background): “A recent report by Cockburn [3] discusses at length the main results, see also [4].”
A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations (2010, neutral): “The advantages of the DG methods over classical ﬁnite diﬀerence and ﬁnite volume methods are well-documented in the literature: 5,6 the DG methods work well on arbitrary meshes, result in stable high order discretizations of the convective and diﬀusive operators, allow for a simple and unambiguous…”
Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems (2007, methodology): “We use the hierarchical orthogonal reconstruction basis functions that are given for example in [10,15] for triangles in 2D and tetrahedrons in 3D.”
Discontinuous Galerkin methods for solving the Signorini problem (2011, background): “[5, 11, 13].”
DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS (2004, background): “[9], [5], [4]).”
Error Estimates of the Discontinuous Galerkin Method for Nonlinear Nonstationary Convection-Diffusion Problems (2005, methodology): “For a survey of various DGFE techniques, see, for example, [8] and [9].”
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems (2001, methodology): “Maybe the most important is the issue of adaptivity and parallelizability; see, for example, the papers on the subject that have appeared in [ 45 ].”