Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
TLDR
This work considers the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations, and shows results displaying the sharpness of the estimates.Abstract:
We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δx only. For example, when polynomials of degree k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k+1/2 in the L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1/2 in L2(Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.read more
Citations
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Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: 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.
Book
Runge-Kutta discontinuous Galerkin methods for convection-dominated problems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: The Runge-Kutta discontinuous Galerkin (RKDG) method as discussed by the authors is one of the state-of-the-art methods for non-linear convection-dominated problems.
Book ChapterDOI
The Development of Discontinuous Galerkin Methods
TL;DR: An overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments is presented.
Journal ArticleDOI
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes
TL;DR: A conservative least-squares polynomial reconstruction operator is applied to the discontinuous Galerkin method, which yields space–time polynomials for the vector of conserved variables and for the physical fluxes and source terms that can be used in a natural way to construct very efficient fully-discrete and quadrature-free one-step schemes.
Journal ArticleDOI
Locally divergence-free discontinuous Galerkin methods for the Maxwell equations
TL;DR: The locally divergence-free discontinuous Galerkin method for numerically solving the Maxwell equations is developed, using the use of approximate solutions that are exactly divergence- free inside each element.
References
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Journal ArticleDOI
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: 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.
Journal ArticleDOI
TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: In this paper, a classe de methodes a elements finis de Galerkin discontinues a variation totale bornee for the resolution des lois de conservation, and the convergence of the convergence is studied.
BookDOI
Time Dependent Problems and Difference Methods
TL;DR: Time-Dependent Problems and Difference Methods, Second Edition as discussed by the authors provides guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems, and provides a more useful analysis of numerical methods.
Journal ArticleDOI
An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Claes Johnson,Juhani Pitkäranta +1 more
TL;DR: In this article, the authors proved Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain.
Book
Superconvergence in Galerkin Finite Element Methods
TL;DR: In this paper, the main technical tools for superconvergence in L 2-projections are presented, including the K-operator, local symmetry, and translation invariant meshes.
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