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28
The development of discontinuous Galerkin methods
, 1999
"... In this paper, we present 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. We show how these methods made their way into the main stream of computational ..."
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Cited by 182 (20 self)
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In this paper, we present 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. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible NavierStokes equations, and HamiltonJacobilike equations.
Arbitrary high order discontinuous Galerkin schemes
 GOUDON & E. SONNENDRUCKER EDS). IRMA SERIES IN MATHEMATICS AND THEORETICAL PHYSICS
, 2005
"... In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartesian and tr ..."
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Cited by 50 (7 self)
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In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler timestepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADERDG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N + 2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.
Locally divergencefree discontinuous Galerkin methods for the Maxwell Equations
 J. Comput. Phys
"... Abstract In this paper, we develop the locally divergencefree discontinuous Galerkin method for numerically solving the Maxwell equations. The distinctive feature of the method is the use of approximate solutions that are exactly divergencefree inside each element. As a consequence, this method h ..."
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Cited by 47 (4 self)
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Abstract In this paper, we develop the locally divergencefree discontinuous Galerkin method for numerically solving the Maxwell equations. The distinctive feature of the method is the use of approximate solutions that are exactly divergencefree inside each element. As a consequence, this method has a smaller computational cost than that of the discontinuous Galerkin method with standard piecewise polynomial spaces. We show that, in spite of this fact, it produces approximations of the same accuracy. We also show that this method is more efficient than the discontinuous Galerkin method using globally divergencefree piecewise polynomial bases. Finally, a postprocessing technique is used to recover (2k þ 1)th order of accuracy when piecewise polynomials of degree k are used.
C.: Positiondependent smoothnessincreasing accuracyconserving (SIAC) filtering for improving discontinuous Galerkin solutions
 SIAM J. Sci. Comput
, 2011
"... Abstract The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. One of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to ..."
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Cited by 15 (9 self)
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Abstract The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. One of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine highorder discretizations on an interelement level while allowing discontinuities between elements. This flexibility, however, generates a plethora of difficulties when one attempts to use DG fields for feature extraction and visualization, as most postprocessing schemes are not designed for handling explicitly discontinuous fields. This work introduces a new method of applying smoothnessincreasing, accuracyconserving filtering on discontinuous Galerkin vector fields for the purpose of enhancing streamline integration. The filtering discussed in this paper enhances the smoothness of the field and eliminates the discontinuity between elements, thus resulting in more accurate streamlines. Furthermore, as a means of minimizing the computational cost of the method, the filtering is done in a onedimensional manner along the streamline.
Local Discontinuous Galerkin Methods for HighOrder TimeDependent Partial Differential Equations
, 2010
"... Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, ..."
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Cited by 12 (1 self)
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Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving highorder timedependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for highorder timedependent PDEs are also be discussed.
PostProcessing for the Discontinuous Galerkin Method over NonUniform Meshes
 SIAM J. Scientific Computing
, 2007
"... Abstract. A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Süli [Proceedings ..."
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Cited by 11 (8 self)
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Abstract. A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Süli [Proceedings
SmoothnessIncreasing AccuracyConserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus HigherOrder Accuracy
, 2012
"... © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract Smoothnessincreasing accuracyconserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order k + 1 to order 2k + 1 for specifi ..."
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Cited by 7 (4 self)
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© The Author(s) 2012. This article is published with open access at Springerlink.com Abstract Smoothnessincreasing accuracyconserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order k + 1 to order 2k + 1 for specific types of translation invariant meshes (Cockburn et 2
SMOOTHNESSINCREASING ACCURACYCONSERVING (SIAC) POSTPROCESSING FOR DISCONTINUOUS GALERKIN SOLUTIONS OVER STRUCTURED TRIANGULAR MESHES ∗
, 1899
"... Abstract. Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothnessincreasing accuracyconserving (SIAC) filtering. Howeve ..."
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Cited by 7 (3 self)
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Abstract. Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothnessincreasing accuracyconserving (SIAC) filtering. However, it is a computationally complex task to perform this in an efficient manner, which becomes an even greater issue considering nonquadrilateral mesh structures. In this paper, we present an extension of this SIAC filter to structured triangular meshes. The basic theoretical assumption in the previous implementations of the postprocessor limits the use to numerical solutions solved over a quadrilateral mesh. However, this assumption is restrictive, which in turn complicates the application of this postprocessing technique to general tessellations. Additionally, moving from quadrilateral meshes to triangulated ones introduces more complexity in the calculations as the number of integrations required increases. In this paper, we extend the current theoretical results to variable coefficient hyperbolic equations over structured triangular meshes and demonstrate the effectiveness of the application of this postprocessor to structured triangular meshes as well as exploring the effect of using inexact quadrature. We show that there is a direct theoretical extension to structured triangular meshes for hyperbolic equations with bounded variable coefficients. This is a challenging first step toward implementing SIAC filters for unstructured tessellations. We show that by using the usual Bspline implementation, we are able to improve on the order of accuracy as well as decrease the magnitude of the errors. These results are valid regardless of whether exact or inexact integration is used. The results here demonstrate that it is still possible, both theoretically and computationally, to improve to 2k+1 over the DG solution itself for structured triangular meshes.
UNIFORM SUPERCONVERGENCE ANALYSIS OF THE DISCONTINUOUS GALERKIN METHOD FOR A SINGULARLY PERTURBED PROBLEM IN 1D
"... Abstract. It has been observed from the authors ’ numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed twopoint boundary problems of the convectiondiffusion type. Especially when using a piecewise polyn ..."
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Cited by 4 (3 self)
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Abstract. It has been observed from the authors ’ numerical experiments (2007) that the Local Discontinuous Galerkin (LDG) method converges uniformly under the Shishkin mesh for singularly perturbed twopoint boundary problems of the convectiondiffusion type. Especially when using a piecewise polynomial space of degree k, the LDG solution achieves the optimal convergence rate k +1 under the L2norm, and a superconvergence rate 2k +1 forthe onesided flux uniformly with respect to the singular perturbation parameter ɛ. In this paper, we investigate the theoretical aspect of this phenomenon under a simplified ODE model. In particular, we establish uniform convergence rates √ ( ) k+1 ln N ɛ for the L N
On the negativeorder norm accuracy of a localstructurepreserving LDG method
 J. Sci. Comput
"... Abstract The accuracy in negativeorder norms is examined for a localstructurepreserving local discontinuous Galerkin method for the Laplace equation [Li and Shu, Methods and Applications of Analysis, v13 (2006), pp.215233]. With its distinctive feature in using harmonic polynomials as local app ..."
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Cited by 3 (0 self)
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Abstract The accuracy in negativeorder norms is examined for a localstructurepreserving local discontinuous Galerkin method for the Laplace equation [Li and Shu, Methods and Applications of Analysis, v13 (2006), pp.215233]. With its distinctive feature in using harmonic polynomials as local approximating functions, this method has lower computational complexity than the standard local discontinuous Galerkin method while keeping the same order of accuracy in both the energy and the L 2 norms. In this note, numerical experiments are presented to demonstrate some accuracy loss of the method in negativeorder norms.