Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo; Marini, L. D.

doi:10.1137/s0036142901384162

Cited by 2,838 publications

(2,929 citation statements)

References 62 publications

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“…(iii) On the other extreme, the class of discontinuous Galerkin methods uses basis functions which are allowed to experience jump discontinuities across the interfaces [173,6]. These are particularly effective basis functions in problems with low regularity, such as the Eikonal equation [221] or problems with shock discontinuities [47,142].…”

Section: Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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Section: Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

146

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“…[5]). On a boundary face F = ∂K∩∂Ω, we set [ We collect all interior (respectively, boundary) faces in the set…”

Section: Model Problem and Dg Discretizationmentioning

confidence: 99%

Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic Problems

Antonietti

Houston

2013

Lecture Notes in Computational Science and Engineering

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“…In order to perform the stability analysis, it is convenient to rewrite the relaxation scheme described in section 3.3 in a more compact form by eliminating the unknown v from the final system: in order to do that, we need to introduce the so-called lifting operators (see [3]). …”

Section: Reformulationmentioning

confidence: 99%

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Cavalli¹,

Naldi²,

Perugia³

2012

SIAM J. Sci. Comput.

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Abstract. In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter ε. When ε → 0 + , the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing ε yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicitexplicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach. Key words. discontinuous Galerkin method, relaxation models, nonlinear diffusion AMS subject classifications. 65M60, 65M12, 35K55DOI. 10.1137/110827752 1. Introduction. Linear and nonlinear diffusion equations come from a variety of diffusion phenomena widely appearing in nature. They are suggested as mathematical models in many fields, such as filtration, phase transition, biochemistry, image analysis, and dynamics of biological groups. In the nonlinear case, the classical solutions to many of these PDEs fail to exist in finite time, even if the initial data are smooth. In such cases, suitable criteria have been introduced, which allow one to select physically relevant weak solutions beyond the singularity time.Recently, relaxation approximations to such PDEs have been introduced. These methods are based on replacing the equation by a semilinear hyperbolic system with stiff relaxation terms, tuned by a relaxation parameter ε. When ε → 0 + , the solution of this system "relaxes" onto the solution of the original PDE. Thus a consistent discretization of the relaxation system for ε = 0 yields a consistent discretization of the original PDE, as can be seen, for instance, in [29] and [2]. The advantage of this procedure is that the numerical scheme obtained in this fashion does not need approximate Riemann solvers for the convective term but possesses the robustness of upwind discretizations. Moreover, the complexity introduced by replacing the original

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Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems

Cited by 2,838 publications

References 62 publications

A review of numerical methods for nonlinear partial differential equations

A review of numerical methods for nonlinear partial differential equations

Preconditioning High–Order Discontinuous Galerkin Discretizations of Elliptic Problems

Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

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