Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations

Rivière, Béatrice

doi:10.1137/1.9780898717440

Cited by 775 publications

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“…Despite the great interest over the last few years in the DGFEM and its application to a wide range of problems (cf. [40,32], for example), the question of developing efficient iterative solvers for the solution of the resulting (linear) system of equations has been addressed only recently and only in the framework of the h-version of the DGFEM (h-DGFEM). For example, a wide class of domain decomposition methods for discontinuous Galerkin approximations of elliptic problems has been proposed and analyzed in [24,34,12,27,3,5,21].…”

Section: Introductionmentioning

confidence: 99%

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

Antonietti

Houston

2010

J Sci Comput

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In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

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Section: Introductionmentioning

confidence: 99%

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

Antonietti

Houston

2010

J Sci Comput

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“…Moreover, the use of standard adaptive mesh refinement techniques based on a posteriori error estimators is marred by the pollution effect [5,27]. Recently, Discontinuous Galerkin (DG) methods [14,24,34] have been increasingly applied to wave propagation problems in general [13] and the Helmholtz equation in particular [2,3,18,19,20,21] including hybridized DG approximations [23]. An a posteriori error analysis of DG methods for standard second order elliptic boundary value problems has been performed in [1,8,10,26,31,35], and a convergence analysis has been 1 provided in [9,25,32].…”

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confidence: 99%

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Hoppe

Sharma

2012

IMA Journal of Numerical Analysis

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Abstract. We are concerned with a convergence analysis of an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method for the numerical solution of acoustic wave propagation problems as described by the Helmholtz equation. The mesh adaptivity relies on a residual-type a posteriori error estimator that does not only control the approximation error but also the consistency error caused by the nonconformity of the approach. As in the case of IPDG for standard second order elliptic boundary value problems, the convergence analysis is based on the reliability of the estimator, an estimator reduction property, and a quasi-orthogonality result. However, in contrast to the standard case, special attention has to be paid to a proper treatment of the lower order term in the equation containing the wavenumber which is taken care of by an Aubin-Nitsche type argument for the associated conforming finite element approximation. Numerical results are given for an interior Dirichlet problem and a screen problem illustrating the performance of the adaptive IPDG method.

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“…The other terms are identical to the ones in the proof of Theorem 2.13 and 3.4 ( [13]). Then the main result is obtained by combining all bounds and using Gronwall's inequality of Lemma2.…”

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confidence: 77%