A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems

Karakashian, Ohannes A.; Pascal, Frédéric

doi:10.1137/s0036142902405217

Cited by 354 publications

(335 citation statements)

References 18 publications

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“…also [7,21,20] for similar constructions). Using this recovery operator, in conjunction with the inconsistent formulation for the IPDG presented in [17] (which ensures that the weak formulation of the problem is defined under minimal regularity assumptions on the analytical solution), we derive efficient and reliable a posteriori estimates of residual type for the IPDG method in the corresponding energy norm.…”

Section: Introductionmentioning

confidence: 99%

“…Using this recovery operator, in conjunction with the inconsistent formulation for the IPDG presented in [17] (which ensures that the weak formulation of the problem is defined under minimal regularity assumptions on the analytical solution), we derive efficient and reliable a posteriori estimates of residual type for the IPDG method in the corresponding energy norm. Some ideas from a posteriori analyses for the Poisson problem presented in [4,21,20,1,9] are also implicitly utilized here in the context of fourth order problems.…”

Section: Introductionmentioning

confidence: 99%

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An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

Georgoulis

Houston

Virtanen

2009

IMA Journal of Numerical Analysis

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We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

Georgoulis

Houston

Virtanen

2009

IMA Journal of Numerical Analysis

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“…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]. However, to the best of our knowledge a convergence analysis for adaptive DG discretizations of the Helmholtz equation is not yet available in the literature.…”

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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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“…Under assumption (2), the variational problem (3) is uniquely solvable. This paper is a continuation of our work on hp-adaptive DG methods for diffusion and convection-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

An a-posteriori error estimate forhp-adaptive DG methods for convection–diffusion problems on anisotropically refined meshes

Giani

Schötzau

Zhu

2014

Computers & Mathematics with Applications

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Citation for published item:qi niD tef no nd h¤ otz uD hominik nd huD vi ng @PHIRA 9en Eposteriori error estim te for hpE d ptive hq methods for onve tion!di'usion pro lems on nisotropi lly re(ned meshesF9D gomputers nd m them ti s with ppli tionsFD TU @RAF ppF VTWEVVUF Further information on publisher's website:httpXGGdxFdoiForgGIHFIHITGjF mw FPHIPFIHFHIS Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computers Mathematics with Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe prove an a-posteriori error estimate for hp-adaptive discontinuous Galerkin methods for the numerical solution of convection-diffusion equations on anisotropically refined rectangular elements. The estimate yields global upper and lower bounds of the errors measured in terms of a natural norm associated with diffusion and a semi-norm associated with convection. The anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure. We present a series of numerical experiments to test the feasibility of this approach within a fully automated hp-adaptive refinement algorithm.

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A Posteriori Error Estimates for a Discontinuous Galerkin Approximation of Second-Order Elliptic Problems

Cited by 354 publications

References 18 publications

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

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

An a-posteriori error estimate forhp-adaptive DG methods for convection–diffusion problems on anisotropically refined meshes

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