Pointwise error estimates  for discontinuous Galerkin methods  with lifting operators for elliptic problems

Guzmán, Johnny

doi:10.1090/s0025-5718-06-01823-0

Cited by 15 publications

(14 citation statements)

References 13 publications

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“…Our modified superapproximation estimate follows. This result, which is due to the authors jointly, has also appeared in a slightly different context in the work [Guz06] of the second author.…”

Section: An Improved Superapproximation Resultsmentioning

confidence: 60%

“…Local finite element projections have been used, for example, in [NS74], [SW77], [SW95], and [AL95] in order to prove local a priori error estimates. The methodology of Lemma 3.3 in which no local projections are used has been employed, for example, in [Dem04] and [Guz06] in order to prove local a priori error estimates and in [XZ00], [LN03], and [Dem07] in order to prove local a posteriori error estimates. allow us to apply Lemma 3.3 with G 0 on the left-hand side of the estimate (3.9) and G 1 on the right-hand side.…”

Section: Taking =mentioning

confidence: 99%

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Local energy estimates for the finite element method on sharply varying grids

2010

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Abstract. Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global "pollution" term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which matches much more closely the typical practical situation. Our chief technical innovation is an improved superapproximation result.

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Section: An Improved Superapproximation Resultsmentioning

confidence: 60%

Section: Taking =mentioning

confidence: 99%

Local energy estimates for the finite element method on sharply varying grids

2010

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“…Local error estimates for the LDG method applied to Laplace's equation were carried out by Chen [5]. Later Guzmán [17] proved similar results for three DG methods, including the LDG method, in primal form.…”

Section: J Guzmánmentioning

confidence: 80%

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

Guzmán

2008

Math. Comp.

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Abstract. We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

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“…Using the local error estimates found in [5] and [14] and the techniques used here, we can prove optimal W 1 ∞ error estimates for various DG methods on convex polyhedral domains.…”

Section: Discussionmentioning

confidence: 92%

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

et al. 2009

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Abstract. A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green's function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in W 1 ∞ . In contrast to previously known results, W 2 p regularity for p > 3, which does not hold for general convex polyhedral domains, is not required. Furthermore, the new Green's function estimates allow us to obtain localized error estimates at a point.

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Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems

Abstract: Abstract. In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.

Cited by 15 publications

References 13 publications

Local energy estimates for the finite element method on sharply varying grids

Local energy estimates for the finite element method on sharply varying grids

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods

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