Pointwise Error Estimates of Discontinuous Galerkin Methods with Penalty for Second-Order Elliptic Problems

Chen, Zhangxin; Chen, Hongsen

doi:10.1137/s0036142903421527

Cited by 64 publications

(49 citation statements)

References 16 publications

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“…For these four methods, a natural question arises: How do these methods behave pointwise? Kanschat and Rannacher [8] gave a quasi-optimal convergence result in L ∞ for the interior penalty (IP) method, and Chen and Chen [6] gave weighted pointwise estimates for the same method, which implies the result in [8]. In this paper, we show weighted pointwise error estimates for the three remaining methods.…”

Section: Introductionmentioning

confidence: 60%

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

Guzmán¹

2006

Math. Comp.

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Section: Introductionmentioning

confidence: 60%

“…Once one has local H 1 estimates, weighted pointwise estimates are easily obtained following the pointwise estimates proof of Schatz [12] for the standard continuous Galerkin method or a similar proof in [6] for the IP method. Therefore, our main contribution is to prove local H 1 estimates for these methods.…”

Section: Introductionmentioning

confidence: 99%

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

Guzmán¹

2006

Math. Comp.

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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: 93%

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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“…The Downloaded 09/28/17 to 128.178. 13 fine scale solver will be applied on a domain ω 1 , slightly larger than ω, ω 1 ⊃⊃ ω, and the overlap is denoted by ω 0 := ω 1 ∩ ω 2 . Figure 1 illustrates possible domain decompositions.…”

Section: Introductionmentioning

confidence: 99%

An Optimization Based Coupling Method for Multiscale Problems

Abdulle¹,

Jecker²,

Shapeev³

2016

Multiscale Model. Simul.

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Abstract.A new multiscale coupling method is proposed for elliptic problems with highly oscillatory coefficients with a continuum of scales in a subset of the computational domain and scale separation in complementary regions of the computational domain. A discontinuous Galerkin (DG) finite element heterogeneous multiscale method (FE-HMM) is used in the region with scale separation, while a continuous standard finite element method is used in the region without scale separation. The use of a DG-FE-HMM method allows for a flexible meshing of the different models in the overlapping region. The unknown boundary conditions at the interfaces are obtained by minimizing the error of the two models in the overlapping region. We prove the well-posedness of both the continuous and discrete coupling problems and establish convergence of the multiscale method towards the fine scale solution. Since in the region with scale separation we obtain an approximation at a cost independent of the smallest scale in the problem, the computational cost of the multiscale method is significantly smaller than a fine scale solver over the whole computational domain, while the algorithm allows us to treat situations for which standard numerical homogenization methods do not apply.

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Pointwise Error Estimates of Discontinuous Galerkin Methods with Penalty for Second-Order Elliptic Problems

Cited by 64 publications

References 16 publications

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

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

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

An Optimization Based Coupling Method for Multiscale Problems

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