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

Guzmán, Johnny; Leykekhman, Dmitriy; Roßmann, Jürgen; Schatz, Alfred H.

doi:10.1007/s00211-009-0213-y

Cited by 53 publications

(43 citation statements)

References 37 publications

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“…Using the Hölder inequality, stability of the Ritz projection in W 1,∞ (Ω) from [12] and the L ∞ error estimate from Lemma 3.1 we have…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

“…Later the result was extended to convex polyhedral domains with some restriction on angles in [2]. This restriction was removed in [12] and even extended to certain graded meshes in [6]. For parabolic problems similar results are rather scarce.…”

Section: Introductionmentioning

confidence: 99%

“…(Ω) we use Lemma 3.7, with α = β = − 1 2 and p = 3, to obtain 24) where in the last step we used stability of the Ritz projection in the W (Ω) seminorm, see [12]. Using the inverse and the triangle inequalities,…”

Section: Introductionmentioning

confidence: 99%

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Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Leykekhman¹,

Vexler²

2017

SIAM J. Numer. Anal.

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Abstract. In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the L ∞ (I; W 1,∞ (Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [15]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.

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“…Using the Hölder inequality, stability of the Ritz projection in W 1,∞ (Ω) from [12] and the L ∞ error estimate from Lemma 3.1 we have…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Leykekhman¹,

Vexler²

2017

SIAM J. Numer. Anal.

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“…Sharper Green's function estimates are derived for two-and three-dimensional convex polygonal and polyhedral domains in [23].…”

Section: Lemma 32 Let G(x Y) Denote the Green's Function For (11)mentioning

confidence: 99%

Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes

et al. 2011

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Abstract. We consider finite element methods for a model second-order elliptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the W 1 ∞ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof technique. This result holds under a condition on the grid which is mildly more restrictive than the shape regularity condition typically enforced in adaptive codes. The second main contribution of this work is a discussion of the properties of and relationships between similar mesh restrictions that have appeared in the literature.

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“…We refer the reader to Remark 3.2 for a more detailed comparison. In a more general context, we note that sharp estimates for continuous Green's functions (or their generalized versions) frequently play a crucial role in a priori and a posteriori error analyses [12,14,27].…”

Section: Introductionmentioning

confidence: 99%

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

Chadha

Kopteva

2010

Adv Comput Math

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Abstract. A singularly perturbed semilinear reaction-diffusion problem in the unit cube, is discretized on arbitrary nonuniform tensor-product meshes. We establish a second-order maximum norm a posteriori error estimate that holds true uniformly in the small diffusion parameter. No mesh aspect ratio condition is imposed. This result is obtained by combining (i) sharp bounds on the Green's function of the continuous differential operator in the Sobolev W 1,1 and W 2,1 norms and (ii) a special representation of the residual in terms of an arbitrary current mesh and the current computed solution. Numerical results on a priori chosen meshes are presented that support our theoretical estimate.

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

Cited by 53 publications

References 37 publications

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes

Maximum norm a posteriori error estimate for a 3d singularly perturbed semilinear reaction-diffusion problem

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