Hölder estimates for Green's functions on convex polyhedral domains and their applications to finite element methods
type: Item
instance of: scholarly article
title: Hölder estimates for Green's functions on convex polyhedral domains and their applications to finite element methods
zbMATH Open document ID: 1165.65068
published in: Numerische Mathematik
publication date: 2009-04-27T00:00:00Z
review text: A model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The authors first obtain sharp Hölder estimates for the corresponding Green's function. Applying these to finite element methods, the best approximation property of the error in \(W_\infty^1\) is shown [see \textit{S. C. Brenner} and \textit{R. L. Scott}, The mathematical theory of finite element methods. 3rd ed. Texts in Applied Mathematics 15. (New York), NY: Springer. (2008; Zbl 1135.65042)]. It is also observed that using local estimates obtained by \textit{Z. Chen} and \textit{Z. Chen} [SIAM J. Numer. Anal. 42, 1146--1166 (2004; Zbl 1081.65102)] and \textit{J. Guzman} [Math. Comp. 75, No.~255, 1067--1085 (2006; Zbl 1092.65096)] and the techniques used here, one can prove optimal \(W_\infty^1\) error estimates for various discontinuous Galerkin methods on convex polyhedral domains.
reviewed by: H. P. Dikshit
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5547474
zbMATH Keywords: Second order elliptic equations
zbMATH Keywords: convex polyhedral domains
zbMATH Keywords: Green's function
zbMATH Keywords: finite element methods
zbMATH Keywords: best approximation property
zbMATH Keywords: optimal error estimates
zbMATH Keywords: discontinuous Galerkin methods
author: Johnny Guzmán
author: Dmitriy Leykekhman
author: J. Rossmann
author: Alfred H. Schatz
MaRDI profile type: Publication
full work available at URL: s00211-009-0213-y
OpenAlex ID: W2067267383
cites work: Asymptotic Error Expansions for the Finite Element Method for Second Order Elliptic Problems in $R^N$, $N\geq2$. I: Local Interior Expansions
cites work: A new approach to Richardson extrapolation in the finite element method for second order elliptic problems
cites work: The Mathematical Theory of Finite Element Methods
cites work: Pointwise Error Estimates of Discontinuous Galerkin Methods with Penalty for Second-Order Elliptic Problems
cites work: Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems
cites work: Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems
cites work: Sharply localized pointwise and $W_\infty^{-1}$ estimates for finite element methods for quasilinear problems
cites work: A Galerkin Method for a Nonlinear Dirichlet Problem
cites work: Polynomial Approximation of Functions in Sobolev Spaces
cites work: Asymptotic $L^\infty $-Error Estimates for Linear Finite Element Approximations of Quasilinear Boundary Value Problems
cites work: Potential Space Estimates for Green Potentials in Convex Domains
cites work: Maximum-norm stability of the finite element Stokes projection
cites work: Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems
cites work: The Green function for uniformly elliptic equations
cites work: Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. I: A smooth problem and globally quasi-uniform meshes
cites work: On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains
cites work: Q4537539
cites work: Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains
cites work: Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains
cites work: Über die punktweise Konvergenz finiter Elemente
cites work: Interior Estimates for Ritz-Galerkin Methods
cites work: Zur \(L^\infty\)-Konvergenz linearer finiter Elemente beim Dirichlet- Problem
cites work: Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations
cites work: Variational Formulation of a Model Free-Boundary Problem
cites work: A Weak Discrete Maximum Principle and Stability of the Finite Element Method in L ∞ on Plane Polygonal Domains. I
cites work: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates
cites work: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. II: Interior estimates
cites work: Q5698746
cites work: Interior Maximum Norm Estimates for Finite Element Methods
cites work: On the Quasi-Optimality in $L_\infty$ of the $\overset{\circ}{H}^1$-Projection into Finite Element Spaces*
cites work: Interior Maximum-Norm Estimates for Finite Element Methods, Part II
cites work: A mixed method for 4th order problems using linear finite elements
cites work: Optimal L ∞ Estimates for the Finite Element Method on Irregular Meshes
cites work: Q3126803
DOI: 10.1007/S00211-009-0213-Y
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description: scientific article; zbMATH DE number 5547474