A sharp error estimate for the numerical solution of a Dirichlet problem for the Poisson equation
type: Item
instance of: scholarly article
title: A sharp error estimate for the numerical solution of a Dirichlet problem for the Poisson equation
zbMATH Open document ID: 0765.65096
DOI: 10.1016/0377-0427(92)90004-H
author: B. Büttgenbach
author: G. Lüttgens
author: R. J. Nessel
author: Henning Esser
published in: Journal of Computational and Applied Mathematics
publication date: 1993-05-16T00:00:00Z
review text: The approximate solution of the Poisson equation with Dirichlet boundary conditions on the unit square using the standard five-point difference scheme on a uniform grid with the mesh space \(h\) is considered. For the class of problems for which the exact solution \(y\) satisfies the condition \(y\in C^{(2)}(\overline{\Omega})\) it is shown that the well- known error bounds for \(y_ h-y\), where \(y_ h\) denotes the discrete solution, are sharp. This is shown by means of well-known facts on grid functions, applying the discrete Green function \(G_ h(\xi,\eta)\) to represent the solution of the discrete problem with zero boundary conditions, thus allowing the representation of the error as a function of the exact solution \(y\), and using abstract moduli of continuity with certain properties.
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 166570
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: finite difference method
zbMATH Keywords: sharp error bounds
zbMATH Keywords: Poisson equation
zbMATH Keywords: discrete Green's function
MaRDI profile type: Publication
cites work: On the sharpness of error bounds in connection with finite difference schemes on uniform grids for boundary value problems of ordinary differential equations
cites work: Q3890207
cites work: Q3126804
cites work: Quantitative extensions of the uniform boundedness principle
cites work: Q4040640
cites work: Q3750065
cites work: Q5667926
cites work: Q5673447
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description: scientific article; zbMATH DE number 166570