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ON THE ORDER OF MAXIMUM ERROR OF THE FINITE DIFFERENCE SOLUTIONS OF LAPLACE’S EQUATION ON RECTANGLES

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  01 July 2008

A. A. DOSIYEV  and
S. CIVAL BURANAY
A. A. DOSIYEV*
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, Turkey (email: [email protected])
S. CIVAL BURANAY
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, Turkey (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C1,1 is analyzed. It is shown that the maximum error for a nine-point approximation is of the order of O(h2(|ln  h|+1)) as a five-point approximation. This order can be improved up to O(h2) when the nine-point approximation in the grids which are a distance h from the boundary is replaced by a five-point approximation (“five and nine”-point scheme). It is also proved that the class of boundary functions C1,1 used to obtain the error estimations essentially cannot be enlarged. We provide numerical experiments to support the analysis made. These results point at the importance of taking the smoothness of the boundary functions into account when choosing the numerical algorithms in applied problems.

Keywords

finite difference methodnonsmooth solutionsuniform error

MSC classification

Secondary: 65N06: Finite difference methods 65N15: Error bounds
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 59 - 73
Copyright
Copyright © Australian Mathematical Society 2008

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