Open Access

The Block-Grid Method for Solving Laplace's Equation on Polygons with Nonanalytic Boundary Conditions

Boundary Value Problems20102010:468594

DOI: 10.1155/2010/468594

Received: 8 April 2010

Accepted: 1 June 2010

Published: 24 June 2010

Abstract

The block-grid method (see Dosiyev, 2004) for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq2_HTML.gif , is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq4_HTML.gif is the error of the approximation of the mentioned integrals, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq5_HTML.gif is the mesh step. For the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq6_HTML.gif -order derivatives ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq7_HTML.gif ) of the difference between the approximate and the exact solution in each "singular" part https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq8_HTML.gif order is obtained, here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq9_HTML.gif is the distance from the current point to the vertex in question, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq10_HTML.gif is the value of the interior angle of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F468594/MediaObjects/13661_2010_Article_927_IEq11_HTML.gif th vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.

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Authors’ Affiliations

(1)
Department of Mathematics, Eastern Mediterranean University

Copyright

© A. A. Dosiyev et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.