IJPAM: Volume 21, No. 3 (2005)

A COMBINED METHOD FOR SOLVING LAPLACE'S
BOUNDARY VALUE PROBLEM WITH SINGULARITIES

A.A. Dosiyev

, S. Cival

$^{1,2}$ Department of Mathematics
Eastern Mediterranean University
Gazimagosa, Mersin 10, CYPRUS, TURKEY

e-mail: adiguzel.dosiyev@emu.edu.tr

e-mail: suzan.buranay@emu.edu.tr

Abstract.A combined Block-Grid Method (see Dosiyev [#!2!#], [#!3!#], [#!4!#], Dosiyev and Cival [#!5!#]) for the solution of the Dirichlet problem on polygons, when a boundary function on some sides is given from $C_{1,1}$ is analized. The obtained uniform estimate for the error of the approximate solution is of order $O\left( h^{2}\left\vert \ln h\right\vert +1\right)$ , whereas it is of order $O(h^{2}(\left\vert \ln h\right\vert +1)/r_{j}^{p-1/\alpha _{j}})$ for the errors of order derivatives in a finite neighbourhood of vertices; here is the mesh step, $r_{j}$ is the distance from the current point to the vertex in question, $\alpha _{j}\pi$ is the value of the angle.