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

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 C2,λ, 0<λ<1, 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 O(h2+ε), where ε is the error of the approximation of the mentioned integrals, h is the mesh step. For the p-order derivatives (p=0,1,…) of the difference between the approximate and the exact solution in each "singular" part O((h2+ε)rj1/αj-p) order is obtained, here rj is the distance from the current point to the vertex in question, αjπ is the value of the interior angle of the jth 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.