A Fourth-Order Block-Grid Method for Solving Laplace's Equation on a Staircase Polygon with Boundary Functions in Ck,?

Abstract

The integral representations of the solution around the vertices of the interior reentered angles (on the singular parts) are approximated by the composite midpoint rule when the boundary functions are from C-4,C-lambda, 0 < lambda < 1. These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the nonsingular part of the polygon by the fourth-order gluing operator. It is proved that the uniform error is of order O(h(4) + epsilon), where epsilon > 0 and h is the mesh step. For the p-order derivatives (p = 0, 1,...) of the difference between the approximate and the exact solutions, in each singular part O((h(4) + epsilon)r(j)(1/alpha j-p)) order is obtained; here r(j) is the distance from the current point to the vertex in question and alpha(j)pi is the value of the interior angle of the jth vertex. Numerical results are given in the last section to support the theoretical results.