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

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 $C^{\mathrm{4},\lambda },$ 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 $O(h^{\mathrm{4}}+\epsilon ),$ where $\epsilon >\mathrm{0}$ and $h$ is the mesh step. For the $p$-order derivatives ($p=\mathrm{0,1},\dots$) of the difference between the approximate and the exact solutions, in each “ singular” part $O((h^{\mathrm{4}}+\epsilon )r_j^{\mathrm{1}/{\alpha }_j-p})$ 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 $j$th vertex. Numerical results are given in the last section to support the theoretical results.