The highly accurate block-grid method in solving Laplace’s equation for nonanalytic boundary condition with corner singularity

Abstract

The highly accurate block-grid method for solving Laplace’s boundary value problems on polygons is developed for nonanalytic boundary conditions of the first kind. The quadrature approximation of the integral representations of the exact solution around each reentrant corner(‘‘singular’’ part) are combined with the 9-point finite difference equations on the ‘‘nonsingular’’ part. In the integral representations, and in the construction of the sixth order gluing operator, the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane which are computed with ε accuracy. It is proved that the uniform error of the approximate solution is of order O(h6+ε), where h is the mesh step. This estimation is true for the coefficients of singular terms also. The errors of p-order derivatives (p = 0, 1, . . .) in the ‘‘singular’’ parts are O((h6 + ε)r1/αj−p j ), rj is the distance from the current point to the vertex in question and αjπ is the value of the interior angle of the jth vertex. Finally, we give the numerical justifications of the obtained theoretical results.