The high accurate block-grid method for solving Laplace's boundary value problem with singularities

Abstract

A high accurate difference-analytical method is introduced for the solution of the mixed boundary value problem for Laplace's equation on graduated polygons. The polygon can have broken sections and be multiply connected. The uniform estimate of the error of the approximate solution is of order O(h(6)), whereas it is of order O(h(6)/r(j)(p-lambdaj)) for the errors of p-order derivatives (p = 1, 2,...) in a finite neighborhood of reentry vertices; here, h is the mesh step, r(j) is the distance from the current point to the vertex in question, lambda(j) = 1/(aalpha(j)), and a = 1 or 2 depending on the types of boundary conditions. Further, alpha(j)pi is the value of the interior angle at the considered vertex. Numerical experiments are illustrated in section 8 to support the analysis made.