A high accurate composite grid method for solving Laplace's boundary value problems with singularities

Abstract

A sixth-order accurate composite grid method for solving a mixed boundary value problem for Laplace's equation on staircase polygons (the polygons may have polygonal cuts and be multiply connected) is constructed and justified. The O(h(6)) order of accuracy for the number of nodes O(h(-2)lnh(-1)) is obtained by using 9-point scheme on exponentially compressed polar and square grids, as well as constructing the sixth-order matching operator connecting the subsystems. This estimate is obtained for requirements on the functions specifying the boundary conditions which cannot be essentially lowered in C-k,(L). Finally, we illustrate the high accuracy of the method in solving the well known Motz problem which has singularity due to abrupt changes in the type of boundary conditions.