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

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(h⁶) 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,λ. 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.