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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  ln h  –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.

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