A high accurate composite grid method for solving Laplace's boundary value problems with singularities
Е. А. Волков, Adiguzel A. Dosiyev · Russian Journal of Numerical Analysis and Mathematical Modelling · 2007
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.