Analysis of defects and harmonic grid generation in domains with angles and cutouts

A survey of works concerning difficulties associated with harmonic grid generation in plane domains with angles and cutouts is given, and some new results are presented. It is well known that harmonic grids produced by standard methods in domains with cutouts or reentrant angles (i.e., interior angles greater than $\pi$) may contain defects, such as self-overlappings or exit beyond the domain boundary. It is established that, near the vertex of a reentrant angle, these defects follow from the asymptotics constructed for the underlying harmonic mapping, according to which the grid line leaving the angle vertex is tangent to one of the angle sides at the vertex (an effect referred to as “adhesion”), except for a special case. A survey of results is given for domains $\mathscr{Z}$ of three types with angles or cutouts ($L$-shaped, horseshoe, and a domain with a rectangular cutout), for which standard methods for harmonic grid generation encounter difficulties. Applying the multipole method to such domains yields a harmonic mapping for them with high accuracy: the a posteriori error estimate of the mapping in the $C(\bar{\mathscr{Z}})$ norm is 10$^{-7}$ in the case of using $120$ approximative functions.