Construction of Harmonic Mapping for a Class of Domains with Curved Boundaries by Applying the Multipole Method

An algorithm based on the multipole method is proposed for harmonic mapping of one class of domains $$\mathfrak{g}$$ with a curved boundary containing reentrant arc corners and narrow slots. The results of a numerical implementation of this algorithm are presented for two domains of this type. The use of several hundred approximation functions (multipoles) ensures accuracy of order 10–4 in the $$C(\bar {\mathfrak{g}})$$ norm. For conformal mapping of the same domains, a similar algorithm of the same accuracy based on the multipole method was presented, together with a numerical implementation, in a previous study. A comparison of previous and present results provides material for comparing the quality of computational grids obtained using conformal and harmonic mappings.