Development of the mapping function at an analytic corner

The treatment of corner and pole-type singularities in numerical conformal mapping techniques

“…Also, the so-called re-parametization method of Hoidn [13] Any boundary singularities of the mapping functions are corner singularities, similar to those that arise in the study of elliptic boundary value problems. The asymptotic form of these singularities can be determined from the results of Lehman [21], which generalize earlier work of Lichtenstein [24], Kellog [19], Warschawski [38] and Lewy [23].…”

A Comparison of some numerical conformal mapping methods for simply and multiply connected domains

“…Koebe-like method compared to the Schwarz-Christoffel map using SC Toolbox [14], which is highly accurate. See [9,10,18,29,30,31,44] for related papers. The domain Ω is normalized inside the unit disk such that 0 ∈ Ω and such that none of the corners lie on the unit circle.…”

“…Note that the convergence rate is nearly independent of N . (The number of conjugate gradient iterations to solve the inner linear systems was fixed at 30 In Figure 24 the Fornberg map is composed with successive applications of our corner smoothing method to produce the map to the exterior of m = 3 rectangles. Table 7 shows the convergence of the successive iterates for near circular domains resulting from domains with corners.…”

On the decrease of Faber polynomials in domains with piecewise analytic boundary

“…A finer description of the mapping function near a corner is possible by its asymptotic expansion due to Lehman [3]; see also Pommerenke [4], p. 58. We need this expansion near a point wj = (p{zj) on ÖA.…”