A Fornberg-like conformal mapping method for slender regions

AbstractA method is presented for approximating the conformal map from the interior of an ellipse to the interior of a simply-connected target region. The map is represented as a truncated Chebyshev series. Conditions that the mapping function be conformal are transplanted from the ellipse to an annulus with the Joukowski map. The resulting conditions on the Laurent coefficients then give a system of equations for the Newton update of the approximate boundary correspondence. This system is a generalization of Fornberg's system for maps from the disk and is solved similarly in O(N log N) operations by the conjugate gradient method. Our numerical experiments demonstrate that the maps from the ellipse to a slender target region of similar aspect ratio can be constructed with far fewer mesh points than are required for maps from the disk, thus circumventing the ill-conditioning due to crowding in these cases