Numerical conformal mapping methods based on Faber series

Methods are presented for approximating the conformal map from the interior of various regions to the interior of simply-connected target regions with a smooth boundary. The methods for the disk due to Fornberg (1980) and the ellipse due to DeLillo and Elcrat (1993) are reformulated so that they may be extended to other new computational regions. The case of a cross-shaped region is introduced and developed. These methods are used to circumvent the severe ill-conditioning due to the crowding phenomenon suffered by conformal maps from the unit disk to target regions with elongated sections while preserving the fast Fourier methods available on the disk. The methods are based on expanding the mapping function in the Faber series for the regions. All of these methods proceed by approximating the boundary correspondence of the map with a Newton-like iteration. At each Newton step, a system of linear equations is solved using the conjugate gradient method. The matrix-vector multiplication in this inner iteration can be implemented with fast Fourier transforms at a cost of O(N log N). It is shown that the linear systems are discretizations of the identity plus a compact operator and so the conjugate gradient method converges superlinearly. Several computational examples are given along with a discussion of the accuracy of the methods

Mathematical Theory of Electromagnetism

These lecture notes treat the mathematical theory of electromagnetism at a level appropriate for graduate students in mathematics. The subject is treated as a continuum theory. The basic organization of these lectures was made by the first author and used in lectures given at Universita di Roma, La Sapienza. The purpose of them is to give a self contained treatment of electromagnetism for students of mathematics in order to lay the foundation for the many applications of these ideas in applied mathematics. Special features in this regard are the treatment of steady currents in conductors, electromagnetic waves, anisotropy, dispersion and nonlinear optics. [DOI: 10.1685/SELN09001] About DO