Numerical Conformal Mapping and Applications

Abstract

Abstract : Progress was made in three principal areas: (1) Conformal mapping of highly elongated polygons. Conventional methods of conformal mapping break down when applied to highly distorted regions, as arise frequently in applications. In the first year of research under this grant, Prof. Trefethen and Louis Howell developed a modified Schwarz-Christoffel formula to handle highly elongated polygons. In the second year this work was completed and written up for publication in the 'SIAM Journal on Scientific and Statistical Computing'; (2) Conformal mapping of circular polygons. Work by Trefethen and Howell is underway on the problem of extending Schwarz-Christoffel methods to the mapping of 'circular polygons' bounded by straight sides and circular arcs; (3) Applications in numerical linear algebra. The efficient solution of large nonsymmetric linear algebra problems Ax = b is an important but incompletely understood area of numerical analysis. Because the eigenvalues of A are generally complex, some algorithms for this problem are based on conformal mapping, complex approximation, and other techniques of complex analysis. One particular algorithm combining both conformal mapping and complex approximation is discussed in a new paper by Trefethen, and this has led to an investigation in greater generality of the behavior of non-normal matrices with complex spectra.