ABSTRACT

This chapter explains how to create transformations of regions with regard to spectral methods using multivariate polynomials on the unit ball. Other than the quite large literature on constructing conformal mappings for planar regions, there is, surprisingly, not much literature about the mapping extension problem. The most useful research appears in the area of automatic grid generation for the solution of partial differential equations, although it differs from our need to produce explicitly the mapping function, one that is easily differentiable. The chapter discusses constructions of pi, harmonic mappings, integration-based mapping formula, integration-based mapping in three dimensions, iteration methods, and iteration algorithm.