Creating Domain Mappings

KENDALL ATKINSON∗ AND OLAF HANSEN† Abstract. Consider being given a mapping φ : Sd−1 1−1 −→ onto ∂Ω, with ∂Ω the (d− 1)-dimensional smooth boundary surface for a bounded open simply-connected region Ω in Rd, d ≥ 2. We consider the problem of constructing an extension Φ : Bd 1−1 −→ onto Ω with Bd the open unit ball in Rd. The mapping is also required to be continuously differentiable with a non-singular Jacobian matrix at all points. We discuss ways of obtaining initial guesses for such a mapping Φ and of then improving it by an iteration method.