CONSTRUCTING DIFFEOMORPHISMS BETWEEN SIMPLY CONNECTED PLANE DOMAINS-PART 2

delta Consider a simply connected domain Q subset of R(2 )with boundary 8 Omega that is given by a smooth function phi : [a, b ] -> R-2. Our goal is to calculate a polynomial P ( n ) : B-2 ->Omega of maximum degree n such that P ( n ) is a diffeomorphism. Here B-2 is the open unit disk in R-2, and n has to be chosen suitably large. The polynomial mapping P ( n ) is given as the L-2-projection of a mapping that is only known for a discrete set of points in B-2. The construction of was given in a previous article of the authors [Electron. Trans. Numer. Anal., 55 (2022), pp. 671-686]. Using P ( n ) we can transform boundary value problems on Q to analogous ones on B2 and then solve them using a Galerkin method. In Section 5 we give numerical examples demonstrating the use of P ( n ) to solve Dirichlet problems for two example regions Omega.