Iterative methods for discrete nonlinear Riemann-Hilbert problems

Abstract

Let D denote the complex unit disk, T:= D , and let F: T× R 2 → R be a given function. The paper presents iterative methods for the computation of functions w = u + i v which are holomorphic in D and satisfy the boundary relation F( t, u( t), v( t)) = 0 for all tϵ T . It is shown that an appropriate discretization of this boundary value problem can be obtained by replacing the class of holomorphic functions with the set of polynomials of degree not exceeding n and imposing the boundary condition at a grid of 2 n uniformly distributed points on the circle T . The proposed iterative methods for the solution of the discrete as well as the nondiscrete problems converge quadratically in a neighborhood of the solution. Error estimates for the iterative solutions are given in the scale of the Sobolev spaces W k 2.