Discrete Riemann–Hilbert problems, interpolation of simply closed curves, and numerical conformal mapping

Abstract

Let G be a simply connected region with 0 ϵ G and with a twice Lipschitz continuously differentiable boundary curve, Γ, and let z μ, μ = 1,…, N, be an even number of N = 2 n equidistant grid points on the unit circle {sfnc zsfnc = 1} with z 1 = 1. Then there exists for all sufficiently large N a polynomial P̂ n of degree n + 1, normalized by the condition that the coefficient p 0 = 0, and the coefficients p 1 and p n+1 are real, such that P̂ n satisfies the interpolation condition P̂ n ( z μ) ϵ Γ for all μ = 1,…, N. In a neighbourhood of the normalized conformal mapping function Φ there is exactly one such interpolating polynomial. The sequence of these P̂ n converges to the conformal mapping function Φ as n → ∞. If Γ is three times Lipschitz continuously differentiable, then the P̂ n are also conformal mappings of the unit circle onto regions, which approximate G. An important tool for the theoretical investigation is a discrete analogon of the Riemann—Hilbert problem. We present two fast procedures for the numerical solution of the discrete Riemann—Hilbert problem: A conjugate gradient method with computational cost O( N log N) and a Toeplitz matrix method with cost O( N log 2 N). Using this, one can calculate the interpolating polynomials by a Newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function.