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

摘要

Let G be a simply connected region with 0 ϵ G and with a twice Lipschitz continuously differentiable boundary curve, Γ, and let zμ, μ = 1,…, N, be an even number of N = 2n equidistant grid points on the unit circle {sfnczsfnc = 1} with z1 = 1. Then there exists for all sufficiently large N a polynomial P̂n of degree n + 1, normalized by the condition that the coefficient p0 = 0, and the coefficients p1 and pn+1 are real, such that P̂n satisfies the interpolation condition P̂n(zμ) 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 P̂n converges to the conformal mapping function Φ as n → ∞. If Γ is three times Lipschitz continuously differentiable, then the P̂n 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 log2N). 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.