Convergence proofs and error estimates for an iterative method for conformal mapping

Abstract

The iterative method as introduced in [8] and [9] for the determination of the conformal mapping ? of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve Γ ofG is parametrized by a function ? with Lipschitz continuous derivative $$\dot \eta \ne 0$$ then the method converges locally in the Sobolev spaceW of 2?-periodic absolutely continuous functions with square integrable derivative. If ? is in a Holder classC 2+μ, the order of convergence is at least 1+μ. If Γ is inC l+1+μ withl?1, 0<μ<1, then the iteration converges inC l+μ. For analytic boundary curves the convergence takes place in a space of analytic functions. For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N 1?l?μ) if Γ is inC l+1+μ, andO (exp (??N/2)) if Γ is an analytic curve. The number ? in the latter formula is bounded by ??logR, whereR is the radius of the largest circle into which ? can be extended analytically such that?'(z)?0 for |z|<R. The results of some test calculations are reported.