Spline approximation of a non-linear Riemann–Hilbert problem†

Abstract

In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H 1(Γ) → H 1(Γ) is defined via a Nemytski operator. We approximate A by A n : H 1(Γ) → H 1(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. Šnirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A n (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation. Dedicated to Ian H. Sloan on the occasion of his 70th birthday.