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

Dedicated to Ian H. Sloan on the occasion of his 70th birthday.

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.

Keywords:

• non-linear Riemann–Hilbert problems
• spline approximation
• quasi-ruled Fredholm operators
• degree of mapping

AMS Subject Classifications:

• 34B16
• 35J65
• 41A15
• 47H10
• 47H11
• 65R20

Acknowledgements

This work was partially done while the authors were guests of the Institute of Biomathematics and Biometry at the Helmholtz Center in Munich, and while the second author was guest professor at the Babeş-Bolyai University in Cluj-Napoca within the Gottfried Herder Program of the German Academic Exchange Service DAAD in 2007. The authors also gratefully acknowledge the stimulating discussions with Prof. Messoud Efendiev.

Notes

Dedicated to Ian H. Sloan on the occasion of his 70th birthday.