Let \(D\) be a domain in the complex plane \(\mathbb C\) bounded by a simple closed curve \(\Gamma\) with a finite number of corner points. The authors consider some numerical methods to solve the so-called Muskhelishvili integral equations on \(\Gamma\) \[ R\varphi(t)\equiv-k\overline{\varphi(t)}-\frac{k}{2\pi i}\int_\Gamma\overline{\varphi(\tau)}d \log\frac{\overline\tau-\overline t}{\tau-t}-\frac{1}{2\pi i}\int_\Gamma\varphi(\tau)d \frac{\overline\tau-\overline t}{\tau-t}=f_0(t). \] Such equations often arise in applications, especially in plane elasticity theory. In the considered case, the integral operator \(R\) is noncompact, which generates difficulties with both Fredholm properties of \(R\) and stability of the approximation methods.
In this paper, the Fredholm properties of the operator \(R\) and their relations to the invertibility of \(R\) are investigated. Necessary and sufficient conditions for the stability of approximation methods based on piecewise constant splines, of the quadrature and Galerkin methods are established.
The study is based on the local principle, originally proposed by G. R. Allan to investigate complex structures. The original proof of that principle essentially uses the theory of analytic functions. The authors provide another version directly applicable in real algebras.