Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. Over th...

We study the homogeneous Schwarz problem for Douglis analytic functions. consider two-dimensional matrices J with a multiple eigenvalue and eigenvector, which is not proportional to real vector. obtain sufficient condition matrix under there exist two linearly independent solutions of defined in certain domain D. present an example.

Krylov-type methods are widely used in order to accelerate the convergence of 10 Schwarz-type methods in the linear case. Authors in [2] have shown that they ac11 celerate without overhead cost the convergence speed of Schwarz methods for dif12 ferent types of transmission conditions. In the nonlinear context, the well-known 13 class of Newton-Krylov-Schwarz methods (cf. [5]) for steady-state p...

This paper focuses on the acceleration of Schwarz method by Aitken’s convergence technique considering special structure Schwarz’s error operator between two consecutive iterates domain decomposition’s interfaces. The new proposed method, called Sparse Aitken-Schwarz builds a low-rank space for each subdomain to approximate true solution its interfaces instead building associated gathered subdo...

In this article, we first give a modified Schwarz–Pompeiu formula in general sector ring with angle \(\vartheta =\frac{\pi }{\alpha },\ \alpha \ge 1/2\) by proper conformal mappings, and obtain the solution of Schwarz problem for Cauchy–Riemann equation explicit forms. Furthermore, construct some integral operators discuss their properties detail. Finally, virtue these operators, problems an in...