On the convergence condition for the Schwarz alternating method for the two-dimensional Laplace equation

Abstract

The Schwarz alternating method makes it possible to construct a solution of the Dirichlet problem for the two-dimensional Laplace equation in a finite union of overlapping domains, provided that this problem has a solution in each domain. The existing proof of the method convergence and estimation of the convergence rate use the condition that the normals to the boundaries of the domains at the intersection points are different. In the paper, it is proved that this constraint can be removed for domains with Hölder continuous normals. Removing the constraint does not affect the rate of convergence.