A note on a relaxation Schwarz alternating method (Q804276)

The paper is dealing with the numerical solution of boundary value problems in terms of a generalized form of the Schwarz alternating method. In this generalization form the domain under consideration is decomposed into a finite arbitrary number of overlapping subdomains. As proved by \textit{L. Badea} [Numer. Math. 55, No.1, 61-68 (1989; Zbl 0633.65029)] an iterative procedure used alternatively over the subdomains is convergent for some classes of boundary value problems. However, the convergence is slow. In order to remove this drawback the author of the present note proposes and analyzes a ``relaxation Schwarz alternating method''. After a proof of the convergence of this method, numerical examples are given which show that the improvement on the convergence speed may be ``dramatic'' if suitable relaxation parameters are in use. However, no method is known to find such parameters in advance. According to a given notice, this difficult subproblem is under serious consideration by the author. In fact, the main message of the present paper is that a good method is coming but is not ready yet.