Overlapping Schwarz algorithms for solving Helmholtz’s equation

In this paper, prepared for the proceedings of the international conference on domain decomposition held in Boulder, CO in August 1997, we give a progress report on the development of a new family of domain decomposition methods for the solution of Helmholtz''s equation. We present three algorithms based on overlapping Schwarz methods; in our favorite method we proceed to the continuous finite element approximation of the Helmholtz''s equation through a sequence of discontinuous iterates. While this is, quite possibly, a new type of overlapping Schwarz methods, we have been inspired to develop this idea by the thesis of Bruno Despr\''{e}s.