615 SOME RECENT RESULTS ON SCHWARZ TYPE DOMAIN DECOMPOSITION ALGORITHMS M. Dryja, O. Widlund September 1992 Numerical experiments have shown that two-level Schwarz methods, for the solution of discrete elliptic problems, often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. A supporting theory is outlined.