581 SOME SCHWARZ METHODS FOR SYMMETRIC AND NONSYMMETRIC ELLIPTIC PROBLEMS O. Widlund, September 1991 This paper begins with an introduction to additive and multiplicative Schwarz methods. A two-level method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multi-level Schwarz methods, are formulated and discussed. The paper is concluded with a discussion of recent joint results with Xiao-Chuan Cai on nonsymmetric and indefinite problems.