The classical Schwarz alternating method has recently been generalized in several directions. This effort has resulted in a number of new powerful domain decomposition methods for solving general elliptic problems, including the nonsymmetric and indefinite cases. In this paper, we present several overlapping Schwarz preconditioned Krylov space iterative methods for solving elliptic boundary value problems with operators that are dominated by the self-adjoint, second-order terms, but need not be either self-adjoint or definite. All algorithms discussed in this paper involve two levels of preconditioning, and one of the critical components is a global coarse grid problem. We show that, under certain assumptions, the algorithms are optimal in the sense that the convergence rates of the preconditioned Krylov iterative methods are independent of the number of unknowns of the linear system and also the number of subdomains. The optimal convergence theory holds for problems in both two- and three-dimensional spaces, and for both structured and unstructured grids. Some numerical results are presented also
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