Parallel Multilevel Elliptic Preconditioners

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In the last few years, there has been an increased interest in the study of multilevel preconditioners for solving elliptic problems in conjunction with the conjugate gradient method. Part of the motivation is that these methods are highly parallelizable, in addition to usually possessing good convergence properties, a combination that most of the classical preconditioners (e.g. SSOR, incomplete factorizations and polynomial preconditioners) lack. The authors' previous work in this area has convinced them that this is a very promising approach for constructing efficient and robust iterative methods for solving large elliptic difference equations on massively parallel computers. Continued research in this area, will include a careful study of implementation issues for massively parallel computers such as the CM.2, a theoretical study of the relationships among different multilevel preconditioners and the class of domain decomposition methods, hybrid multilevel methods which retain the nice properties of incomplete factorization methods with respect to non.smooth coefficients, and extensions beyond self-adjoint second order elliptic operators (e.g.indefinite, convection-diffusion and biharmonic operators.)