Parallel Multiscale Iterative Methods

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We propose to carry out research on the analysis and development of iterative methods that possess good convergence rates when applied to solving linear and nonlinear systems arising from 3D partial differential equations and that also possess sufficient inherent degree of parallelism for efficient implementation on advanced parallel computers. We focus on two representative types of iterative algorithms, one fine grain (multiscale preconditioners) and one coarse grain (domain decomposition), that we feel offer the best hope as a basis for developing efficient algorithms for their respective architectures. We shall pay particular attention to problems with rough and discontinuous coefficients, convection-diffusion problems and model Navier-Stokes problems. In addition, new classes of parallel machines have also emerged (e.g. CMX and MASPAR), and we propose to investigate the suitability of some of these new architectures for these two classes of iterative methods.