Local Multiplicative Schwarz Algorithms for Convection-diiusion Equations

We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diiusion equations discretized by nite element or nite diierence methods. The precon-ditioners consist of two components, namely, the usual two-level additive Schwarz precon-ditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the ow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the ow than either the classical multiplicative Schwarz algorithms , and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for uid ow applications than the classical additive or multiplicative Schwarz algorithms.