Local Multiplicative Schwarz Algorithms for Steady and Unsteady Convection Di usion Equations

In this paper we develop a new class of overlapping Schwarz type algorithms for solving scalar steady and unsteady convection di usion equations discretized by nite element or nite di erence methods The preconditioners consist of two components namely the usual additive Schwarz preconditioner and the sum of some second order 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 For the steady case 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 For the unsteady case we show the algorithms are optimal without having a coarse space as long as the time step and the subdomain size satisfy a certain condition We show by numerical examples that the new algorithms are less sensitive to the direction of the ow than 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 and multiplicative Schwarz algorithms