Local Multiplicative Schwarz Algorithms for Convection-Di usion Equations

We develop a new class of overlapping Schwarz type algorithms for solving scalar convectiondi usion equations discretized by nite element or nite di erence methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner 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. 1 The work was supported in part by the NSF grant ASC-9457534, the National Aeronautics and Space Administration under NASA contract NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering, the NSF Grand Challenges Applications Group grant ASC-9217394 and the NASA HPCC Group grant NAG5-2218. 2 The work was supported in part by the NSF grant ASC-9406582, and in part by the NSF Grand Challenges Applications Group grant ASC-9217394 and by the NASA HPCC Group grant NAG5-2218. i