An E cient Iterative Method for the Generalized Stokes Problem

This paper presents an e cient iterative scheme for the generalized Stokes problem, which arises frequently in the simulation of time-dependent Navier-Stokes equations for incompressible uid ow. The general form of the linear system is A B B 0 ! u p ! = f 0 ! (1) where A = M + T is an n n symmetric positive de nite matrix, in which M is the mass matrix, T is the discrete Laplace operator, and are positive constants proportional to the inverses of the time-step t and the Reynolds number Re respectively, and B is the discrete gradient operator of size n k (k < n). Even though the matrix A is symmetric and positive de nite, the system is inde nite due to the incompressibility constraint (Bu = 0). This causes di culties both for iterative methods and commonly used preconditioners. Moreover, depending on the ratio = , A behaves like the mass matrix M at one extreme and the Laplace operator T at the other, causing problems for the common iterative methods employed to solve this system. The generalized Stokes problem is one of the most time-consuming steps in the large scale simulation of incompressible uid ows. Iterative methods, which are indispensable for solving this problem in realistic situations, rely heavily on e ective preconditioners that can be e ciently implemented on multiprocessors. Therefore, the issues of e cient iterative algorithms and robust, e ective and parallelizable preconditioners for the generalized Stokes problem must be resolved satisfactorily. Previous e orts to solve the linear system in Eq. 1 can be broadly classi ed as follows: This research has been supported in part by the NSF under the grant NSF/CDA 9396332-001. Dept. of Computer Science, Univ. of Minnesota, Twin Cities Dept. of Computer Science, Univ. of Illinois, Urbana-Champaign