Schur complement preconditioners for the Navier-Stokes equations - Loghin, D; Wathen, A (2002)
Mixed finite element formulations of fluid flow problems lead to large systems of equations of saddlepoint type for which iterative solution methods are mandatory for reasons of efficiency. A successful approach in the design of solution methods takes into account the structure of the problem; in particular, it is well-known that an efficient solution can be obtained if the associated Schur complement problem can be solved efficiently and robustly. In this work we present a preconditioner for the Schur complement for the Oseen problem which was introduced in Kay and Loghin (Technical Report 99/06, Oxford University Computing Laboratory, 1999). We show that the spectrum of the preconditioned system is independent of the mesh parameter; moreover, we demonstrate that the number of GMRES iterations grows like the square-root of the Reynolds number. We also present convergence results for the Schur complement of the Jacobian matrix for the Navier-Stokes operator which exhibit the same mesh independence property and similar growth with the Reynolds number. Copyright © 2002 John Wiley and Sons, Ltd.