Schur complement preconditioners for the Navier-Stokes equations

• Published in: International journal for numerical methods in fluids 2002-09, Vol.40 (3-4), p.403-412
• Main Authors: Loghin, D., Wathen, A. J.
• Format: Article
• Language: English
• Subjects: Computational methods in fluid dynamics; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Physics
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.296

Mixed finite element formulations of fluid flow problems lead to large systems of equations of saddle‐point 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 & Sons, Ltd.