Minimum Residual Methods for the Navier-Stokes Equations: Indefiniteness vs Asymmetry

Abstract

this paper we describe some competing iterative approaches for the solution of time-dependent Stokes systems. Although the matrix in (1.1) is symmetric, since it is not positive definite, the conjugate gradient method may break down. Positive definiteness can be achieved, however, by simply replacing the bottom left block B with GammaB: this corresponds to using an equation of the form divu = Gammag (often zero) to represent the continuity equation rather than -divu = g. The coefficient matrix is now positive definite but at the expense of destroying symmetry. An applicable iterative method is therefore the nonsymmetric Generalised Minimum Residual algorithm (GMRES). When the residual norm is used for convergence, GMRES will be better than any other nonsymmetric Krylov subspace solver in terms of iteration counts. Here we compare this technique with that of solving the fully coupled symmetric indefinite system (1.1) via the Minimum Residual algorithm (MINRES). Note that when using a Uzawa-type solver, which decouples the velocity and pressure equations, the choice of SigmaB is unimportant. A comparison of MINRES and Uzawa for the case of simple preconditioners is given in [1]. The main features of each iterative method are outlined in section 2 while section 3 deals with the issue of preconditioning. Finally, the results of various numerical experiments for a test problem are presented in section 4.