Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Abstract

Summary.We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state NavierStokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

“…Cases with several clusters of eigenvalues were also considered in the same paper. A similar analysis can be found in [12]. We now illustrate our new analytic model of superlinear convergence for GMRES (given in detail in section 4).…”

“…The described phenomenon of superlinear convergence of Krylov subspace methods has been widely observed, and some models explaining this behavior have been proposed; see [2], [38], [39], and also [9], [10], [12]. The analysis proposed in most of the cited papers relies on the polynomial representation of the approximate solution in K m ; see section 2.…”

Comparison of Some Preconditioners for the Incompressible Navier-Stokes Equations

“…For the two-by-two block system arising in the incompressible Navier-Stokes equations, several state-of-art approximations of the Schur complement are proposed and analysed, c.f., [6,7,11,13,14,17,20,24,26,27,31]. In this report we choose the SIMPLER, augmented Lagrangian and 'grad-div' preconditioners for study and furthermore propose some improvements.…”

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

“…As is known, saddle-point problems correspond to the Kuhn-Tucker conditions for linearly constrained quadratic programming problems, which typically result from mixed or hybrid finite element approximations of second-order elliptic problems, elasticity problems, or the Stokes equations (see, e.g., Brezzi and Fortin [13]) and from Lagrange multiplier methods (see, e.g., Fortin and Glowinski [17]). A number of structured preconditioners [15,16,25,11,10] and iterative methods [14,22,4,2,9] have been studied in the literature for these problems. See also [27,21,20,26,19,18,6,8] and the references therein.…”