Fast Nonsymmetric Iterations and Preconditioning for Navier–Stokes Equations

Abstract

Abstract. Discretization and linearization of the steady-state Navier-Stokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner.

Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations

“…The real and imaginary parts of µ show a weak dependence on h, which according to the theory must disappear in the limit of h → 0; note that the minimum of the real part of µ is already h-independent even for these rather coarse grids. On the other hand, there is a strong dependence of µ on the viscosity ν, as already observed in [17].…”

“…Following the same argument as in [17], we can prove that the largest and smallest real and imaginary parts of µ are independent of h, but depend on ν. The resulting eigenvalue bounds for λ are very similar to those for the LBB-stable case (C = 0) from Section 2 and [8].…”

“…Following the argument in [17], it can be shown that the µ's are contained in a rectangle which lies in the positive half plane Re(z) > 0 and which does not depend on h. Easy manipulations give…”

Modeling, Simulation and Optimization of Electrorheological Fluids

“…(3.7) tells us that e (ν) p → 0 as ν → ∞. It is well-known from the theory of linear iterative schemes that the convergence can be significantly improved by preconditioning (cf., e.g., BANK et al [1990], BRAMBLE et al [1997], ELMAN [2002], ELMAN and GOLUB [1994], ELMAN and SILVESTER [1996], KLAWONN [1998], RUSTEN and WINTHER [1992]). In terms of the Richardson iteration (3.3), we may use…”

A note on an SOR-like method for augmented systems

“…where A ∈ R m×m is symmetric and positive definite (SPD), and B ∈ R m×n , appears in many different applications such as the finite-element method for solving the NavierStokes equation (Elman & Golub, 1994;Elman & Silvester, 1996;Elman et al, 1997;Fischer et al, 1998). When A and B are large and sparse, iterative methods for solving system (1.1) are effective because of storage requirements and preservation of sparsity.…”