Boundary conditions are analyzed for a class of preconditioners used for the incompressible Navier-Stokes equations. We consider pressure convection-diffusion preconditioners [SIAM J. Sci. Comput., 24 (2002), pp. 237–256] and [J. Comput. Appl. Math., 128 (2001), pp. 261–279] as well as least-square commutator methods

The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized Navier-Stokes equations (Oseen’s problem). We utilize the so-called augmented Lagrangian approach, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of two-by-two block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the block-preconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an element-by-element fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block. Keywords: Navier-Stokes equations, saddle point systems, augmented Lagrangian, finite elements, approximation of mass matrixiterative methods, preconditioning 1

We investigate the performance of smoothers based on the Hermitian/skew-Hermitian (HSS) and augmented Lagrangian (AL) splittings applied to the Marker-and-Cell (MAC) discretization of the Oseen problem. Both steady and unsteady flows are considered. Local Fourier analysis and numerical experiments on a two-dimensional lid-driven cavity problem indicate that the proposed smoothers result in h-independent convergence and are fairly robust with respect to the Reynolds number. A direct comparison shows that the new smoothers compare favorably to coupled smoothers of Braess–Sarazin

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. Key words. saddle point problems, matrix splittings, iterative methods, preconditioning, Stokes problem, Oseen problem, stretched grids

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the

This survey paper is based on three talks given by the second author at the London Mathematical

www.elsevier.com/locate/apnum A dimensional split preconditioner for Stokes and linearized

Navier-Stokes equations with variable viscosity

Robust preconditioning methods for algebraic problems, arising in multiphase flow models