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. * This work was supported by the U. S. Army Research Office under grant DAAL-0392-G0016 and the U. S. National Science Foundation under grant ASC-8958544 at the University of Maryland, and the Science and Engineering Research Council of Great Britain V...

Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dual-threshold-based ILUT preconditioners. In particular, tests with some convection-diffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.

Standard preconditioning techniques based on incomplete LU (ILU) factorizations offer a limited degree of parallelism, in general. A few of the alternatives advocated so far consist of either using some form of polynomial preconditioning, or applying the usual ILU factorization to a matrix obtained from a multicolor ordering. In this paper we present an incomplete factorization technique based on independent set orderings and multicoloring. We note that in order to improve robustness, it is necessary to allow the preconditioner to have an arbitrarily high accuracy, as is done with ILUs based on threshold techniques. The ILUM factorization described in this paper is in this category. It can be viewed as a multifrontal version a Gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism. The emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on un...

. We consider the design of robust and accurate finite element approximation methods for solving convection--diffusion problems. We develop some two--parameter streamline diffusion schemes with piecewise bilinear (or linear) trial functions and show that these schemes satisfy the necessary conditions for L 2 -uniform convergence of order greater than 1=2 introduced by Stynes and Tobiska. For smooth problems, the schemes satisfy error bounds of the form O(h)juj2 in an energy norm. In addition, extensive numerical experiments show that they effectively reproduce boundary layers and internal layers caused by discontinuities on relatively coarse grids, without any requirements on alignment of flow and grid. Key words. Convection--diffusion, streamline diffusion, crosswind diffusion, boundary layer, characteristic layer. AMS(MOS) subject classifications. primary 65N30, 65F10 1. Introduction. Consider the two--dimensional convection--diffusion equation \Gamma"\Deltau + fi \Delta ru = f ...

this paper we are interested in efficient preconditioning techniques for the finite element discretization of the advection-diffusion equation in two dimensions \Gamma"\Deltau + b:ru + cu = f (1.1) (1.2) in the case r:b = 0, with a view to eventually preconditioning the incompressible Navier-Stokes equations (see [6] for an overview of preconditioning techniques for this problem). It is well known (see for example [11], [15], [10], [24]) that as " reduces, so that the problem becomes singularly perturbed, the numerical solution will suffer from instabilities. This will also be the case with the preconditioners employed [5]. Generally speaking, the methods employed to remedy this shortcoming fall into two categories: ffl Petrov-Galerkin methods on isotropic, relatively coarse meshes; ffl standard Galerkin methods on special (adaptive, locally refined) meshes. Both approaches have been shown to produce smooth solutions by giving a good representation of the continuous operator. One expects then the performance of preconditioners like ILU to be much improved (see [1]), although see [7] for some negative results on ILU preconditioning. However, preconditioning strategies for linear systems of equations cannot fully address the issue of efficiency in a black-box format. A preconditioner, as a good approximation to the inverse of a matrix, should exhibit specific properties of the continuous problem. The Green's function of an operator, as a weak inverse, can provide useful insight in this respect since its properties are faithfully preserved by the discrete operator . (This is due, as we shall see, to the fact that the finite element method constructs increasingly better approximations to the Green's function as well as to the solution itself.) In the case of the a...

In this paper, we study the computational cost of solving the convection-diffusion equation using various discretization strategies and iteration solution algorithms. The choice of discretization influences the properties of the discrete solution and also the choice of solution algorithm. The discretizations considered here are stabilized low order finite element schemes using streamline diffusion, crosswind diffusion and shock-capturing. The latter, shock-capturing discretizations lead to nonlinear algebraic systems and require nonlinear algorithms. We compare various preconditioned Krylov subspace methods including Newton-Krylov methods for nonlinear problems, as well as several preconditioners based on relaxation and incomplete factorization. We find that although enhanced stabilization based on shock-capturing requires fewer degrees of freedom than linear stabilizations to achieve comparable accuracy, the nonlinear algebraic systems are more costly to solve than those derived from a judicious combination of streamline diffusion and crosswind diffusion. Solution algorithms based on GMRES with incomplete block-matrix factorization preconditioning are robust and efficient.

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. Key words. Navier-Stokes, iterative methods, preconditioners, Krylov subspace AMS subject classifications. 65F10, 65N12, 65N22, 65M60 1. Introduction. Consider the steady-state Navier-Stokes problem: given data f, find the velocity u and pressure p satisfying (1.1) 1-v V2u + u(div u) + u. Vu + grad p f