A Review of Preconditioning Techniques for Steady Incompressible Flow

Abstract: Simulation of the motion of an incompressible fluid remains an important and very challenging computational problem. The resources required for accurate modelling of three-dimensional flow test even the most ad- vanced computer hardware. Mixed finite element approximation of the underlying PDEs leads to symmetric indefinite or unsymmetric indefinite linear systems of equa- tions. In the talk we will review a generic block preconditioning strategy which have the property that the eigenvalues of the preconditioned matrices are contained in intervals that are bounded independently of the mesh size. Although the strategy is well established (original papers by Rusten & Winther, Silvester & Wathen, and Elman & Silvester appeared in the early 1990’s) there have been some important and exciting developments in the last couple of years. Two such developments are discussed in this talk. First, we will present numerical results showing the effectiveness of an algebraic multigrid implementation of our preconditioning strategy when modelling ground-water flow in porous media that exhibit random spatial variability [1]. Second, we will discuss improvements to the “textbook” methodology, see [2,chap. 8], in the context of solving steady flow problems modelled by the Navier-Stokes equations. References [1] Oliver Ernst, Catherine Powell, David Silvester, and Elisabeth Ullmann. Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput., 31:1424–1447, 2009. [2] Howard Elman, David Silvester, and Andy Wathen. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford, 2005. xiv+400 pp. ISBN: 978-0-19-852868-5; 0-19-852868-X.