AILU preconditioning for the finite element formulation of the incompressible Navier–Stokes equations Y.S. Nam a , H.G. Choi b , J.Y. Yoo a, * a School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea b BK21 Project, Mechanical Engineering Research Division, Seoul National University, San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea Received 19 February 2002; received in revised form 15 May 2002 Abstract In this paper, the effect of a variable reordering method on the performance of ‘‘adapted incomplete LU (AILU)’’ preconditioners applied to the P2P1 mixed finite element discretization of the three-dimensional unsteady incom- pressible Navier–Stokes equations has been studied through numerical experiments, where eigenvalue distribution and convergence histories are examined. It has been revealed that the performance of an AILU preconditioner is improved by adopting a variable reordering method which minimizes the bandwidth of a globally assembled saddle-point type matrix. Furthermore, variants of the existing AILU(1) preconditioner have been suggested and tested for some three- dimensional flow problems. It is observed that the AILU(2) outperforms the existing AILU(1) with a little extra computing time and memory. Ó 2002 Elsevier Science B.V. All rights reserved. 1. Introduction Preconditioning is an essential device for the efficiency and robustness of Krylov iterative solvers, par- ticularly, of ill-conditioned matrices. There are many kinds of preconditioners proposed in the literature, among which incomplete LU (ILU) type preconditioners based on incomplete Cholesky decomposition are most widely used. They have been extensively used in order to solve a system of linear equations which results from discretizing an elliptic partial differential equation [1–5]. The ILU type preconditioner has been effectively used together with Krylov iterative solvers (Conjugate Gradient, GMRES, etc.) to solve elliptic type pressure equations of incompressible computational fluid dynamics problems [4,5]. An elliptic type pressure equation is derived in segregated algorithms of the incompressible Navier–Stokes equations such as SIMPLE, proposed by Patankar [6], and splitting (projection) method, originally devised by Chorin [7]. As is well known, a separate pressure equation is obtained from the divergence-free constraint of the Comput. Methods Appl. Mech. Engrg. 191 (2002) 4323–4339 www.elsevier.com/locate/cma * Corresponding author. Tel.: +82-2-880-7112; fax: +82-2-883-0179. E-mail address: jyyoo@plaza.snu.ac.kr (J.Y. Yoo). 0045-7825/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0045-7825(02)00369-9