A Robust Algebraic Preconditioner for Nite Diierence Approximations of Convection{diiusion Equations

Stable nite di erence approximations of convection-di usion equations lead to large sparse linear systems of equations whose coe cient matrix is an M{matrix, which is highly non symmetric when the convection dominates. For an e cient iterative solution of such systems, it is proposed to consider in the non symmetric case an algebraic multilevel preconditioning method formerly proposed for pure di usion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis which applies to constant coe cient problems with periodic boundary conditions whenever using an \idealized" version of the two-level preconditioner. Within this setting, it is proved that any eigenvalue of the preconditioned system satis es j 1 1 i cj 12 for some real constant c such that jcj 1 4 . This result holds independently of the grid size and uniformly with respect to the ratio between convection and di usion. Extensive numerical experiments are conducted to assess the convergence of practical twoand multi-level schemes. These experiments, that include problems with highly variable and rotating convective ow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non uniform (stretched) grids. keywords: iterative methods for linear systems, acceleration of convergence preconditioning. AMS classification : 65F10, 65B99, 65N20.