A Robust Algebraic Preconditioner for Finite Difference Approximations of Convection-Diffusion Equations

Yvon Notay
Universite Libre de Bruxelles
Service de Metrologie Nucleaire (CP 165)
50 av. F.D.Roosevelt
B-1050 Bruxelles


Stable finite difference approximations of convection-diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M-matrix, which is highly non symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion 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 coefficient problems with periodic boundary conditions whenever using an `idealized' version of the two-level preconditioner. Within this setting, it is proved that any eigenvalue lambda of the preconditioned system satisfies

      |    1                    |             1 
      |  ------   -   1  - i c  |      <      -
      |  lambda                 |      =      2
for some real constant c such that |c| is not larger than 0.25. This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two- and multi-level schemes. These experiments, that include problems with highly variable and rotating convective flow, 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.

Contributed September 22, 1999.