Optimal Order Preconditioning of Nite Diierence Matrices

A new multilevel preconditioner is proposed for the iterative solution of two dimensional discrete second order elliptic PDEs. It is based a recursive block incomplete factorization of the system matrix partitioned in a two-by-two block form, in which the submatrix related to the rst block of unknowns in approximated by a MILU(0) factorization, and the Schur complement computed from a diagonal approximation of the latter submatrix. It is shown that this technique, combined with a simple W cycle scheme, leads to optimal order preconditioning of ve point nite di erence matrices. This result holds independently of possible anisotropy or jumps in the PDE coe cients as long as the latter are piecewise constant on the coarsest mesh. Numerical results illustrate the e ciency and the robustness of the proposed method. keywords: iterative methods for linear systems, acceleration of convergence preconditioning. AMS classification : 65F10, 65B99, 65N20.