Robustness of MILU with respect to
                         the Coefficients of the PDE

                            Maria Elizabeth G. Ong
                     University of California, San Diego

                                 Tony F. Chan
                    University of California, Los A ngeles

                               Tarek P. Mathew
                            University of Wyoming

Preconditioned conjugate gradients is a popular method for solving symmetric
positive definite linear systems; the preconditioner is used to accelerate
convergence.  One indicator of the rate of convergence is the condition number
of the preconditioned coefficient matrix.  Good preconditioners reduce the
condition number and/or cluster the eigenvalues of the coefficient matrix.  In
addition, they should be simple to implement and applicable to a wide class of
problems; that is, they should be robust.

A popular preconditioner is the modified incomplete LU (MILU) factorization.
In this talk, we show the robustness of MILU for problems with anisotropic or
discontinuous coefficients.  We demonstrate robustness by showing that the
condition number of the preconditioned coefficient matrix is independent of
the coefficients of the underlying partial differential equation.