Preconditioned iterative methods for the nine-point approximation to the convection–diffusion equation

Iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth-order finite difference schemes for convection–diffusion problems. Simple recurrences for implementing the ILU(0) factorization of the nine-point scheme are derived. Different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices. In terms of algebraic properties of the preconditioned matrices, our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses. Comparison of the convergence rates of GMRES applied to the preconditioned linear systems is done with respect to the field of values, Ritz and harmonic Ritz values of the preconditioned matrices. Numerical results show that the GMRES residual norm decreases rapidly when the difference between the Ritz and harmonic Ritz values becomes small. We also describe the results of experiments when some well-known Krylov subspace methods are used to solve the linear system arising from the compact fourth-order discretizations.