Iterative solution of cyclically reduced systems arising from discretization of the three-dimensional convection-diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection-diffusion model problem. This system is typically nonsymmetric. We show that performing one step of cyclic reduction, followed by reordering of the unknowns, yields a system of equations for which the block Jacobi method generally converges faster than for the original system, using lexicographic ordering. The matrix representing the system of equations can be symmetrized for a large range of the coefficients of the underlying partial differential equation, and the associated iteration matrix has a smaller spectral radius than the one associated with the original system. In this sense, the three-dimensional problem is similar to the one-dimensional and two-dimensional problems, which have been studied by Elman and Golub. The process of reduction, the suggested orderings, and bounds on the spectral radii of the associated iteration matrices are presented, followed by a comparison of the reduced system with the full system and by details of the numerical experiments.