ITERATIVE SOLUTION OF CYCLICALLY REDUCED SYSTEMS ARISING FROM DISCRETIZATION OF THE 3-DIMENSIONAL CONVECTION-DIFFUSION EQUATION

We consider the system of equations arising from finite difference dis cretization of a three-dimensional convection-diffusion model problem. This system is typically nonsymmetric. We show that performing one st ep of cyclic reduction, followed by reordering of the unknowns, yields a system of equations for which the block Jacobi method generally con verges faster than for the original system, using lexicographic orderi ng. The matrix representing the system of equations can be symmetrized for a large range of the coefficients of the underlying partial diffe rential equation, and the associated iteration matrix has a smaller sp ectral radius than the one associated with the original system. In thi s sense, the three-dimensional problem is similar to the one-dimension al and two-dimensional problems, which have been studied by Elman and Golub. The process of reduction, the suggested orderings, and bounds o n the spectral radii of the associated iteration matrices are presente d, followed by a comparison of the reduced system with the full system and by details of the numerical experiments.