On methods of incomplete LU decompositions for solving Poisson's equation in annular regions

The application of the finite difference method to discretize Poisson's equation in an annular region produces an algebraic system which reflects the periodic nature of the problem. The two natural orderings of the unknowns yield coefficient matrices which are either block cyclic tridiagonal or block tridiagonal with diagonal blocks which are cyclic tridiagonal. The use of methods based on incomplete LU decompositions for solving these systems are examined. An explicit treatment of the periodicity in the incomplete LU factorization of the coefficient matrix is avoided by using the method of deletion and complementation. The method, when used as a preconditioner for the minimal residual method, is found to be competitive with other methods based on factorization ideas and to be more robust.