An incomplete factorization method with the fast Fourier transform for discrete Poisson equations with different boundary conditions

For the discrete Laplacian on a rectangular grid with Dirichlet and Dirichlet-Neumann boundary conditions, a spectral equivalent preconditioner of incomplete block-factorization type is constructed. The inversion of this preconditioner is realized with the help of the fast Fourier transform with $O(N\ln N)$ arithmetical operations.