Geodesic
An incomplete factorization method with the fast Fourier transform for discrete Poisson equations with different boundary conditions
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 229-242

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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. 
@article{SJVM_2001_4_3_a1,
     author = {I. A. Blatov and E. V. Kitaeva},
     title = {An incomplete factorization method with the fast {Fourier} transform for discrete {Poisson} equations with different boundary conditions},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {229--242},
     publisher = {mathdoc},
     volume = {4},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a1/}
}
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I. A. Blatov; E. V. Kitaeva. An incomplete factorization method with the fast Fourier transform for discrete Poisson equations with different boundary conditions. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 4 (2001) no. 3, pp. 229-242. http://geodesic.mathdoc.fr/item/SJVM_2001_4_3_a1/