The paper

Buzbee, B. L., Golub G. H., and Nelson C. W. "On Direct Methods for Solving Poisson's Equations." SIAM Journal on Numerical Analysis. 7:627-56, 1970.

Abstract: Some efficient and accurate direct methods are developed for solving certain elliptic partial difference equations over a rectangle with Dirichlet, Neumann, or periodic boundary conditions. Generalizations to higher dimensions and to L-shaped regions are included.

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