A new procedure of cyclic reduction is proposed, whereby instead of performing a step of elimination on the original cartesian mesh using a two-color ordering and a standard 5-point or 7-point operator, we perform the decoupling step on the reduced mesh associated with one color, using non-standard operators that are better aligned with that mesh. This yields a cartesian mesh and box shaped 9-point (in 2D) or 27-point (in 3D) operators that are easy to deal with. Convergence analysis for multi-line and multi-plane orderings is carried out. Numerical experiments demonstrate the merits of the approach taken.
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