Analysis of a recursive 5-point/9-point factorization method
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Nested recursive two-level factorization methods for nine-point difference matrices are analyzed. Somewhat similar in construction to multilevel methods for finite element matrices, these methods use recursive red-black orderings of the meshes, approximating the nine-point stencils by five-point ones in the red points and then forming the reduced system explicitly. As this Schur complement is again a nine-point matrix (on a skew grid this time), the process of approximating and factorizing can be applied anew.
Progressing until a sufficiently coarse grid has been reached, this gives a multilevel preconditioner for the original matrix. Solving the levels in V-cycle order will not give an optimal order method, but we show that using certain combinations of V-cycles and W-cycles will give methods of both optimal order of numbers of iterations and computational complexity.
KeywordsAlgebraic multilevel Chebyshev polynomial approximation nine-point differences optimal order preconditioners
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