Linear iterations as smoothers in multigrid methods: Theory with applications to incomplete decompositions

Abstract In the present paper we discuss smoothing and the construction of smoothers in a general framework. The main results apply to general symmetric smoothers. They yield proofs of the smoothing smoperty for point- and blockwise incomplete decompositions, in particular for 5-point, 7-point, and block ILU applied to the anisotropic model problem. Further we give proofs for the V cycle with pointwise ILU smoothers, we present strategies to construct a smoother from a general symmetric iterative scheme, and we establish a general connection between convergent iterations and smoothers.

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