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multigrid; aggressive coarsening; optimal convergence result
We analyze a general multigrid method with aggressive coarsening and polynomial smoothing. We use a special polynomial smoother that originates in the context of the smoothed aggregation method. Assuming the degree of the smoothing polynomial is, on each level $k$, at least $C h_{k+1}/h_k$, we prove a convergence result independent of $h_{k+1}/h_k$. The suggested smoother is cheaper than the overlapping Schwarz method that allows to prove the same result. Moreover, unlike in the case of the overlapping Schwarz method, analysis of our smoother is completely algebraic and independent of geometry of the problem and prolongators (the geometry of coarse spaces).
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