Fj f fNjdx Ajj f VNjVNjdx J nJ n

Finally, we need to specify sufficient conditions for the approximate solvers (smoothers) on the different grids to ensure convergence. Let SJ denote a smoother. As an example, the relaxed Jacobi method would correspond to SJ = wD-1, with DJ being the diagonal of AJ. As before, these smoothers need to be convergent; that is p(SjAj) < a, J e [0,L], (4.99)

where a C (0,2). The other condition basically says that the smoother has to be close to A-1 for the high frequency components. This can be stated as

(S-1v, v) < a(Ajv,v), Vv e (I - Qj-1)Vj, J e [1,L] (4.100)

where a > 0. This condition can be seen as a generalization of the eigenvalue result for the relaxed Jacobi (4.96).

Finally, we state the V-cycle multigrid algorithm. Notice that there are a number of generalizations of this algorithm, e.g., W-cycle and full multigrid; see e.g., [15] and [135].

The Multigrid V-cycle Algorithm

I: Bi =


0 0

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