Recursive Krylov-based multigrid cycles

We consider multigrid cycles based on the recursive use of a two–grid method, in which the coarse–grid system is solved by μ ≥ 1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at levels of given multiplicity, a V–cycle formulation being used at all other levels. For symmetric positive definite systems and symmetric multigrid schemes, we consider a flexible (or generalized) conjugate gradient method as Krylov subspace solver for both inner and outer iterations. Then, based on some algebraic (block–matrix) properties of the V–cycle multigrid viewed as preconditioner, we show that the method can have optimal convergence properties if μ is chosen sufficiently large. We also formulate conditions that guarantee both, optimal complexity and convergence, bounded independently of the number of levels. Numerical results illustrate that the method can be faster than standard V– or W–cycles, and actually more robust than predicted by the theory.