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The multigrid V-cycle method is a popular method for solving systems oflinear equations. It computes an approximate solution by using smoothing onfine levels and solving a system of linear equations on the coarsest level.Solving on the coarsest level depends on the size and difficulty of theproblem. If the size permits, it is typical to use a direct method based on LUor Cholesky decomposition. In settings with large coarsest-level problems,approximate solvers such as iterative Krylov subspace methods, or directmethods based on low-rank approximation, are often used. The accuracy of thecoarsest-level solver is typically determined based on the experience of theusers with the concrete problems and methods.  In this paper we present an approach to analyzing the effects of approximatecoarsest-level solves on the convergence of the V-cycle method for symmetricpositive definite problems. Using these results, we derive coarsest-levelstopping criterion through which we may control the difference between theapproximation computed by a V-cycle method with approximate coarsest-levelsolver and the approximation which would be computed if the coarsest-levelproblems were solved exactly. The coarsest-level stopping criterion may thus beset up such that the V-cycle method converges to a chosen finest-level accuracyin (nearly) the same number of V-cycle iterations as the V-cycle method withexact coarsest-level solver. We also utilize the theoretical results to discusshow the convergence of the V-cycle method may be affected by the choice of atolerance in a coarsest-level stopping criterion based on the relative residualnorm.

The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods