  The multigrid V-cycle method is a popular method for solving systems of
linear equations. It computes an approximate solution by using smoothing on
fine levels and solving a system of linear equations on the coarsest level.
Solving on the coarsest level depends on the size and difficulty of the
problem. If the size permits, it is typical to use a direct method based on LU
or Cholesky decomposition. In settings with large coarsest-level problems,
approximate solvers such as iterative Krylov subspace methods, or direct
methods based on low-rank approximation, are often used. The accuracy of the
coarsest-level solver is typically determined based on the experience of the
users with the concrete problems and methods.
  In this paper we present an approach to analyzing the effects of approximate
coarsest-level solves on the convergence of the V-cycle method for symmetric
positive definite problems. Using these results, we derive coarsest-level
stopping criterion through which we may control the difference between the
approximation computed by a V-cycle method with approximate coarsest-level
solver and the approximation which would be computed if the coarsest-level
problems were solved exactly. The coarsest-level stopping criterion may thus be
set up such that the V-cycle method converges to a chosen finest-level accuracy
in (nearly) the same number of V-cycle iterations as the V-cycle method with
exact coarsest-level solver. We also utilize the theoretical results to discuss
how the convergence of the V-cycle method may be affected by the choice of a
tolerance in a coarsest-level stopping criterion based on the relative residual
norm.
