Two-sided bounds on the convergence rate of two-level methods

In this paper, we prove two-sided bounds on the convergence rate of a standard two-level subspace correction method. We then apply these estimates to show that a two-level method with point-wise smoother for variational problem in H₀ (curl) does not have optimal convergence rate. This result justifies the conclusion, observed numerically and reported in the literature, that a point relaxation as a smoother does not lead to an optimal multigrid method. In fact, we show that for such problems using a well-conditioned smoother will always lead to a method that is not optimal.