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 H 0 (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.