Abstract
The analysis of the convergence behavior of the multilevel methods is in the literature typically carried out under the assumption that the problem on the coarsest level is solved exactly. This assumption is, however, not satisfied in practical computation either due to the use of an iterative solver on the coarsest level, or due to the finite precision arithmetic, or both.
In this talk we present an abstract description of the multilevel methods which allows inexact solve on the coarsest level and discuss its convergence behavior. In particular, we show that even under these weaker assumptions it is still possible to derive a bound on the rate of convergence, which is independent of the number of levels.
Further, we consider application of the multilevel methods to the elliptic partial differential equations and their finite element discretization. We discuss both exact and inexact solvers on the coarsest level.
We show that the convergence behavior of the multilevel method with inexact solver on the coarsest level may depend on the mesh size of the initial triangulation.