Multilevel and Local Grid Refinement Methods

In this chapter, we describe iterative methods for solving the discretization of a self adjoint and coercive elliptic equation on a grid with a multilevel structure. Such grids are obtained by the successive refinement of an initial coarse grid, either globally or locally. When the refinement is global, the resulting grid is quasi-uniform, while if the refinement is restricted to subregions, the resulting grid will not be quasi-uniform. We describe preconditioners formulated using multigrid methodology [BR22, HA4, HA2, BR36]. Multilevel preconditioners can yield optimal order performance, like multigrid methods, however, they are convergent only with Krylov space acceleration.