ABSTRACT
In the previous section we introduced a two level algorithm for a model problem and proved, under certain natural conditions, its uniform convergence. Keeping in mind the example of Section 2, we shall introduce in this section an abstract setting in which the levels are defined by a nested set of subspaces of the space M where the solution of (1.1) lies. First, we shall present one of the simplest multilevel algorithms, generalizing the two level algorithm of Section 2. Later in the section more general algorithms are introduced within the present framework. They will allow for more smoothing or correction steps. Various abstract conditions will be introduced along the way and a number of theorems concerning the convergence properties of the algorithms will be proved. The meanings of the conditions will be illustrated by specific examples in Sections 6, 7 and 8.
