Hierarchical Preconditioners for Elliptic Partial Differential Equations

Abstract

Abstract In this chapter we shall present an introductory survey of the hierarchical basis preconditioners for elliptic partial differential equations. In the past decade this class of methods has drawn much interest. Hierarchical basis methods, [4], [5], [19], [20], and [25], are closely related to multigrid methods [16], [17], and [22], and to the BPX class of methods, [8], [9], [23], [27], and [28). They share with them a rapid rate of convergence, although hierarchical basis methods have a logarithmic-like dependence on the size of the problem, whereas multigrid and BPX have no h-dependence and are essentially optimal in their rates of convergence, [21] and [7].