Abstract
A framework for algebraic multilevel preconditioning methods is presented for solving large sparse systems of linear equations with symmetric positive definite coefficient matrices, which arise in the discretization of second order elliptic boundary value problems by the finite element method. This framework covers not only all known algebraic multilevel preconditioning methods, but yields also new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional(3-D) problem domains, and their relatives condition numbers are bounded uniformly with respect to the numbers of both the levels and the nodes.
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