Abstract
A framework for parallel algebraic multilevel preconditioning methods is presented for solving large sparse systems of linear equations with symmetric positive definite coefficient matrices, which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields 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 relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.
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