Algebraic Multilevel Preconditioning Methods, II - Axelsson, O.; Vassilevski, P. S. (SIAM journal on numerical analysis, 1990)
In a previous paper Axelsson and Vassilevski presented a multilevel method to solve second-order self-adjoint elliptic problems which has an optimal order of computational complexity under mild restrictions of the finite-element mesh if the coefficients are smooth within elements of the coarsest initial mesh. The preconditioner is based on an approximate factorization of matrices partitioned in two-by-two block form where the inverses of the arising Schur complements are approximated by certain matrix polynomials, involving the inverse of the preconditioner on the next coarser mesh level, and the Schur complement itself (version (i)) or the stiffness matrix on the next coarser level (version (ii)). In this way the preconditioner is defined recursively from one level to the next until the coarsest mesh, where it is assumed that the matrix is solved by a direct solution method, for instance. There is a relation,$ν > (1 - γ^2)^{-1/2}$, between the degree (ν) of these polynomials and the constant (γ) in the method for the fundamental inequality, the extended Cauchy-Bunyakowski-Schwarz (CBS) inequality for the elliptic inner product and corresponding hierarchical finite-element basis functions, and this implies an optimal rate of convergence of the corresponding iterative method. In the first paper it was assumed that the systems to be eliminated on each level containing the major block matrices (denoted A(k) 11), of the given matrix were solved exactly. In the present paper an implementation of version (ii) is considered, where this matrix is also approximated. It is shown that the same relation between nu and γ holds for the extended version also, for an optimal rate of convergence.