Parallel hybrid algebraic multilevel iterative methods
An optimal preconditioner for systems of linear equations with a symmetric positive definite block coefficient matrix is presented. This preconditioner is blockdiagonal, and each block is defined by an algebraic multilevel iteration (AMLI) preconditioner [see, e.g., \textit{O. Axelsson} and \textit{P. S. Vassilevski}, SIAM J. Numer. Anal. 27, No. 6, 1569-1590 (1990; Zbl 0715.65091); \textit{P. S. Vassilevski}, Math. Comput. 58, No. 198, 489-512 (1992; Zbl 0765.65044)]. Within the AMLI preconditioners a polynomial of order \(\nu > 1\) is used for the approximation of the corresponding Schur complement on the levels \(k = sk_0 + 1\) (\(k_0\) is an appropriately chosen parameter, \(s = 1,2, \ldots , l/k_0 - 1\), \(l\) is the number of levels). On all other levels a polynomial of first order is utilized. It is shown that the convergence rate of iterative solvers with this preconditioner is independent of the number of levels used. The cost of arithmetical work is proportional to the number of unknowns on the finest level. Because of the block structure of the preconditioner its parallelization is obvious.
scientific article; zbMATH DE number 1089501
scientific article