A class of new parallel hybrid algebraic multilevel iterations (Q2770170)

Efficient iterative solvers for systems of linear equations \(Ax=b\) are discussed. It is supposed that the matrix \(A\) is a symmetric, positive definite matrix having a block structure \(A = \text{block}\{ A_{ij} \}\), \(i,j =1,2, \ldots ,\alpha\). The paper is a continuation of the papers of \textit{Z. Bai} [Linear Algebra Appl. 267, 281-315 (1997; Zbl 0891.65033)] and of \textit{D. Wang} and \textit{Z. Bai} [Linear Algebra Appl. 250, 317-347 (1997; Zbl 0871.65023)]. NEWLINENEWLINENEWLINEIn the present paper new block diagonal preconditioners are proposed. The construction of each block is based on a hybrid algebraic multilevel iteration method [see, e.g., \textit{P. S. Vassilevski}, Math. Comput. 58, No. 198, 489-512 (1992; Zbl 0765.65044)]. New possibilities for the approximation of the Schur complement are used. The author proves that the convergence rate of the presented preconditioned iterative solvers is independent of the number of unknowns and the number of levels used. Furthermore, the computational cost per iteration step is proportional to the number of unknowns. The algorithms are well-suited for implementing on parallel computers.