A parallel version of some block preconditionings

The block preconditioned conjugate gradient (BPCG) methods, even if very effective for solving the linear systems deriving from the discretization of partial differential equations, are not efficient for parallel computers. The bottleneck for their parallel implementation is represented by the solution of the linear system to obtain the preconditioned residual. Here, we present a parallel version of some BPCG algorithms. They are based on an approximate solver for tridiagonal systems, which utilizes an incomplete 2 X 2 block odd-even reduction. The stability properties of the approximate solver are investigated. Some numerical tests, on a net of transputers, are also included. 0 1991 Academic press, Inc.