A Parallel Fast Direct Solver for Block Tridiagonal Systemswith

A parallel fast direct solver based on the Divide & Conquer method for linear systems with separable block tridiagonal matrices is considered. Such systems appear, for example, when discretizing the Poisson equation in a rectangular domain using the ve{point nite diierence scheme or the piecewise linear nite elements on a triangulated rectangular mesh. The Divide & Conquer method has the arithmetical complexity O(N log N), and it is closely related to the cyclic reduction, but instead of using the matrix polynomial factorization the so{called partial solution technique is employed. The method is presented and analyzed in a general base q framework and based on this analysis, the base four variant is chosen for parallel implementation using the MPI standard. The generalization of the method to the case of arbitrary block dimension is described. The numerical experiments show the sequential eeciency and numerical stability of the considered method compared to the well{known BLKTRI{implementation of the generalized cyclic reduction. The good scalability properties of the parallel Divide & Conquer method are demonstrated in a distributed memory Cray T3E computer.