A Parallel Fast Direct Solver for the Discrete Solution of Separable Elliptic Equations

A parallel fast direct solver based on the Divide & Conquer method is considered for linear systems with separable block tridiagonal matrices. Such systems are obtained, for example, by discretizing the two{dimensional Poisson equation posed on rectangular domains with the continuous piecewise linear nite elements on nonuniform triangulated rectangular meshes. 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 parallel implementation using the MPI standard is described and a good parallel scalability of the proposed method is demonstrated on an IBM SP2 parallel computer. Also, the sequential performance is compared with the well{known BLKTRI{implementation of the generalized cyclic reduction method using a single processor of IBM SP2.