A Parallel Block Cyclic Reduction Algorithm for the Fast Solution of Elliptic Equations

This paper presents an adaptation of the Block Cyclic Reduction (BCR) algorithm for a multi-vector processor. The main bottleneck of BCR lies in the solution of linear systems whose coefficient matrix is the product of tridiagonal matrices. This bottleneck is handled by expressing the rational function corresponding to the inverse of this product as a sum of elementary fractions. As a result the solution of this system leads to parallel solutions of tridiagonal systems. Numerical experiments performed on an Alliant FX/8 are reported. The numerical solution of linear elliptic partial differential equations is a problem of major importance in many fields of science and engineering. Several techniques have developed in the past two decades to solve separable elliptic problems much faster than the traditional iterative methods [1,4,5,14]. The most commonly used of these Rapid Elliptic Solvers (RES) are based on either applying the Fast Fourier Transform (FFT) to decouple the block tridiagonal systems into multiple scalar ones, or on using the general Block Cyclic Reduction (BCR) as described in [18], or finally, a combination of the above approaches as is the case in the FACR algorithm, first introduced by Hockney. For example, the solution of Poisson's equation on a rectangle discretized with an n × n grid, entails an asymptotic operation count of O(n 3 log n) and O(n 2 log2n) for the (iterative) SOR and ADI methods respectively, but only O(n 2 log n) for the cyclic reduction and FFT-based methods. With the advent of vector and parallel architectures, researchers have concentrated their efforts on exploiting the computational resources to achieve the best possible performance of these solvers [7,11]. The FFT-based methods are very suitable for such a task because of the amenability of the FFT to parallel computation and because of the immediate decoupling of the equations into multiple scalar tridiagonal systems. Thus, the development of highly efficient * A preliminary version of this work appeared in reference [3] and was presented at I.