A parallel iterative solver for almost block-diagonal linear systems

Abstract

Many applications in science and engineering give rise to large sparse linear system of equations. An important class of such applications is that which involves the numerical handling of partial differential equations via finite difference or finite element discretization. In many cases, proper ordering of grid points, or proper permutations of the rows and columns of the linear system, yields a sparse matrix in which the nonzero elements are contained in a band of width much smaller than the number of equations. In such situations, the matrix can be organized in the block-tridiagonal, or almost block-diagonal form. In this paper we present an iterative scheme for solving general linear systems of this form, which is suitable for parallel computers. First, we illustrate our approach by applying the scheme to diagonally dominant systems in which any diagonal block is nonsingular. The effectiveness of the scheme is demonstrated using the Generalized Minimal Residual (GMRES) algorithm, [7]. This parallel scheme is also generalized to handle general systems in which any diagonal block raay be singular. This parallel scheme has its roots in the direct solver introduced in [10] for tridiagonal systems, and those in [2], [4], and [9] for block-tridiagonal systems. Fig-1 gives an example of the systems being considered in this paper, in which the system is partitioned into four diagonal blocks, with three small upper and lower off-diagonal blocks. This example can be generalized easily into a matrix A with p diagonal blocks A1, A2, . . . , A v, and (p 1) connecting off diagonal blocks Vi and Wi's. For simplicity we assume that all the Ai's are of the same size hi, where nt divides n, the dimension of A, and the Vi's and Wds are each of size m, where m << nt. In addition, we assume that the A ' s are sparse and of no particular structure.