Projection methods for the numerical solution of non‐self‐adjoint elliptic partial differential equations

Abstract

AbstractWe compare the relative performances of two iterative schemes based on projection techniques for the solution of large sparse nonsymmetric systems of linear equations, encountered in the numerical solution of partial differential equations. The Block–Symmetric Successive Over‐Relaxation (Block‐SSOR) method and the Symmetric–Kaczmarz method are derived from the simplest of projection methods, that is, the Kaczmarz method. These methods are then accelerated using the conjugate gradient method, in order to improve their convergence. We study their behavior on various test problems and comment on the conditions under which one method would be better than the other. We show that while the conjugate‐gradient‐accelerated Block‐SSOR method is more amenable to implementation on vector and parallel computers, the conjugate‐gradient accelerated Symmetric–Kaczmarz method provides a viable alternative for use on a scalar machine.