Introduction

The finite element method has been an established method for approximating incompressible flow for a number of years. The primitive variable, velocity/pressure formulation, is the most popular way to implement the method although it is not the only possibility. Much theoretical work has been done to establish convergence and error estimates and there is a large amount of literature on the topic, see for example Temam [1984], Thomasset [1981], Girault and Raviart [1986], Gunzburger [1989]. Effort has been concentrated on two-dimensional flow and although mathematically three-dimensional flow is no more difficult, in practice the current state of computer hardware makes the implementation of three-dimensional elements much more problematic. It is only the availability of modern supercomputers that has allowed the approximation of such flows to be attempted. However, an element and method of solution that works well on a large vector processor may be quite inefficient on a fine-grained parallel computer thus the concept of the “best element” or even a “good element” may be highly dependent on the computer on which it is to be implemented.

Most methods of solution involve at least one iteration of one form or another, the innermost loop being the solution of a set of linear equations. For practical three dimensional flow problems a direct method of solution is unlikely to be a feasible proposition for almost all situations and this inner system of equations will have to be solved iteratively. “How accurately do we need to solve this system?” and “What are the interactions between the various iterations taking place?” are two questions that have to be faced by anyone implementing the finite element method for three-dimensional flow.