(b) (a) 6 Conclusions and Further Research

23 the advection problem. Experiments with the incompressible Navier Stokes equations for ows with closed streamlines need to be performed, employing distributive Gauss-Seidel relaxation, as shown in 4]. These will no doubt raise further questions. It is anticipated that the techniques investigated here will carry over to three dimensions, although the implementation will be considerably more complicated. This is supported by the fact that simple experiments performed with the overweighting methods of 5] in three dimensions exhibited the expected performance. An obvious drawback of the entire approach is that it is inherently sequential , and eecient parallel implementations are hard to envisage. Some par-allelization might be achievable by performing a downstream line Gauss Sei-del relaxation. Also conceivable is a domain decomposition approach which leaves several lines of residuals per vortex. Another drawback of the present approach is that it is not directly applicable for ows with signiicant additional viscosity, since this entails using discretizations that are not purely upstream. Methods that deal with such ows as well are presently being investigated. (unpublished). 22 W = 2. In the \realistic" examples 3-5, one saves at most 25% of the time spent in relaxation by using the optimal value rather than 2, and usually much less. This would also be the case in Example 1 if we were to employ the \symmetric" relaxation. Even with this nominal value the convergence rates are comparable to those of elliptic problems. An experimental approach that is hoped to eventually lead to a fully eecient solver for general high-Reynolds ows has been introduced, analyzed and tested on the advection-diiusion problem in the inviscid limit. The numerical tests mostly match the predictions very well, indicating that the main cause for slow convergence of the usual multigrid algorithms for recirculating ows has indeed been understood, and a way to eliminate it has been found. The multigrid V-cycle, using downstream relaxation and upstream dis-cretization, was shown to yield an eecient solver for the tested problem in several simple situations of closed characteristics and in a mixed en-tering/recirculation problem. The tests were performed with the classical standard rst-order upstream discretization scheme and also with a novel rst-order upstream discretization, that was shown to preclude the spurious solutions reported in 3]. The present approach is cheaper to implement than that developed in 5], and can straightforwardly be applied to mixed entering/recirculating ows. More important, there is potential of success with high-order …