3D multigrid on partially ordered sets of grids

In this paper we discuss different possibilities of using partially ordered sets of grids in multigrid algorithms. Because, for a classical sequence of regular grids the number of degrees of freedom grows much faster with the refinement level for 3D than for 2D, it is more difficult to find sufficiently effective relaxation procedures. Therefore, we study the possibility of using different families of (regular rectangular) grids. Semi-coarsening is one technique in which a partially ordered set of grids is used. In this case still a unique discrete fine-grid problem is solved. On the other hand, sparse grid techniques are more efficient if we compare the accuracy obtained with the number of degrees of freedom used. However, in the latter case it is not always straightforward to identify an appropriate discrete equation that should be solved. The different approaches are compared. The relation between the different approaches is described by looking at hierarchical bases and by considering fulI approximation (FAS). We show that, by lack of a semi-orthogonality property, the 3D situation is essentially more difficult than the 2D case. We also describe different multigrid strategies. Numerical results are given for a transonic Euler-flow over the ONERA M6-wing