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 nmch 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 fine-grid discrete 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 fre(Ylom used. However, in the latter case it is not always feasible 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 ba.c;es and by considering full approximation (FAS). We show that in some ca.c:;es the 3D situation is essentially more difficult than the 2D ease. We also describe different rnultigrid strategies. Numerical results are given for a transonir: Euler-flow over the ONERA M6-winp;. Note: In essence, this papP.r will be publiRhecl in the Pror.c:edinp;s of the Fifth European Multigrid Conference, BirkhauRer, BaHel