Discretization spaces with edge-oriented degrees of freedom, as for instance nonconforming rotated multilinear finite elements, staggered grid discretizations or certain finite volume variants, are often used in practical applications, especially in Computational Fluid Dynamics, due to their seemingly excellent numerical properties. Even for complex situations which require highly anisotropic meshes with large aspect ratios, i.e., which are needed for resolving boundary layers, there are some theoretical results in the mathematical and engineerical community which address the robustness and efficiency of corresponding multigrid tools.
It is clear that special smoothing operators - ILU, line methods, etc. - are required. However, we have recently figured out the following (surprising) result which is partially contradicting to some of (our own) older statements: Multigrid for nonconforming finite elements with standard intergrid operators is not stable!
Additional modifications have to be done in practise for the grid transfer and coarse grid operators. To be precise, we demonstrate that the following ingredients are necessary for robust and efficient multigrid on such highly complex meshes:
Numerical examples show that without these local modifications, multigrid for such discretization spaces may fail! Finally, we demonstrate how these techniques can be incorporated into a self-adaptive framework for various finite element spaces.