Multigrid techniques for finite elements on locally refined meshes

We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast to a standard multigrid algorithm, where the hierarchy of meshes is generated by global refinement, we suppose that the finest mesh results from an adaptive refinement algorithm using bisection and ‘hanging nodes’. We discuss the additional difficulties introduced by these meshes and investigate two different algorithms. The first algorithm uses merely the local refinement regions per level, leading to optimal solver complexity even on strongly locally refined meshes, whereas the second one constructs the lower level meshes by agglomeration of cells. In this note, we are mainly interested in implementation details and practical performance of the two multigrid schemes.