We consider symmetric positive definite variational problems with tensor product structure. These problems typically arise from coordinate transformations of standard elliptic problems and cannot be solved by standard multigrid techniques due to lack of ellipticity in some coordinate directions. We introduce a hierarchy of finite element discretizations of these problems using tensor product base functions and give a multigrid algorithm that makes use of block smoothers in the problematic coordinate directions and semicoarsening in the remaining directions. Using a variant of standard Fourier analysis based on the investigation of invariant subspaces, we can prove level-independent convergence estimates that are stable with respect to the operators corresponding to the problematic directions. Using standard Fourier techniques, an optimal convergence bound for the two-dimensional case is derived.
Numerical examples demonstrate that our algorithm works for more general settings, too.