  We present a robust and efficient multigrid method for single-patch
isogeometric discretizations using tensor product B-splines of maximum
smoothness. Our method is based on a stable splitting of the spline space into
a large subspace of interior splines which satisfy a robust inverse inequality,
as well as one or several smaller subspaces which capture the boundary effects
responsible for the spectral outliers which occur in Isogeometric Analysis. We
then construct a multigrid smoother based on an additive subspace correction
approach, applying a different smoother to each of the subspaces. For the
interior splines, we use a mass smoother, whereas the remaining components are
treated with suitably chosen Kronecker product smoothers or direct solvers.
  We prove that the resulting multigrid method exhibits iteration numbers which
are robust with respect to the spline degree and the mesh size. Furthermore, it
can be efficiently realized for discretizations of problems in arbitrarily high
geometric dimension. Some numerical examples illustrate the theoretical results
and show that the iteration numbers also scale relatively mildly with the
problem dimension.
