  In the past decades, multigrid methods for linear systems having multilevel
Toeplitz coefficient matrices with scalar entries have been largely studied. On
the other hand, only few papers have investigated the case of block entries,
where the entries are small generic matrices instead of scalars. In that case
the efforts of the researchers have been mainly devoted to specific
applications, focusing on algorithmic proposals but with very marginal
theoretical results.
  In this paper, we propose a general two-grid convergence analysis proving an
optimal convergence rate independent of the matrix size, in the case of
positive definite block Toeplitz matrices with generic blocks. In particular,
the proof of the approximation property has not a straightforward
generalization of the scalar case and in fact we have to require a specific
commutativity condition on the block symbol of the grid transfer operator.
Furthermore, we define a class of grid transfer operators satisfying the
previous theoretical conditions and we propose a strategy to insure fast
multigrid convergence even for more than two grids.
  Among the numerous applications that lead to the block Toeplitz structure,
high order Lagrangian finite element methods and staggered discontinuous
Galerkin methods are considered in the numerical results, confirming the
effectiveness of our proposal and the correctness of the proposed theoretical
analysis.
