  In the literature, there exist several studies on symbol-based multigrid
methods for the solution of linear systems having structured coefficient
matrices. In particular, the convergence analysis for such methods has been
obtained in an elegant form in the case of Toeplitz matrices generated by a
scalar-valued function. In the block-Toeplitz setting, that is, in the case
where the matrix entries are small generic matrices instead of scalars, some
algorithms have already been proposed regarding specific applications and a
first rigorous convergence analysis has been performed in [7]. However, with
the existent symbol-based theoretical tools, it is still not possible to prove
the convergence of many multigrid methods known in the literature. This paper
aims to generalize the previous results giving more general sufficient
conditions on the symbol of the grid transfer operators.In particular, we treat
matrix-valued trigonometric polynomials which can be non-diagonalizable and
singular at all points and we express the new conditions in terms of the
eigenvectors associated with the ill-conditioned subspace. Moreover, we extend
the analysis to the V-cycle method proving a linear convergence rate under
stronger conditions, which resemble those given in the scalar case. In order to
validate our theoretical findings, we present a classical block structured
problem stemming from a FEM approximation of a second order differential
problem. We focus on two multigrid strategies that use the geometric and the
standard bisection grid transfer operators and we prove that both fall into the
category of projectors satisfying the proposed conditions. In addition, using a
tensor product argument, we provide a strategy to construct efficient V-cycle
procedures in the block multilevel setting.
