  The main focus of this paper is the study of efficient multigrid methods for
large linear systems with a particular saddle-point structure. Indeed, when the
system matrix is symmetric, but indefinite, the variational convergence theory
that is usually used to prove multigrid convergence cannot be directly applied.
However, different algebraic approaches analyze properly preconditioned
saddle-point problems, proving convergence of the Two-Grid method. In
particular, this is efficient when the blocks of the coefficient matrix possess
a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient
conditions for convergence and provide optimal parameters for the
preconditioning of the saddle-point problem in terms of the associated
generating symbols. In this paper, we propose a symbol-based convergence
analysis for problems that have a hidden block Toeplitz structure. Then, they
can be investigated focusing on the properties of the associated generating
function f, which consequently is a matrix-valued function with dimension
depending on the block size of the problem. As numerical tests we focus on the
matrix sequence stemming from the finite element approximation of the Stokes
problem. We show the efficiency of the methods studying the hidden 9-by-9 block
multilevel structure of the obtained matrix sequence. Moreover, we propose an
efficient algebraic multigrid method with convergence rate independent of the
matrix size. Finally, we present several numerical tests comparing the results
with state-of-the-art strategies.
