The main focus of this paper is the study of efficient multigrid methods for
large linear system with a particular saddle-point structure. In particular, 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
of the problem. As numerical tests we focus on the matrix sequence stemming
from the finite element approximation of the Stokes equation. We show the
efficiency of the methods studying the hidden 9×9 block structure of
the obtained matrix sequence proposing an efficient algebraic multigrid method
with convergence rate independent of the matrix size. Moreover, we present
several numerical tests comparing the results with different known strategies