  Multigrid methods are popular solution algorithms for many discretized PDEs,
either as standalone iterative solvers or as preconditioners, due to their high
efficiency. However, the choice and optimization of multigrid components such
as relaxation schemes and grid-transfer operators is crucial to the design of
optimally efficient algorithms. It is well--known that local Fourier analysis
(LFA) is a useful tool to predict and analyze the performance of these
components. In this paper, we develop a local Fourier analysis of monolithic
multigrid methods based on additive Vanka relaxation schemes for mixed
finite-element discretizations of the Stokes equations. The analysis offers
insight into the choice of "patches" for the Vanka relaxation, revealing that
smaller patches offer more effective convergence per floating point operation.
Parameters that minimize the two-grid convergence factor are proposed and
numerical experiments are presented to validate the LFA predictions.
