  Numerical solution of discrete PDEs corresponding to saddle point problems is
highly relevant to physical systems such as Stokes flow. However, scaling up
numerical solvers for such systems is often met with challenges in efficiency
and convergence. Multigrid is an approach with excellent applicability to
elliptic problems such as the Stokes equations, and can be a solution to such
challenges of scalability and efficiency. The degree of success of such
methods, however, is highly contingent on the design of key components of a
multigrid scheme, including the hierarchy of discretizations, and the
relaxation scheme used. Additionally, in many practical cases, it may be more
effective to use a multigrid scheme as a preconditioner to an iterative Krylov
subspace solver, as opposed to striving for maximum efficacy of the relaxation
scheme in all foreseeable settings. In this paper, we propose an efficient
symmetric multigrid preconditioner for the Stokes Equations on a staggered
finite-difference discretization. Our contribution is focused on crafting a
preconditioner that (a) is symmetric indefinite, matching the property of the
Stokes system itself, (b) is appropriate for preconditioning the SQMR iterative
scheme, and (c) has the requisite symmetry properties to be used in this
context. In addition, our design is efficient in terms of computational cost
and facilitates scaling to large domains.
