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Numerical solution of discrete PDEs corresponding to saddle point problems ishighly relevant to physical systems such as Stokes flow. However, scaling upnumerical solvers for such systems is often met with challenges in efficiencyand convergence. Multigrid is an approach with excellent applicability toelliptic problems such as the Stokes equations, and can be a solution to suchchallenges of scalability and efficiency. The degree of success of suchmethods, however, is highly contingent on the design of key components of amultigrid scheme, including the hierarchy of discretizations, and therelaxation scheme used. Additionally, in many practical cases, it may be moreeffective to use a multigrid scheme as a preconditioner to an iterative Krylovsubspace solver, as opposed to striving for maximum efficacy of the relaxationscheme in all foreseeable settings. In this paper, we propose an efficientsymmetric multigrid preconditioner for the Stokes Equations on a staggeredfinite-difference discretization. Our contribution is focused on crafting apreconditioner that (a) is symmetric indefinite, matching the property of theStokes system itself, (b) is appropriate for preconditioning the SQMR iterativescheme, and (c) has the requisite symmetry properties to be used in thiscontext. In addition, our design is efficient in terms of computational costand facilitates scaling to large domains.

A Symmetric Multigrid-Preconditioned Krylov Subspace Solver for Stokes Equation