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.

翻译：对应于鞍点问题的离散偏微分方程（PDE）的数值求解与诸如斯托克斯流等物理系统密切相关。然而，将此类系统的数值求解器进行规模扩展通常面临效率和收敛性方面的挑战。多重网格方法对椭圆问题（如斯托克斯方程）具有出色的适用性，可有效解决此类可扩展性与效率的挑战。但此类方法的成功程度高度依赖于多重网格方案关键组件的设计，包括离散化层级结构和松弛策略。此外，在许多实际场景中，将多重网格方案作为迭代Krylov子空间求解器的预处理算子，可能比在所有可预见情况下追求松弛方案的最大效能更为有效。本文针对交错有限差分离散下的斯托克斯方程，提出了一种高效的对称多重网格预处理算子。我们的贡献集中于设计一种预处理算子，使其具备以下特性：(a) 对称不定性，与斯托克斯系统自身的性质相匹配；(b) 适用于预处理SQMR迭代方案；(c) 拥有在此类应用中必需的对称性质。此外，我们的设计在计算成本方面具有高效性，并支持向大规模计算域扩展。