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：离散偏微分方程对应鞍点问题的数值求解与Stokes流等物理系统高度相关。然而，扩展此类系统的数值求解器常面临效率与收敛性的挑战。多重网格方法对Stokes方程等椭圆型问题具有极佳的适用性，可解决可扩展性与效率方面的挑战。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方案的选择。此外，在许多实际场景中，相比在所有可预见情况下追求松弛方案的最大效能，将多重网格方案作为Krylov子空间迭代求解器的预条件子可能更为有效。本文针对交错有限差分离散格式下的Stokes方程，提出一种高效对称多重网格预条件子。我们的贡献聚焦于设计一种预条件子，使其满足：(a) 对称不定性，与Stokes系统本身的特性相匹配；(b) 适用于预条件SQMR迭代格式；(c) 具备该应用场景所需的对称性质。此外，该设计在计算成本上具有高效性，并支持向大规模计算域扩展。