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数值解与Stokes流等物理系统高度相关。然而，对此类系统进行数值求解器的规模化扩展常面临效率与收敛性方面的挑战。多重网格方法对于Stokes方程这类椭圆问题具有极佳的适用性，可成为解决可扩展性与效率挑战的有效方案。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及所采用的松弛格式。此外，在许多实际场景中，相较于追求松弛格式在所有可预见场景下的最大效能，将多重网格方案作为迭代式Krylov子空间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化的Stokes方程，提出一种高效的对称多重网格预条件子。我们的贡献主要在于构建满足以下特性的预条件子：(a) 具有对称不定性，与Stokes系统自身特性匹配；(b) 适用于SQMR迭代方案的预条件处理；(c) 具备在该应用场景下所需的对称性质。此外，我们的设计在计算成本方面具有高效性，并能促进大规模计算域的扩展。