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A Symmetric Multigrid-Preconditioned Krylov Subspace Solver for Stokes Equations

Yutian Tao, Eftychios Sifakis

TL;DR

The paper targets scalable solving of Stokes saddle-point systems discretized with a MAC staggered grid by introducing a symmetric multigrid preconditioner suitable for the SQMR Krylov method. It develops two symmetry-preserving smoothing strategies—symmetric distributive and symmetric Vanka—and combines them through a boundary/interior partitioning to maintain overall symmetry while achieving fast convergence. The preconditioner is applied within a V-cycle as a symmetric operator, with theoretical and practical considerations including adjoint restriction/prolongation, a coarsening by re-discretization, and a potential penalty-based stabilization. Extensive 2D and 3D numerical experiments show competitive convergence with classical MG and robust performance in challenging geometries, highlighting the method’s scalability and potential for GPU acceleration.

Abstract

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.

A Symmetric Multigrid-Preconditioned Krylov Subspace Solver for Stokes Equations

TL;DR

The paper targets scalable solving of Stokes saddle-point systems discretized with a MAC staggered grid by introducing a symmetric multigrid preconditioner suitable for the SQMR Krylov method. It develops two symmetry-preserving smoothing strategies—symmetric distributive and symmetric Vanka—and combines them through a boundary/interior partitioning to maintain overall symmetry while achieving fast convergence. The preconditioner is applied within a V-cycle as a symmetric operator, with theoretical and practical considerations including adjoint restriction/prolongation, a coarsening by re-discretization, and a potential penalty-based stabilization. Extensive 2D and 3D numerical experiments show competitive convergence with classical MG and robust performance in challenging geometries, highlighting the method’s scalability and potential for GPU acceleration.

Abstract

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.
Paper Structure (27 sections, 6 theorems, 28 equations, 18 figures, 1 table, 1 algorithm)

This paper contains 27 sections, 6 theorems, 28 equations, 18 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multigrid for the Stokes equations
  3. Design of smoothing operator and preserving symmetry
  4. Symmetric distributive smoother
  5. Symmetric Vanka smoother
  6. Symmetric integration of smoother
  7. Preconditioning
  8. Discrete Domain Design
  9. Numerical Convergence Experiments
  10. 2D Numerical Examples
  11. Driven Cavity Example
  12. Poiseuille Flow around A Cylinder Example
  13. Poiseuille Flow around Hollow Square Example
  14. Brancher Example
  15. 3D Numerical Examples
  16. ...and 12 more sections

Key Result

Lemma 1

Consider a sequence $\mathbf{x}^{(k+1)}=\mathbf{A}\mathbf{x}^{(k)}+\mathbf{b}$ with $\mathbf{x}^{(0)} = \mathbf{0}$. The general formula is $\mathbf{x}^{(n)} = \sum_{i=0}^{n-1}\mathbf{A}^i\mathbf{b}$.

Figures (18)

  • Figure 1: The staggered grid discretization in 2D (left) and 3D (right)
  • Figure 2: 2D discretization on fine and coarse level; just the horizontal component $u$ of the velocity field $\mathbf{u}=(u,v)$ is illustrated on the left, the placement of vertical velocity components $v$ is analogous, but on faces perpendicular to the $y$-axis.
  • Figure 3: 2D restriction operator; just the horizontal component $u$ of the velocity field $\mathbf{u}=(u,v)$ is illustrated on the left, the placement of vertical velocity components $v$ is analogous, but on faces perpendicular to the $y$-axis.
  • Figure 4: Vanka Smoother on 2D example
  • Figure 5: 2D Discrete Domain Example
  • ...and 13 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Theorem 4