Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations

TL;DR

This novel monolithic algebraic multigrid solver for the discrete Stokes problem discretized with stable mixed finite elements not only meets but frequently surpasses the performance of inexact Uzawa preconditioners, demonstrating the versatility and robust performance across a diverse spectrum of problem sets.

Abstract

We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The algorithm is based on the use of the lower-order $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator within a defect-correction setting, in combination with AMG construction of interpolation operators for velocities and pressures. The preconditioning framework is primarily algebraic, though the $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator must be provided. We investigate two relaxation strategies in this setting. Specifically, a novel block factorization approach is devised for Vanka patch systems, which significantly reduces storage requirements and computational overhead, and a Chebyshev adaptation of the LSC-DGS relaxation is developed to improve parallelism. The preconditioner demonstrates robust performance across a variety of 2D and 3D Stokes problems, often matching or exceeding the effectiveness of an inexact block-triangular (or Uzawa) preconditioner, especially in challenging scenarios such as elongated-domain problems.

Figures (12)

• Figure 1: Mesh refinements. (a) structured 2D triangular mesh; (b) the of (a); (c) barycentric mesh (Alfeld split), obtained via of (a); and (d) of (c).
• Figure 1: (Left) one Vanka patch for the $\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$ discretization; (middle) one Vanka patch for the $\pmb{\mathbb{P}}_2/\mathbb{P}_1$ discretization; and (right) one Vanka patch for the $\pmb{\mathbb{P}}_1 \text{iso}\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$ discretization. The pressure DoF(s) (red circles, ) and the velocity DoFs directly algebraically connected to that pressure DoF (gray squares, ) are marked. Displayed patches are on 2D triangular meshes.
• Figure 1: Solver diagram for defect-correction preconditioners. If relaxation is performed on all levels, the diagram corresponds to the DC-all solver.
• Figure 1: Test problems for Section \ref{['sec:application_problems']}
• Figure 2: Meshes used to construct the Stokes problems. The boundaries are marked with three colors (red, blue, and gray), corresponding to three types of velocity field boundary conditions (BCs). Red corresponds to non-zero Dirichlet boundaries. Blue corresponds to Neumann boundary conditions. Gray indicates zero Dirichlet BCs.

Theorems & Definitions (4)

• Remark 3.1: $\pmb{\mathbb{P}}_2/\mathbb{P}_1$ to $\pmb{\mathbb{P}}_1 \text{iso}\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$ Grid-Transfer Operators
• Remark 3.2: Applying \ref{['alg:stokes_mg_setup']} directly to $K_0$
• Remark 5.1: DLSC relaxation-based AMG Preconditioners
• Remark 5.2: Velocity Mass Matrix Scaling in DLSC