Generalized Optimal AMG Convergence Theory for Stokes Equations Using Smooth Aggregation and Vanka Relaxation Strategies

TL;DR

This paper develops a generalized optimal AMG convergence theory for Stokes equations, applying it to smooth aggregation AMG with Vanka relaxation strategies, and providing accurate convergence rate bounds for saddle-point systems.

Contribution

It introduces a rigorous convergence analysis framework for AMG applied to saddle-point problems, specifically for Stokes equations, incorporating Vanka relaxation strategies.

Findings

The theory accurately predicts AMG convergence rates for Stokes problems.

Vanka relaxation strategies improve AMG solver effectiveness for saddle-point systems.

Multiplicative Vanka relaxation effectively handles velocity-pressure coupling.

Abstract

This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Although symmetry can be restored by eliminating the partition of unity, this compromises convergence. Alternatively, multiplicative Vanka relaxation updates velocity and pressure sequentially within each patch, propagating updates multiplicatively across the domain and effectively addressing velocity-pressure coupling, ensuring a symmetric relaxation. We demonstrate that the generalized optimal AMG theory consistently provides accurate lower bounds on the convergence rate for SA-AMG applied to Stokes equations. These findings suggest potential avenues for further enhancement in AMG solver design for saddle-point systems.