Towards Matrix-Free Patch Smoothers for the Stokes Problem: Evaluating Local p-Multigrid Solvers

TL;DR

This paper investigates matrix-free, p-multigrid-based vertex-patch smoothers for the Stokes problem, demonstrating that a single local iteration can achieve convergence comparable to exact solvers, thus improving efficiency.

Contribution

It introduces a fully iterative, matrix-free local solver approach for vertex-patch smoothers in multigrid methods for the Stokes problem, showing effectiveness with minimal local solves.

Findings

Single local iteration achieves comparable convergence to exact solves.

Braess-Sarazin preconditioner shows high resilience across scenarios.

Method effective on distorted meshes and with viscosity jumps.

Abstract

Vertex-patch smoothers offer an effective strategy for achieving robust geometric multigrid convergence for the Stokes equations, particularly in the context of high-order finite elements. However, their practical efficiency is often limited by the computational cost of solving the local saddle-point problems, especially when explicit matrix factorizations are not feasible. We explore a fully iterative, matrix-free-compatible approach to the local patch solve using $p$-multigrid techniques. We evaluate different local solver configurations: Braess-Sarazin and block-triangular preconditioners. Our numerical experiments suggest that the Braess-Sarazin approach is particularly resilient. We find that a single iteration of the local solver yields global convergence rates comparable to those obtained with exact local solvers, even on distorted meshes and in the presence of large viscosity jumps.