A restricted additive Vanka smoother for geometric multigrid

TL;DR

This paper introduces a restricted additive Vanka smoother for geometric multigrid methods, improving efficiency and scalability in solving saddle-point problems like the Stokes equations, with comparable convergence to classical methods.

Contribution

The paper proposes a novel restricted additive Vanka smoother that enhances multigrid efficiency and scalability for saddle-point problems, demonstrating competitive convergence and reduced computational cost.

Findings

Achieves convergence rates similar to classical Vanka smoothers

Reduces computational cost per iteration

Faster overall solution runtimes for Stokes problems

Abstract

The solution of saddle-point problems, such as the Stokes equations, is a challenging task, especially in large-scale problems. Multigrid methods are one of the most efficient solvers for such systems of equations and can achieve convergence rates independent of the problem size. The smoother is a crucial component of multigrid methods and significantly affects its overall efficiency. We propose a Vanka-type smoother that we refer to as Restricted Additive Vanka and investigate its convergence in the context of adaptive geometric multigrid methods for the Stokes equations. The proposed smoother has the advantage of being an additive method and provides favorable properties in terms of algorithmic complexity, scalability and applicability to high-performance computing. We compare the performance of the smoother with two variants of the classical Vanka smoother using numerical benchmarks for the Stokes problem. We find that the restricted additive smoother achieves comparable convergence rates to the classical multiplicative Vanka smoother while being computationally less expensive per iteration, which results in faster solution runtimes.