A local Fourier analysis of additive Vanka relaxation for the Stokes equations

TL;DR

This paper develops a local Fourier analysis to optimize additive Vanka relaxation schemes in multigrid methods for Stokes equations, providing insights into patch size effects and parameter choices for improved convergence.

Contribution

It introduces a local Fourier analysis framework for additive Vanka relaxation in multigrid methods applied to Stokes equations, guiding optimal patch selection and parameter tuning.

Findings

Smaller patches improve convergence efficiency.

Proposed parameters minimize the two-grid convergence factor.

Numerical experiments validate LFA predictions.

Abstract

Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid-transfer operators is crucial to the design of optimally efficient algorithms. It is well--known that local Fourier analysis (LFA) is a useful tool to predict and analyze the performance of these components. In this paper, we develop a local Fourier analysis of monolithic multigrid methods based on additive Vanka relaxation schemes for mixed finite-element discretizations of the Stokes equations. The analysis offers insight into the choice of "patches" for the Vanka relaxation, revealing that smaller patches offer more effective convergence per floating point operation. Parameters that minimize the two-grid convergence factor are proposed and numerical experiments are presented to validate the LFA predictions.