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

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.