Improving patch selection for monolithic multigrid solvers for high-order {T}aylor-{H}ood discretizations

TL;DR

This paper introduces an enhanced patch construction for monolithic multigrid solvers applied to high-order Taylor-Hood discretizations, significantly improving convergence and efficiency in fluid flow simulations.

Contribution

It proposes a novel overlap strategy in pressure degrees of freedom within Vanka patches, boosting multigrid solver performance for incompressible flow discretizations.

Findings

Reduced multigrid iteration counts

Faster time-to-solution in linear Stokes problems

Improved nonlinear iteration efficiency in Navier-Stokes simulations

Abstract

Numerical simulation of incompressible fluid flows has been an active topic of research in Scientific Computing for many years, with many contributions to both discretizations and linear and nonlinear solvers. In this work, we propose an improved relaxation scheme for higher-order Taylor-Hood discretizations of the incompressible Stokes and Navier-Stokes equations, demonstrating its efficiency within monolithic multigrid preconditioners for the linear(ized) equations. The key to this improvement is an improved patch construction for Vanka-style relaxation introducing, for the first time, overlap in the pressure degrees of freedom within the patches. Numerical results demonstrate significant improvement in both multigrid iterations and time-to-solution for the linear Stokes case, on both triangular and quadrilateral meshes. For the nonlinear Navier-Stokes case, we show similar improvements, including in the number of nonlinear iterations needed in an inexact Newton method. These improvements enable three-dimensional numerical simulations, which also presented.