A robust multigrid method for the time-dependent Stokes problem

TL;DR

This paper introduces a new coupled multigrid method for the time-dependent Stokes problem that avoids solving sub-problems during smoothing, demonstrating robust convergence regardless of grid size or time step length.

Contribution

The paper presents a novel coupled multigrid approach that eliminates the need for solving Poisson sub-problems, enhancing efficiency and robustness for time-dependent Stokes simulations.

Findings

The method converges robustly across different grid sizes.

It maintains efficiency without solving Poisson sub-problems.

Convergence is independent of time step length.

Abstract

In the present paper we propose a coupled multigrid method for generalized Stokes flow problems. Such problems occur as subproblems in implicit time-stepping approaches for time-dependent Stokes problems. The discretized Stokes system is a large-scale linear system whose condition number depends on the grid size of the spatial discretization and of the length of the time step. Recently, for this problem a coupled multigrid method has been proposed, where in each smoothing step a Poisson problem has to be solved (approximately) for the pressure field. In the present paper, we propose a coupled multigrid method where the solution of such sub-problems is not needed. We prove that the proposed method shows robust convergence behavior in the grid size of the spatial discretization and of the length of the time-step.