A robust all-at-once multigrid method for the Stokes control problem

TL;DR

This paper introduces a multigrid method for the Stokes control problem that achieves robust convergence rates unaffected by grid size or regularization parameters, improving computational efficiency.

Contribution

It develops an all-at-once multigrid approach with proven convergence robustness for large-scale optimality systems in Stokes control problems.

Findings

Convergence rates are independent of grid size.

Convergence rates are independent of regularization parameters.

Method outperforms existing preconditioners in robustness.

Abstract

In this paper we present an all-at-once multigrid method for a distributed Stokes control problem (velocity tracking problem). For solving such a problem, we use the fact that the solution is characterized by the optimality system (Karush-Kuhn-Tucker-system). The discretized optimality system is a large-scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem. Recently, block-diagonal preconditioners have been proposed, which allow to solve the problem using a Krylov space method with convergence rates that are robust in both, the grid size and the regularization parameter or cost parameter. In the present paper, we develop an all-at-once multigrid method for a Stokes control problem and show robust convergence, more precisely, we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter.