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

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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.

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Acknowledgments

The author thanks Markus Kollmann for providing parts of the code used to compute the numerical results presented in this paper. Moreover, the support of the numerical analysis group of the Mathematical Institute, University of Oxford, is gratefully acknowledged. Finally, the author thanks the referees for very valuable remarks that have allowed to improve the paper significantly.

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  1. Mathematical Institute, University of Oxford, Oxford, UK

    Stefan Takacs

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  1. Stefan Takacs
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Correspondence to Stefan Takacs.

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The research was funded by the Austrian Science Fund (FWF): J3362-N25.

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Takacs, S. A robust all-at-once multigrid method for the Stokes control problem. Numer. Math. 130, 517–540 (2015). https://doi.org/10.1007/s00211-014-0674-5

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  • DOI: https://doi.org/10.1007/s00211-014-0674-5

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