"A robust all-at-once multigrid method for the Stokes control problem"
, in Numerische Mathematik, Vol. 180, Nummer 3, Springer Berlin Heidelberg, Seite(n) 517-540, 7-2015, ISBN: 0945-3245, ISSN: 0945-3245
A robust all-at-once multigrid method for the Stokes control problem
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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.