# An All-at-Once Multigrid Method Applied to a Stokes Control Problem

Stefan Takacs^{*}

In this talk we consider a Stokes control problem of tracking type (velocity tracking problem). The discretisation of the optimality system (KKT system) characterising the solution of such a PDE-constrained optimisation problem leads to a large-scale sparse linear system. This system is symmetric but not positive definite. Therefore, standard iterative solvers are typically not the best choice. The KKT system is a linear system for two blocks of variables: the primal variables (velocity field, pressure distribution and control) and the Lagrange multipliers introduced to incorporate the partial differential equation. Based on this natural block-structure, we can verify that this system has a saddle point structure where the $(1, 1)$-block and the $(2, 2)$-block are positive semidefinite. Contrary to the case of elliptic optimal control problems, the $(1, 2)$-block is not positive definite but a saddle point problem itself.\newline We are interested in fast iteration schemes with convergence rates bounded away from one by a constant which is independent of the discretisation parameter (the grid size) and of problem parameters, like in the regularisation parameter appearing in the tracking functional of the model problem. To achieve this goal, we propose an all-at-once multigrid approach. In the talk we will discuss the choice of an appropriate smoother and we will give convergence theory.

Mathematics Subject Classification: 65N55 35Q93

Keywords: Multigrid; KKT system; Stokes; Optimal control problem

Minisymposion: Preconditioning for PDE-Constrained Optimization