Patch Smoothers for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

Abstract

We consider a multigrid method for solving the discretized optimality system of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a class of elliptic optimal control problems. A rigorous multigrid convergence analysis yields level-independent convergence rates. Numerical experiments indicate that the convergence rates are also independent of the involved regularization parameter.