Optimal multigrid preconditioned semi-monotonic augmented Lagrangians applied to the Stokes problem

We propose an optimal computational complexity algorithm for the solution of quadratic programming problems with equality constraints arising from partial differential equations. The algorithm combines a variant of the semi–monotonic augmented Lagrangian (SMALE) method with adaptive precision control and a multigrid preconditioning for the Hessian of the cost function and for the inner product on the space of Lagrange variables. In our approach there is no need for preconditioning of the constraints. The optimality of the algorithm is theoretically proven and confirmed by numerical experiments for the 2–dimensional Stokes problem. Copyright c © 2000 John Wiley & Sons, Ltd.