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. The update rule for penalty parameter acts as preconditioning of constraints. The optimality of the algorithm is theoretically proven and confirmed by numerical experiments for the two‐dimensional Stokes problem. Copyright © 2007 John Wiley & Sons, Ltd.