  The first order condition of the constrained minimization problem leads to a
saddle point problem. A multigrid method using a multiplicative Schwarz
smoother for saddle point problems can thus be interpreted as a successive
subspace optimization method based on a multilevel decomposition of the
constraint space. Convergence theory is developed for successive subspace
optimization methods based on two assumptions on the space decomposition:
stable decomposition and strengthened Cauchy-Schwarz inequality, and
successfully applied to the saddle point systems arising from mixed finite
element methods for Poisson and Stokes equations. Uniform convergence is
obtained without the full regularity assumption of the underlying partial
differential equations. As a byproduct, a V-cycle multigrid method for
non-conforming finite elements is developed and proved to be uniform convergent
with even one smoothing step.
