Submitted

Multigrid Methods for Constrained Minimization Problems and Application to Saddle Point Problems

Long Chen

Submitted

arXiv   Bibtex

ABSTRACT:

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.