Nonsmooth Schur Newton Methods and Applications

The numerical simulation of the coarsening of binary alloys based on the

Cahn-Larch`e equations requires fast, reliable and robust solvers for Cahn-Hilliard equations

with logarithmic potential. After semi-implicit time discretization (cf.  Blowey and Elliott 92),

the resulting spatial problem can be reformulated as a non-smooth pde-constrained

optimal control problem with cost functional induced by the Laplacian. The associated

Karush-Kuhn-Tucker conditions take the form of a nonsmooth saddle point problem

degenerating to a variational inclusion in the deep quench limit.

Our considerations are based on recent work of Gr¨aser & Kornhuber 09 and the

upcoming dissertation of Gr¨aser 10. The starting point is the elimination of the

primal variable leading to a nonlinear Schur complement which turns out to be the

Fr´ech`et derivative of a convex functional. Now so-called nonsmooth Schur-Newton

methods can be derived as gradient-related descent methods applied to this functional.

In the discrete case we can show global convergence for an exact and

an inexact version independent of any regularization parameters. Local quadratic

convergence or finite termination can be shown for piecewise smooth nonlinearities

or in the deep quench limit respectively. The algorithm can be reinterpreted as a

preconditioned Uzawa method and generalizes the well-known primal-dual active

set strategy by Kunisch, Ito, and Hinterm¨uller 03. A (discrete) Allen-Cahn-type

problem and a linear saddle point problem have to be solved (approximately) in

each iteration step. In numerical computations we observe mesh-independent local

convergence for initial iterates provided by nested iteration. In the deep quench

limit, the numerical complexity of the (approximate) solution of the arising linear

saddle point problem dominates the detection of the actual active set.