  The discretization of Cahn-Hilliard equation with obstacle potential leads to
a block 2 by 2 non-linear system, where the p1, 1q block has a non-linear and
non-smooth term. Recently a globally convergent Newton Schur method was
proposed for the non-linear Schur complement corresponding to this non-linear
system. The solver may be seen as an inexact Uzawa method which has the falvour
of an active set method in the sense that the active sets are first identified
by solving a quadratic obstacle problem corresponding to the p1, 1q block of
the block 2 by 2 nonlinear system, and a new decent direction is obtained after
discarding the active set region. The problem becomes linear on nonactive set,
and corresponds to solving a linear saddle point problem on truncated domains.
For solving the quadratic obstacle problem, various optimal multigrid like
methods have been proposed. In this paper solvers for the truncated saddle
point problem is considered. Three preconditioners are considered, two of them
have block diagonal structure, and the third one has block tridiagonal
structure. One of the block diagonal preconditioners is obtained by adding
certain scaling of stiffness and mass matrices, whereas, the remaining two
involves Schur complement. Eigenvalue bound and condition number estimates are
derived for the preconditioned untruncated problem. It is shown that the
extreme eigenvalues of the preconditioned truncated system remain bounded by
the extreme eigenvalues of the preconditioned untruncated system. Numerical
experiments confirm the optimality of the solvers.
