We investigate the use of piecewise linear systems, whose coefficient matrix
is a piecewise constant function of the solution itself. Such systems arise,
for example, from the numerical solution of linear complementarity problems and
in the numerical solution of free-surface problems. In particular, we here
study their application to the numerical solution of both the (linear)
parabolic obstacle problem and the obstacle problem. We propose a class of
effective semi-iterative Newton-type methods to find the exact solution of such
piecewise linear systems. We prove that the semiiterative Newton-type methods
have a global monotonic convergence property, i.e., the iterates converge
monotonically to the exact solution in a finite number of steps. Numerical
examples are presented to demonstrate the effectiveness of the proposed
methods.