  While constraints arise naturally in many physical models, their treatment in
mathematical and numerical models varies widely, depending on the nature of the
constraint and the availability of simulation tools to enforce it. In this
paper, we consider the solution of discretized PDE models that have a natural
constraint on the positivity (or non-negativity) of the solution. While
discretizations of such models often offer analogous positivity properties on
their exact solutions, the use of approximate solution algorithms (and the
unavoidable effects of floating -- point arithmetic) often destroy any
guarantees that the computed approximate solution will satisfy the (discretized
form of the) physical constraints, unless the discrete model is solved to much
higher precision than discretization error would dictate. Here, we introduce a
class of iterative solution algorithms, based on the unigrid variant of
multigrid methods, where such positivity constraints can be preserved
throughout the approximate solution process. Numerical results for one- and
two-dimensional model problems show both the effectiveness of the approach and
the trade-off required to ensure positivity of approximate solutions throughout
the solution process.
