A positivity-preserving unigrid method for elliptic PDEs

Abstract: 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.