Bounded and Positive Finite Element Approximations of Reaction-Diffusion Equations
Traditionally, the numerical analysis of differential equations focuses on stability, convergence and accuracy of the numerical methods employed. However in practice, there are other features of the numerical methods that we are interested in. Generally, when we discretize a differential equation, we would like to preserve certain intrinsic characteristics of the model, for example, positivity and boundedness of the quantities of interest. Consider-able progress has been made in this area for the Finite Difference Method(FDM), for example, the Non Standard Finite Difference Method (NSFDM). However, little work has been done on preserving positivity and bounded-ness for the numerical solutions of the Finite Element Method (FEM). In this work we develop a general framework for achieving this aim, which is based on the careful application of the Standard Galerkin FEM with 'Mass Lump-ing'. For concreteness, we present results for the Heat Equation, The Fisher reaction-diffusion equation, and a coupled predator-prey reaction-diffusion of Rosenzweig-MacArthur form.