Bounded and Positive Finite Element Approximations of Reaction-Diffusion Equations

Abstract

Traditionally, the numerical analysis of di erential equations focusses 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 di erential 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 Di erence Method (FDM), for example, the Non Standard Finite Di erence 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-di usion equation, and a coupled predator-prey reaction-di usion of Rosenzweig-MacArthur form.