Finite element methods for nonlinear reaction-diffusion systems with applications in biology.

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The world is nonlinear and in order to make sense of such a complicated system we use mathematical models to capture the key features of what we study. The most successful mathematical models employ nonlinear partial differential equations, which can be tailored to fit a wide variety of situations. An example of this type are reaction-diffusion equations, which attempt to capture the essential behaviour of a number of different interacting 'species' that also disperse in space. The 'species' in this case can be chemical, biological, or some physical quantity whose density varies in space and time. For example, we may be interested in tracking the fluctuating numbers of foxes and rabbits in a forest, or understand how certain chemicals react and diffuse to produce patterns in a beaker during a laboratory experiment. The problem with such models is that they are based on a continuous representation of time and space, and are usually highly nonlinear. Thus these models almost never have exact solutions that can be worked out on paper. Instead, we must approximate these continuum models using discrete algorithms that can be programmed on a computer. It is then the job of the numerical analyst to rigorously prove that the discrete algorithm approximates the original continuum model with sufficient accuracy, provided we make our computer programs work hard enough. The most widely used, successful and flexible family of discrete algorithms used for approximating nonlinear reaction-diffusion equations are called finite element methods. The research proposal addresses how to construct efficient and accurate finite element methods for specific applications, without sacrificing the key physical features of the model concerned. The physical situations that this work applies to are varied, and include metapopulation dynamics (populations living in fragmented landscapes), pattern formation models (e.g., models designed to understand how skin patterns develop), and stochastic differential equations (e.g., models that incorporate random environmental factors).