Mathematical Sciences: Adaptive Numerical Methods for Singularly-Perturbed Reaction-Diffusion Equations

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Estep proposes to analyze and implement adaptive finite element methods for systems of singularly-perturbed reaction-diffusion equations and dissipative ordinary differential equations. Solutions of such problems typically evolve on multiple scales, and to be useful, numerical results need to be uniformly accurate for the physically meaningful range of the parameters. He is attempting to overcome these difficulties by constructing a theory of adaptive error control based on a posteriori and a priori error analyses. A posteriori results measure the error in terms of the regularity of the approximation and the stability of the solution and provide a basis for adapting the discretization. A priori results bound the error in terms of the regularity of the solution and the stability of the scheme and guarantee convergence. Estep is also examining the dynamical properties of numerical schemes in the context of producing approximations with the correct dynamical behavior. Finally, he is studying the implementation of adaptive error control in finite element codes for parallel computers. The ultimate goal of this project is the public release of a parallel code that can solve systems of reaction-diffusion equations in two and three dimensions with minimal user interface. Many models in applied science result in nonlinear reaction-diffusion differential equations that contain source terms balanced against terms that diffuse energy. This balance is usually delicate and difficult to handle mathematically, consequently numerical approximation is an important tool. Yet, solutions of such problems typically evolve on several scales, i.e. some interesting behavior occurs in very localized regions in space-time while other behavior evolves over long times and over larger regions in space. The use of a uniform numerical discretization for a real application results in huge computations that tax even the largest computers. Estep's goal is to produce numerical schemes that adapt themselves to the localized behavior of the target solution so as to make the computations both as accurate as desired and as efficient as possible. Mathematically, he is trying to understand how to use the information provided by an approximation to adapt the discretization, that is, make the computations self-governing. He is also working on the implementation of this theory into a parallel code that solves very general problems with minimum user input. Success of this project will lead to the public release of the code, to the great benefit of the engineering and scientific communities.