9805748 Estep Systems of quasi-linear reaction-diffusion-convection differential equations have widespread importance in modeling physical phenomena in such diverse fields as biology, chemistry, metallurgy, and combustion. Some examples are shear flow in fluids with temperature dependent viscosity; Hodgkin-Huxley type models of nerve bundles; and models of the spread of disease in populations. Because such problems are typically highly nonlinear and exhibit complex behavior, numerical solutions of the differential equations have become the main tool for investigation of their properties. However, these same qualities give rise to some fundamental scientific questions: How can reliable and accurate estimates of the accuracy of information computed from numerical solutions be obtained and how can a desired range of accuracy be achieved? The project will address these question on three levels: theoretical analysis; implementation in code; and application to practical problems. The proposed approach is based on developing an a posteriori theory of error estimation in which the error is estimated in terms of quantities depending on the numerical solution that can be computed or approximated. The theory will then be used to develop methods of computational error estimation and adaptive error control. The PI will also analyze the reliability and accuracy of the computational error estimates and, because stability has a direct impact on the accuracy of numerical solutions, the preservation of stability properties of the differential equation under discretization. Practical goals of this project include; a publicly-accessible code for solving general reaction-diffusion problems using computational error estimation and adaptive error control, applications to the practical models like those mentioned above, and an educational component in the form of a textbook on the finite element method for the basic nonlinear models of science.
