Jeff Sandelin

Advisor: Dr. Don Estep

Degree Conferred: Fall 2006

After Graduation: Post Doc - Colorado State University

Website: http://www.math.colostate.edu/

Thesis Title:

Global Estimate and Control of Model, Numerical, and Parameter Error

Abstract:

Ordinary differential equations (ODEs) are used extensively in the modeling of natural phenomena in nearly all scientific and engineering fields. Since numerical simulations are a core scientific tool for the investigation of new phenomena, it necessary to quantify the uncertainties in numerical solutions of differential equations. In this thesis, we derive a posteriori error estimates based on adjoint analysis for global error estimation of numerical solutions of ODEs and design and implement various global error control mechanisms. Our solution methods are based on continuous and discontinuous Galerkin (cG and dG) finite element methods. We take a truly global approach to error estimation and control by treating the time dependent equations in a manner similar to elliptic problems and solving the equations for the entire time period, then applying a posteriori error estimates based on duality, the adjoint problem, the generalized Green's function, and variational analysis. We develop and explore new approaches to adaptive error control based on probability and weighted zones. These methods are implemented and compared to classic error control based on ``Equidistribution of Error'' using a variety of test problems. Finally, we provide analysis that shows that modeling error cannot be ignored.