Oliver Roehrle Abstract

PhD, September 2004

Multilevel First-Order System Least Squares for Quasilinear Elliptic Partial Differential Equations

Advisors: Steve McCormick, Tom Manteuffel, Jeffrey Heys, John Ruge and Kamran Mohseni (ASEN)
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The goal of this research is to develop efficient multilevel solvers for two-dimensional elliptic systems of nonlinear partial differential equations (PDEs), where the nonlinearity is of the type uv.

The Navier-Stokes equations are an important representative of this class and are at the focus of this study. Using a first-order system least-squares (FOSLS) approach and introducing a new variable for v, for this class of PDEs we obtain a formulation in which the nonlinearity appears as a product of two different variables. The result is a system that is linear within each variable but nonlinear in the cross terms.

A nested Newton-FOSLS-Multigrid method strategy is analyzed for solving this class of PDEs, and a new multilevel method is developed for handling the nonlinearity directly. Optimal performance is established numerically and theoretically for both approaches.