Type of Document	Dissertation
Author	Krueger, Denise A.
Author's Email Address	dkrueger@vt.edu
URN	etd-08022004-131444
Title	Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations
Degree	PhD
Department	Mathematics
Advisory Committee	Advisor Name Title King, Belinda B. Committee Chair Borggaard, Jeffrey T. Committee Member Burns, John A. Committee Member Iliescu, Traian Committee Member Zietsman, Lizette Committee Member
Keywords	Stabilized Finite Elements Convection-Diffusion Equation Linear Quadratic Regulator Problems Non-normal
Date of Defense	2004-07-20
Availability	unrestricted
Abstract We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for this effort comes from the observation that when linearization is applied to fluid flow control problems the resulting equations have the form of a convection diffusion equation. This effort is focused on the specific problem of computing the feedback functional gains that are the kernels of the feedback operators defined by solutions of operator Riccati equations. We develop a stabilization scheme based on the Galerkin Least Squares (GLS) method and compare this scheme to the standard Galerkin finite element method. We use cubic B-splines in order to keep the higher order terms that occur in GLS formulation. We conduct a careful numerical investigation into the convergence and accuracy of the functional gains computed using stabilization. We also conduct numerical studies of the role that the stabilization parameter plays in this convergence. Overall, we discovered that stabilization produces much better approximations to the functional gains on coarse meshes than the unstabilized method and that adjustments in the stabilization parameter greatly effects the accuracy and convergence rates. We discovered that the optimal stabilization parameter for simulation and steady state analysis is not necessarily optimal for solving the Riccati equation that defines the functional gains. Finally, we suggest that the stabilized GLS method might provide good initial values for iterative schemes on coarse meshes.
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