Title page for ETD etd-08022004-131444

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
  • Stabilized Finite Elements
  • Convection-Diffusion Equation
  • Linear Quadratic Regulator Problems
  • Non-normal
Date of Defense 2004-07-20
Availability unrestricted
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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