Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction
equations arise in many science and engineering applications such as shape optimization of the technological
devices, identification of parameters in environmental processes and flow control problems.
A characteristic feature of convection dominated optimization problems is the presence of sharp layers.
In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions.
Hence, these problems require stabilization techniques to resolve boundary and interior layers accurately.
The Streamline Upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilization
technique for solving convection dominated OCPs.
The focus of this thesis is the application and analysis of the SUPG method for distributed and
boundary OCPs governed by evolutionary diffusion-convection-reaction equations. There are two approaches
for solving these problems: optimize-then-discretize and discretize-then-optimize. For the
optimize-then-discretize method, the time-dependent OCPs is transformed to a biharmonic equation,
where space and time are treated equally. The resulting optimality system is solved by the finite
element package COMSOL. For the discretize-then-optimize approach, we have used the so called allv
at-once method, where the fully discrete optimality system is solved as a saddle point problem at once
for all time steps. A priori error bounds are derived for the state, adjoint, and controls by applying
linear finite element discretization with SUPG method in space and using backward Euler, Crank-
Nicolson and semi-implicit methods in time. The stabilization parameter is chosen for the convection
dominated problem so that the error bounds are balanced to obtain L2 error estimates. Numerical examples
with and without control constraints for distributed and boundary control problems confirm the
effectiveness of both approaches and confirm a priori error estimates for the discretize-then-optimize
approach.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12615399/index.pdf |
| Date | 01 February 2013 |
| Creators | Seymen, Zahire |
| Contributors | Karasozen, Bulent |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for public access |
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