Many real-life applications such as the shape optimization of technological devices, the identification
of parameters in environmental processes and flow control problems lead to optimization
problems governed by systems of convection diusion partial dierential equations
(PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit
layers on small regions where the solution has large gradients. Hence, it requires special numerical
techniques, which take into account the structure of the convection. The integration
of discretization and optimization is important for the overall eciency of the solution process.
Discontinuous Galerkin (DG) methods became recently as an alternative to the finite
dierence, finite volume and continuous finite element methods for solving wave dominated
problems like convection diusion equations since they possess higher accuracy.
This thesis will focus on analysis and application of DG methods for linear-quadratic convection
dominated optimal control problems. Because of the inconsistencies of the standard stabilized
methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion
optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to
the same discrete optimality systems. The other DG methods such as nonsymmetric interior
penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield
the same discrete optimality systems when penalization constant is taken large enough. We
will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained
and control constrained optimal control problems. In convection dominated optimal
control problems with boundary and/or interior layers, the oscillations are propagated downwind
and upwind direction in the interior domain, due the opposite sign of convection terms in
state and adjoint equations. Hence, we will use residual based a posteriori error estimators to
reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis
will be confirmed by several numerical examples with and without control constraints
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12614523/index.pdf |
| Date | 01 July 2012 |
| Creators | Yucel, Hamdullah |
| 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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