Adaptive discontinuous Galerkin methods for convection dominated optimal control problems

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Uygulamalı Matematik Enstitüsü, Türkiye

Tezin Onay Tarihi: 2012

Öğrenci: HAMDULLAH YÜCEL

Danışman: BÜLENT KARASÖZEN

Özet:

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 di usion partial di erential equations (PDEs). When convection dominates di usion, 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 e ciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite di erence, finite volume and continuous finite element methods for solving wave dominated problems like convection di usion 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 di usion 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