Discontinuous galerkin methods for time-dependent convection dominated optimal control problems

Akman, Tuğba
Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.


Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models
Yücel, Hamdullah; BENNER, Peter (2013-11-11)
In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component prob...
Discontinuous galerkin finite elements method with structure preserving time integrators for gradient flow equations
Sarıaydın Filibelioğlu, Ayşe; Karasözen, Bülent; Department of Scientific Computing (2015)
Gradient flows are energy driven evolutionary equations such that the energy decreases along solutions. There have been surprisingly a large number of well-known partial differential equations (PDEs) which have the structure of a gradient flow in different research areas such as fluid dynamics, image processing, biology and material sciences. In this study, we focus on two systems which can be modeled by gradient flows;Allen-Cahn and Cahn-Hilliard equations. These equations model the phase separation in mat...
Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations
UZUNCA, Murat; Karasözen, Bülent; Manguoğlu, Murat (2014-09-04)
In this work, we apply the adaptive discontinuous Galerkin (DGAFEM) method to the convection dominated non-linear, quasi-stationary diffusion convection reaction equations. We propose an efficient preconditioner using a matrix reordering scheme to solve the sparse linear systems iteratively arising from the discretized non-linear equations. Numerical examples demonstrate effectiveness of the DGAFEM to damp the spurious oscillations and resolve well the sharp layers occurring in convection dominated non-line...
Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints
Yücel, Hamdullah (2013-08-30)
We study the numerical solution of control constrained optimal control problems governed by a system of convection diffusion equations with nonlinear reaction terms, arising from chemical processes. Control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method or by adding a Moreau-Yosida-type penalty function to the cost functional. An adaptive mesh refinement indicated by a posteriori error estimates is applied for both approaches.
Yücel, Hamdullah; Benner, Peter (2018-01-01)
In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. ...
Citation Formats
T. Akman, “Discontinuous galerkin methods for time-dependent convection dominated optimal control problems,” M.S. - Master of Science, Middle East Technical University, 2011.