Multigrid methods for optimal control problems governed by convection-diffusion equations

Arslantaş, Özgün Murat
Linear-quadratic optimal control problems governed by partial differential equations proved themselves important through their use in many real life applications. In order to solve the large scale linear system of equations that results from optimality conditions of the optimization problem, efficient solvers are required. For this purpose, multigrid methods, with an ordering technique to deal with the dominating convection, can be good candidates. This thesis investigates an application of the multigrid methods for the linear-quadratic optimal control problems governed by convection-diffusion equation, discretized by a discontinuous Galerkin method, namely, symmetric interior penalty Galerkin (SIPG) method. Further, an ordering technique called Downwind Numbering is proposed to reduce the number of iteration in multigrid approach


Numerical solution of semi-linear advection-diffusion-reaction equations by discontinuous galerkin methods
Yıldız, Süleyman; Karasözen, Bülent; Department of Scientific Computing (2016)
IIn this thesis, we study splitting methods for semi-linear advection-diffusion-reaction (ADR) equations which are discretized by the symmetric interior penalty Galerkin (SIPG) method in space. For the time integration Rosenbrock methods are used with Strang splitting. The linear system of equations are solved iteratively by preconditioned generalized minimum residual method (GMRES). Numerical experiments for ADR equations with different type nonlinearities demonstrate the effectiveness of the proposed appr...
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...
Implementation of different flux evaluation schemes into a two-dimensional Euler solver
Eraslan, Elvan; Aksel, Mehmet Haluk; Department of Mechanical Engineering (2006)
This study investigates the accuracy and efficiency of several flux splitting methods for the compressible, two-dimensional Euler equations. Steger-Warming flux vector splitting method, Van Leer flux vector splitting method, The Advection Upstream Splitting Method (AUSM), Artificially Upstream Flux Vector Splitting Scheme (AUFS) and Roe’s flux difference splitting schemes were implemented using the first- and second-order reconstruction methods. Limiter functions were embedded to the second-order reconstruc...
Ciydem, Mehmet; Koç, Seyit Sencer (2014-03-01)
Numerical methods in space-time have long been used to solve Maxwell's partial differential equations (PDEs) accurately. Finite Difference Time Domain (FDTD), one of the most widely used method, solves Maxwell's PDEs directly in computational grid. In FDTD, grid spacings (Delta x, Delta y, Delta z) are selected to properly sample field quantities to avoid aliasing and maximum allowable time-step (Delta t) is determined to ensure numerical stability of algorithm. Due to discretization of PDEs, FDTD inherentl...
Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints
Yücel, Hamdullah; Karasözen, Bülent (2014-01-02)
In this paper, we study a posteriori error estimates of the upwind symmetric interior penalty Galerkin (SIPG) method for the control constrained optimal control problems governed by linear diffusion-convection-reaction partial differential equations. Residual based error estimators are used for the state, the adjoint and the control. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented for convection dominated problems to illustrate the theoretic...
Citation Formats
Ö. M. Arslantaş, “Multigrid methods for optimal control problems governed by convection-diffusion equations,” M.S. - Master of Science, Middle East Technical University, 2015.