Hamdullah Yücel

Distributed Optimal Control of Diffusion-Convection-Reaction Equations Using Discontinuous Galerkin Methods

We discuss the symmetric interior penalty Galerkin (SIPG) method, the nonsymmetric interior penalty Galerkin (NIPG) method, and the incomplete interior penalty Galerkin (IIPG) method for the discretization of optimal control problems governed by linear diffusion-convection-reaction equations. For the SIPG discretization the discretize-then-optimize (DO) and the optimize-then-discretize (OD) approach lead to the same discrete systems and in both approaches the observed L 2 convergence for states and controls is \(O({h}^{k+1})\), where k is the degree of polynomials used. The situation is different for NIPG and IIPG, where the the DO and the OD approach lead to different discrete systems. For example, when standard penalization is used, the L 2 error in the controls is only O(h) independent of k. However, if superpenalization is used, the lack of adjoint consistency is reduced and the observed convergence for NIPG and IIPG is essentially equal to that of the SIPG method in the DO and OD approach.

A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations

In this paper, we analyze the symmetric interior penalty Galerkin (SIPG) for distributed optimal control problems governed by unsteady convection diffusion equations with control constraint bounds. A priori error estimates are derived for the semi- and fully-discrete schemes by using piecewise linear functions. Numerical results are presented, which verify the theoretical results.

A discontinuous Galerkin method for optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms

In this paper, we study the numerical solution of optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms, arising from chemical processes. The symmetric interior penalty Galerkin (SIPG) method with upwinding for the convection term is used as a discretization method. We use a residual-based error estimator for the state and the adjoint variables. An adaptive mesh refinement indicated by a posteriori error estimates is applied. The arising saddle point system is solved using a suitable preconditioner. Numerical results are presented to illustrate the performance of the proposed error estimator.

Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models

Abstract 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 problems.

Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations

We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau–Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented to illustrate the effectiveness of the adaptivity for both regularization techniques.

Energy Stable Discontinuous Galerkin Finite Element Method for the Allen–Cahn Equation

Allen--Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals is discretized using symmetric interior penalty discontinuous Galerkin (SIPG) finite elements in space. We show that the energy stable average vector field (AVF) method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for the fully discrete scheme. The numerical results for one and two dimensional Allen-Cahn equation with periodic boundary condition, using adaptive time stepping, reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed and the ripening time is detected correctly.

Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints

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.

Adaptive Symmetric Interior Penalty Galerkin Method for Boundary Control Problems

We investigate an a posteriori error analysis of adaptive finite element approximations of linear-quadratic boundary optimal control problems under bilateral box constraints, which act on a Neumann boundary control. We use a symmetric interior Galerkin method as discretization technique. An efficient and reliable residual-type error estimator is introduced by invoking data oscillations. We then derive local upper and lower a posteriori error estimates for the boundary control problem. Adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical results are presented to illustrate the performance of the adaptive finite element approximation.

Optimal Control of Diffusion-Convection-Reaction Equations Using Upwind Symmetric Interior Penalty Galerkin (SIPG) Method

We discuss the numerical solution of linear quadratic optimal control problem with distributed and Robin boundary controls governed by diffusion convection reaction equations. The discretization is based on the upwind symmetric interior penalty Galerkin (SIPG) methods which lead to the same discrete scheme for the optimize-then-discretize and the discretize-then-optimize.