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

Abstract

a b s t r a c tIn 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.

Reduced order optimal control of the convective FitzHugh–Nagumo equations

“…The dG methods have several advantages compared to other numerical techniques such as finite volume and finite element methods; the trial and test spaces can be easily constructed, inhomogeneous boundary conditions and curved boundaries can be handled easily. The dG methods were successfully applied to linear steady state, time dependent and semi-linear optimal control problems with convectiondiffusion-reaction equations [11,12,13]; to the semi-linear steady state OCPs [14]. There are two approaches for solving OCPs with PDE constraints.…”

An interpolation method for the optimal control problem governed by the elliptic convection–diffusion equation

“…Specially, it is also an important step toward optimal control problems for fluid flows. Therefore, many researchers have solved it using many numerical solutions such as hybridizable discontinuous Galerkin method [4, 33], the residual‐free bubbles method [25], embedded discontinuous Galerkin [32], characteristic‐mixed finite element method [10], Raviart–Thomas mixed finite element method (FEM) and the discontinuous Galerkin method [27], adaptive symmetric interior penalty Galerkin method [29], mass‐conservative characteristic finite element scheme [11], local discontinuous Galerkin approximation [33], symmetric interior penalty Galerkin method [30], stabilized finite element method [26].…”

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

“…Many different numerical methods have been investigated for this type of problem including approaches based on the finite element method [1-3, 10-14, 17], mixed finite elements [13,26,28], and discontinuous Galerkin (DG) methods [14,18,24,25,27,29,30]. Also, hybridizable discontinuous Galerkin (HDG) methods have recently been explored for various optimal control problems for the Poisson equation [16,31] and the above convection diffusion equation [15].…”

“…1 arXiv:1801.00082v1 [math.NA] 30 Dec 2017Optimal control problems for convection diffusion equations arise in applications [21] and are also an important step towards optimal control problems for fluid flows. Therefore, researchers have developed many different numerical methods for this type of problem including approaches based on finite differences [3], standard finite element discretizations [14][15][16], stabilized finite elements [2,19], the symmetric stabilization method [4], the SUPG method [13,17], the edge-stabilization method [5,28], mixed finite elements [16,29,31], and discontinuous Galerkin (DG) methods [17,20,26,27,30,32,33].DG methods are well suited for problems with convection, but they often have a higher computational cost compared to other methods. Hybridizable discontinuous Galerkin (HDG) methods keep the advantages of DG methods, but have a lower number of globally coupled unknowns.…”