Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints

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

“…Further, we derive an upper bound of the error measured in terms of the natural energy norm and semi-norm associated with the convective terms defined in (11), provided that the state system (2) has homogeneous boundary conditions, i.e., g u = g v = 0, as proven for a single convection-diffusion equation in [33] and for the linear optimal control problems in [20,21,23].…”

“…We would like to refer to [16][17][18][19] for discontinuous Galerkin methods in details. DG discretizations have been used in [20][21][22][23][24] for distributed linear optimal control problems governed by convection dominated problems. Our aim here is to extend the adaptive mesh refinement in [21,23], which yields more narrowly refined regions around the layers than the SUPG discretization does, to the optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms as in (1)- (2).…”

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

“…In the following, we will use the bound v L 2 ( ) |v| a few times, which is possible since r 0 > 0. We note that this assumption is also made for analysis of optimal control problems governed by convection dominated equations [1,20,27,37,38,40].…”

“…We discretize our optimal control problem using a DG method in which we choose the SIPG discretization for the diffusion and an upwind discretization for the convection, see e.g., [23,27,33,[38][39][40]. We assume that the domain is polygonal such that the boundary is exactly represented by boundaries of triangles.…”

“…Further, an a posteriori error estimator containing only computable quantities is proposed in [31] for elliptic control problems with control and state constraints. Recent results derived in [38] for unconstrained optimal control problems and in [40,41] for control constrained optimal control problems governed by convection dominated equations show that the residual based a posteriori error estimators based on discontinuous Galerkin (DG) discretization yield more accurate results since the errors in layers do not propagate into the entire domain [27]. Our goal here is to extend the results in [38,40] to the state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau-Yosida and Lavrentiev-based techniques.…”