Local Error Estimates for SUPG Solutions of Advection-Dominated Elliptic Linear-Quadratic Optimal Control Problems

Abstract

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

“…A detailed analysis of the SUPG scheme applied to the optimal control of convection-diffusion equation is found in [12]. There, the following main issues are raised.…”

“…This scheme leads to a symmetric optimality system with an optimal convergence order. While commenting on the issues raised in [12] on the SUPG scheme, in [21], using the LPS scheme proposed in [7], a symmetric and consistent optimality system with both the optimize-then discretize and discretize-then optimize approaches, is presented. We employ LPS by introducing a projection operator π h as discussed in [7,21].…”

Variational multiscale method for the optimal control problems of convection--diffusion--reaction equations

“…Moreover, local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems were studied in [9]. Similarly, the local (DG) for the optimal control problem governed by convection-diffusion equations was analyzed in [19].…”

Brezzi-Pitkaranta stabilization and a priori error analysis for the Stokes Control

“…In [25], a discontinuous galerkin finite element method (DG) with interior penalties for the optimal control problem of the convection-diffusion equation was studied and in [18], an edge stabilized galerkin finite element method for the same optimal control system was considered. Moreover, local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems was studied in [17]. Similarly, the local (DG) for optimal control problem governed by convection-diffusion equations was analyzed in [27].…”