A robust numerical method for a control problem involving singularly perturbed equations

We consider an unconstrained linear-quadratic optimal control problem governed by a singularly perturbed convection-reaction-diffusion equation. We discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by Duran and Lombardi in (Duźan and Lombardi 2005, 2006). We prove convergence of this scheme. In addition, when the state equation is a singularly perturbed reaction-diffusion equation, we derive quasi-optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes. We present several numerical experiments when the state equation is both a reaction-diffusion and a convection-reaction-diffusion equation. These numerical experiments reveal a competitive performance of the proposed solution technique.