A Priori Error Estimates for Optimal Control Problems Governed by Transient Advection-Diffusion Equations

Abstract

In this paper, we investigate a characteristic finite element approximation of quadratic optimal control problems governed by linear advection-dominated diffusion equations, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. We derive some a priori error estimates for both the control and state approximations. It is proved that these approximations have convergence order $\mathcal{O}(h_{U}+h+k)$ , where h U and h are the spatial mesh-sizes for the control and state discretization, respectively, and k is the time increment. Numerical experiments are presented, which verify the theoretical results.