Mixed methods for optimal control problems

In this paper, we investigate a posteriori error estimates of a mixed finite element method for elliptic optimal control problems with an integral constraint. The gradient for our method belongs to the square integrable space instead of the classical $H(\mathrm{div};\Omega)$ space. The state and co-state are approximated by the $P^2_0$-$P_1$ (velocity-pressure) pair, and the control variable is approximated by piecewise constant functions. Using a duality argument method and an energy method, we derive residual a posteriori error estimates for all variables.