A two-grid $P_0^2$- $P_1$ mixed finite element scheme for semilinear elliptic optimal control problems

This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $P_0^2$- $P_1$ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $P_0^2$- $P_1$ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $P_0^2$- $P_1$ mixed finite element method if the two mesh sizes satisfy $h=H^2$. Finally, a numerical example demonstrating our theoretical results is presented.