Variational discretization and rectangle mixed finite element methods for quadratic semilinear elliptic optimal control problems

In this paper, we investigate a variational discretization and rectangle mixed finite element methods for the quadratic optimal control problems governed by semilinear elliptic equations. The state and the co-state are approximated by the lowest order Raviart-Thomas rectangle mixed finite element spaces and the control is not discretized. Optimal error estimates are established for the state and control variable. As a result, it can be proved that the discrete solutions possess the convergence property of order h. A numerical example is presented to confirm our theoretical results.