$L^\infty$-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

In this paper, we investigate $L^\infty$-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive $L^\infty$-error estimates of optimal order for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, we present numerical tests which confirm our theoretical results.