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In this paper, we discuss the superconvergence of mixed finite element methods for a semilinear elliptic control problem with an integral  constraint. The state and costate  are approximated by the order $k=1$  Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximation of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that this approximation has convergence  order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.

journal of applied analysis and computation

A GLOBAL SUPERCONVERGENT &lt;i&gt;L&lt;/i&gt;&lt;sup&gt;∞&lt;/sup&gt;-ERROR ESTIMATE OF MIXED FINITE ELEMENT METHODS FOR SEMILINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS