Abstract
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.
Highlights
It is well known that the finite element approximation plays an important role in the numerical treatment of optimal control problems
The aim of this paper is to investigate the superconvergence property of mixed finite element approximation for a semilinear elliptic control problem with an integral constraint
The discretization was already described in previous sections: the control function u was discretized by piecewise constant functions, whereas the state (y, p) and the co-state (z, q) were approximated by the order k = 1 Raviart-Thomas mixed finite element functions
Summary
It is well known that the finite element approximation plays an important role in the numerical treatment of optimal control problems. Hou & Chen [7] derived a superconvergent L2-error estimates of RT1 mixed methods for semilinear elliptic optimal control problems. In [15], Hou investigated the RT0 mixed finite element methods for a semilinear elliptic optimal control problem with a pointwise control constraint, he derive a superconvergence result for the control variable and L∞-error estimates for all variables even for the divergence of the vector-valued functions. As far as we know, there is no superconvergent L∞-error estimates of RT mixed finite element method for semilinear elliptic optimal control problems in the literature. The aim of this paper is to investigate the superconvergence property of mixed finite element approximation for a semilinear elliptic control problem with an integral constraint. In addition C denotes a general positive constant independent of h, where h is the spatial mesh-size for the control and state discretization
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