Abstract:
In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by order $k=1$ Raviart–Thomas mixed finite element spaces, and the control variable is approximated by piecewise constant functions. Approximations 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 these approximations have convergence order $h^2$. Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.
Key words:elliptic equations, optimal control problems, superconvergence, a posteriori error estimates, mixed finite element methods, postprocessing.
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