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JIMO

In this paper, we investigate the superconvergence property of a
quadratic elliptic control problem with pointwise control
constraints. The state and the co-state variables are approximated
by the Raviart-Thomas mixed finite element of order
$k=1$ and the control variable is discretized by piecewise linear
but discontinuous functions. Approximations of the optimal solution
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 $h^{2}$.

JIMO

In this paper, a finite element method for a parabolic optimal
control problem is introduced and analyzed. For the discretization
of a quadratic convex optimal control problem, the state and
co-state are approximated by piecewise linear functions and the
control is approximated by piecewise constant functions. As a
result, it is proved in this paper that the difference between a
suitable interpolation of the control and its finite element
approximation has superconvergence property in order $O(h^2)$.
Finally, two numerical examples are presented to confirm our
theoretical results.

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