Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements
Tianliang Hou Yanping Chen
Journal of Industrial & Management Optimization 2013, 9(3): 631-642 doi: 10.3934/jimo.2013.9.631
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 $h^{2}$.
keywords: postprocessing. discontinuous functions mixed finite element methods Elliptic equations optimal control problems superconvergence
Superconvergence property of finite element methods for parabolic optimal control problems
Chunjuan Hou Yanping Chen Zuliang Lu
Journal of Industrial & Management Optimization 2011, 7(4): 927-945 doi: 10.3934/jimo.2011.7.927
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.
keywords: superconvergence property. parabolic equation finite element methods Optimal control problems

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