Superconvergence property of finite element methods for parabolicoptimal control problems

Chunjuan Hou; Yanping Chen; Zuliang Lu

doi:10.3934/jimo.2011.7.927

Article Contents

2011, Volume 7, Issue 4: 927-945. Doi: 10.3934/jimo.2011.7.927

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Superconvergence property of finite element methods for parabolic optimal control problems

1.
Huashang College Guangdong University Of Business Studies, Guangzhou 511300
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631
3.
School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000

Received: September 30, 2010

Revised: May 31, 2011

Published: September 2011

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Abstract

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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Mathematics Subject Classification: 49J20, 65N30.

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References

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