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Superconvergence of mixed finite element methods for optimal control problems

Author: Yanping Chen
Journal: Math. Comp. 77 (2008), 1269-1291
MSC (2000): Primary 49J20, 65N30
Published electronically: February 28, 2008
MathSciNet review: 2398768
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Abstract: In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive $ L^2$ superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global $ L^2$ superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

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Additional Information

Yanping Chen
Affiliation: Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan 411105, Hunan, China

Keywords: Quadratic optimal control problems, mixed finite elements, rectangular partition, superconvergence, $L^2$ error estimates.
Received by editor(s): December 28, 2005
Received by editor(s) in revised form: June 25, 2007
Published electronically: February 28, 2008
Additional Notes: This work was supported by the Program for New Century Excellent Talents in University of China State Education Ministry NCET-04-0776, the National Science Foundation of China, the National Basic Research Program under the Grant 2005CB321703, and the Scientific Research Fund of the Hunan Provincial Education Department.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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