A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS

doi:10.4208/jcm.1210-m3966

Journal of Computational Mathematics ›› 2013, Vol. 31 ›› Issue (1): 88-106.DOI: 10.4208/jcm.1210-m3966

A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS

Hongfei Fu¹, Hongxing Rui², Hui Guo¹

1. Department of Computational and Applied Mathematics, China University of Petroleum, Qingdao 266580, China;
2. School of Mathematics, Shandong University, Jinan 250100, China;
3. Department of Computational and Applied Mathematics, China University of Petroleum, Qingdao 266580, China

Received:2011-12-03 Revised:2012-10-25 Online:2013-01-15 Published:2013-01-17
Supported by:
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions on an earlier version of this paper. The first author was supported by the National Natural Science Foundation of China (No.11126086,11201485) and the Fundamental Research Funds for the Central Universities (No.12CX04083A); The second author was supported by the National Natural Science Foundation of China (No.11171190); The third author was supported by the National Natural Science Foundation of China (No.11101431).

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Hongfei Fu, Hongxing Rui, Hui Guo. A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS[J]. Journal of Computational Mathematics, 2013, 31(1): 88-106.

In this paper, we develop a priori error estimates for the solution of constrained convection-diffusion-reaction optimal control problems using a characteristic finite element method. The cost functional of the optimal control problems consists of three parts: The first part is about integration of the state over the whole time interval, the second part refers to final-time state, and the third part is a regularization term about the control. We discretize the state and co-state by piecewise linear continuous functions, while the control is approximated by piecewise constant functions. Pointwise inequality function constraints on the control are considered, and optimal a L²-norm priori error estimates are obtained. Finally, we give two numerical examples to validate the theoretical analysis.

Key words: Characteristic finite element method, Constrained optimal control, Convectiondiffusionreaction equations, Pointwise inequality constraints, A priori error estimates

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A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS

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