Commun. Comput. Phys., 15 (2014), pp. 1068-1090.


Numerical Analysis of an Adaptive FEM for Distributed Flux Reconstruction

Mingxia Li 1, Jingzhi Li 2*, Shipeng Mao 3

1 School of Science, China University of Geosciences (Beijing), Beijing 100083, China.
2 Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, China.
3 LSEC, Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences (CAS), Beijing 100190, China.

Received 5 March 2013; Accepted (in revised version) 21 June 2013
Available online 21 January 2014
doi:10.4208/cicp.050313.210613s

Abstract

This paper studies convergence analysis of an adaptive finite element algorithm for numerical estimation of some unknown distributed flux in a stationary heat conduction system, namely recovering the unknown Neumann data on interior inaccessible boundary using Dirichlet measurement data on outer accessible boundary. Besides global upper and lower bounds established in [23], a posteriori local upper bounds and quasi-orthogonality results concerning the discretization errors of the state and adjoint variables are derived. Convergence and quasi-optimality of the proposed adaptive algorithm are rigorously proved. Numerical results are presented to illustrate the quasi-optimality of the proposed adaptive method.

AMS subject classifications: 65N21, 65N50, 65N30

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Key words: Inverse problems, adaptive finite element method, a posteriori error estimates, quasi-orthogonality, convergence analysis.

*Corresponding author.
Email: limx@lsec.cc.ac.cn (M. Li), li.jz@sustc.edu.cn (J. Li), maosp@lsec.cc.ac.cn (S. Mao)
 

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