Numerical Analysis of an Adaptive FEM for Distributed Flux Reconstruction
Mingxia Li 1, Jingzhi Li 2*, Shipeng Mao 31 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
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 , 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.
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