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Numerical Analysis of an Adaptive FEM for Distributed Flux Reconstruction

Published online by Cambridge University Press:  03 June 2015

Mingxia Li ,
Jingzhi Li  and
Shipeng Mao
Mingxia Li*
Affiliation:
School of Science, China University of Geosciences (Beijing), Beijing 100083, China
Jingzhi Li*
Affiliation:
Faculty of Science, South University of Science and Technology of China, Shenzhen 518055, China
Shipeng Mao*
Affiliation:
LSEC, Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences (CAS), Beijing 100190, China
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
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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.

Keywords

65N2165N5065N30Inverse problemsadaptive finite element methoda posteriori error estimatesquasi-orthogonalityconvergence analysis
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1068 - 1090
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Alifanov, O. M., Inverse Heat Transfer Problems, Springer, Berlin, 1994.Google Scholar
[2]Babuška, I. and Rheinboldt, C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), 736–754.Google Scholar
[3]Bangerth, W. and Joshi, A., Adaptive finite element methods for the solution of inverse problems in optical tomography, Inverse Problems, 24(3) (2008), 1–22.Google Scholar
[4]Becker, R. and Vexler, B., A posteriori error estimation for finite element discretization of parameter identification problems, Numer. Math., 96 (2004), 435–459.Google Scholar
[5]Becker, R. and Mao, S., An optimally convergent adaptive mixed finite element method, Numer. Math., 111 (2008), 35–54.Google Scholar
[6]Becker, R. and Mao, S., Convergence and quasi-optimal complexity of a simple adaptive finite element method, Math. Model. Numer. Anal., 43 (2009), 1203–1219.Google Scholar
[7]Becker, R., Mao, S. and Shi, Z.-C., A convergent nonconforming adaptive finite element method with optimal complexity, SIAM J. Numer. Anal., 47(6) (2010), 4639–4659.Google Scholar
[8]Becker, R. and Mao, S., Convergence of a simple adaptive finite element method for optimal control, Tech. report, INRIA, Pau, 2008.Google Scholar
[9]Beilina, L. and Johnson, C., A posteriori error estimation in computational inverse scattering, Math. Model Method Appl. Sci., 15 (2005), 23–35.Google Scholar
[10]Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), 219–268.Google Scholar
[11]Binev, P., Dahmen, W., DeVore, R. and Petruchev, P., Approximation classes for adaptive methods, Serdica Math. J., 28 (2002), 391–416.Google Scholar
[12]Carstensen, C. and Hoppe, R. H. W., Convergence analysis of an adaptive nonconforming finite element methods, Numer. Math., 103 (2006), 251–266.Google Scholar
[13]Carstensen, C. and Hoppe, R. H. W., Error reduction and convergence for an adaptive mixed finite element method, Math. Comput., 75 (2006), 1033–1042.Google Scholar
[14]Cascon, J. M., Kreuzer, Ch., Nochetto, R. H. and Siebert, K. G., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), 2524–2550.CrossRefGoogle Scholar
[15]Chen, L., Holst, M. and Xu, J., Convergence and optimality of adaptive mixed finite element methods, Math. Comput., 78 (2009), 35–53.Google Scholar
[16]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.Google Scholar
[17]DeVore, R., Nonlinear approximation, Acta Numer., 7 (1998), 51–150.Google Scholar
[18]Divo, E. and Kapat, J. S., Multi-dimensional heat flux reconstruction using narrow-band ther-mochromic liquid crystal thermography, Inverse Prob. Sci. Eng., 9(5) (2001), 537–559.Google Scholar
[19]orfler, W. D, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33 (1996), 1106–1124.Google Scholar
[20]Feng, T., Yan, N., and Liu, W., Adaptive finite element methods for the identification of distributed parameters in elliptic equation, Adv. Comput. Math., 29 (2008), 27–53.CrossRefGoogle Scholar
[21]Gaevskaya, A., Hoppe, R., Iliash, Y., and Kieweg, M., Convergence analysis of an adaptive finite element method for distributed control problems with control constraints, Control of Coupled Partial Differential Equations, Birkhäuser, Basel, 2007, 47–68.Google Scholar
[22]Hadamard, J., Lectures on Cauchy’s Problem in Linear Differential Equations, Yale University Press, New Haven, CT, 1923.Google Scholar
[23]Li, J., Xie, J. and Zou, J., An adaptive finite element method for distributed heat flux reconstruction, Inverse Problems, 27 (2011), 075009.Google Scholar
[24]Li, M., Li, J. and Mao, S., A priori error estimates of a finite element method for distributed heat flux reconstruction, J. Comput. Math., to appear.Google Scholar
[25]Li, R., Liu, W., Ma, H., and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), 1321–1349.Google Scholar
[26]Liu, W., Ma, H., Tang, T., and Yan, N., A posteriori error estimates for discontinuous galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal., 42 (2004), 1032–1061.Google Scholar
[27]Liu, W. and Yan, N., A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), 497–521.Google Scholar
[28]Mekchay, K. and Nochetto, R. H., Convergenceof adaptivefinite element methods for general second order linear elliptic PDE, SIAM J. Numer. Anal., 43 (2005), 1803–1827.Google Scholar
[29]Mitchell, W. F., A comparison of adaptive refinement techniques for elliptic problems, ACM Transactions on Mathematical Software (TOMS) archive, 15 (1989), 326–347.CrossRefGoogle Scholar
[30]Morin, P., Nochetto, R. H. and Siebert, K. G., Data oscillation and convergence of adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466–488.CrossRefGoogle Scholar
[31]Morin, P., Nochetto, R. H. and Siebert, K. G., Convergence of adaptive finite element methods, SIAM Rev., 44 (2002), 631–658.Google Scholar
[32]Tang, T. and Xu, J., Adaptive Computations: Theory and Algorithms, Science Press, Beijing, 2007.Google Scholar
[33]Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54 (1990), 483–493.Google Scholar
[34]Stevenson, R., Optimality of a standard adaptive finite element method, Found. Comput. Math., 7(2) (2007), 245–269.Google Scholar
[35]Stevenson, R., The completion of locally refined simplicial partitions created by bisection, Math. Comput., 77 (2008), 227–241.Google Scholar
[36]Xie, J. and Zou, J., Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504–1535.Google Scholar
[37]Zabaras, N. and Kang, S., On the solution of anill-posed inverse design solidification problem using minimization techniques in finite and infinite dimensional spaces, Int. J. Numer. Meth. Eng., 36 (1993), 3973–3990.Google Scholar
[38]Zabaras, N. and Liu, J., An analysis of two-dimensional linear inverse heat transfer problems using an integral method, Numer. Heat Trans., 13 (1988), 527–533.Google Scholar