Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-12T09:56:20.742Z Has data issue: false hasContentIssue false
  • English
  • Français

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:limx@lsec.cc.ac.cn
Corresponding author.Email:li.jz@sustc.edu.cn
Email:maosp@lsec.cc.ac.cn
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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