Convergence of an adaptive finite element method for distributed flux reconstruction

Xu, Yifeng; Zou, Jun

doi:10.1090/mcom/2961

Convergence of an adaptive finite element method for distributed flux reconstruction
HTML articles powered by AMS MathViewer

by Yifeng Xu and Jun Zou PDF

Math. Comp. 84 (2015), 2645-2663 Request permission

Abstract:

We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux, state and costate variables. The sequence of discrete solutions produced by the adaptive algorithm is proved to converge to the true triplet satisfying the optimality conditions in the energy norm, and the corresponding error estimator converges to zero asymptotically.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
O. M. Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994.
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. Math. 44 (1984), no. 1, 75–102. MR 745088, DOI 10.1007/BF01389757
Wolfgang Bangerth and Amit Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography, Inverse Problems 24 (2008), no. 3, 034011, 22. MR 2421948, DOI 10.1088/0266-5611/24/3/034011
Wolfgang Bangerth and Rolf Rannacher, Adaptive finite element methods for differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2003. MR 1960405, DOI 10.1007/978-3-0348-7605-6
Roland Becker, Hartmut Kapp, and Rolf Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept, SIAM J. Control Optim. 39 (2000), no. 1, 113–132. MR 1780911, DOI 10.1137/S0363012999351097
Roland Becker and Boris Vexler, A posteriori error estimation for finite element discretization of parameter identification problems, Numer. Math. 96 (2004), no. 3, 435–459. MR 2028723, DOI 10.1007/s00211-003-0482-9
Larisa Beilina and Claes Johnson, A posteriori error estimation in computational inverse scattering, Math. Models Methods Appl. Sci. 15 (2005), no. 1, 23–35. MR 2110450, DOI 10.1142/S0218202505003885
Larisa Beilina and Michael V. Klibanov, A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem, Inverse Problems 26 (2010), no. 4, 045012, 27. MR 2608625, DOI 10.1088/0266-5611/26/4/045012
Larisa Beilina and Michael V. Klibanov, Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems 26 (2010), no. 12, 125009, 30. MR 2737743, DOI 10.1088/0266-5611/26/12/125009
L. Beilina, M. V. Klibanov, and M. Yu. Kokurin, Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem, J. Math. Sci. (N.Y.) 167 (2010), no. 3, 279–325. Problems in mathematical analysis. No. 46. MR 2839023, DOI 10.1007/s10958-010-9921-1
Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046, DOI 10.1137/07069047X
P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235
Albert Cohen, Ronald DeVore, and Ricardo H. Nochetto, Convergence rates of AFEM with $H^{-1}$ data, Found. Comput. Math. 12 (2012), no. 5, 671–718. MR 2970853, DOI 10.1007/s10208-012-9120-1
E. Divo and J. S. Kapat, Multi-dimensional heat flux reconstruction using narrow-band thermochromic liquid crystal thermography, Inverse Problems in Science and Engineering, 9 (2001), 537-559.
Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
Tao Feng, Ningning Yan, and Wenbin Liu, Adaptive finite element methods for the identification of distributed parameters in elliptic equation, Adv. Comput. Math. 29 (2008), no. 1, 27–53. MR 2420863, DOI 10.1007/s10444-007-9035-6
A. Gaevskaya, R. H. W. Hoppe, Y. Iliash, and M. Kieweg, Convergence analysis of an adaptive finite element method for distributed control problems with control constraints, Control of coupled partial differential equations, Internat. Ser. Numer. Math., vol. 155, Birkhäuser, Basel, 2007, pp. 47–68. MR 2328601, DOI 10.1007/978-3-7643-7721-2_{3}
Anke Griesbaum, Barbara Kaltenbacher, and Boris Vexler, Efficient computation of the Tikhonov regularization parameter by goal-oriented adaptive discretization, Inverse Problems 24 (2008), no. 2, 025025, 20. MR 2408562, DOI 10.1088/0266-5611/24/2/025025
Michael Hintermüller and Ronald H. W. Hoppe, Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations, SIAM J. Control Optim. 48 (2010), no. 8, 5468–5487. MR 2745781, DOI 10.1137/090761823
Michael Hintermüller, Ronald H. W. Hoppe, Yuri Iliash, and Michael Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints, ESAIM Control Optim. Calc. Var. 14 (2008), no. 3, 540–560. MR 2434065, DOI 10.1051/cocv:2007057
Jingzhi Li, Jianli Xie, and Jun Zou, An adaptive finite element reconstruction of distributed fluxes, Inverse Problems 27 (2011), no. 7, 075009, 25. MR 2817425, DOI 10.1088/0266-5611/27/7/075009
Ruo Li, Wenbin Liu, Heping Ma, and Tao Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim. 41 (2002), no. 5, 1321–1349. MR 1971952, DOI 10.1137/S0363012901389342
Wenbin Liu and Ningning Yan, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math. 15 (2001), no. 1-4, 285–309 (2002). A posteriori error estimation and adaptive computational methods. MR 1887737, DOI 10.1023/A:1014239012739
J.-L. Lions, Optimal control of systems governed by partial differential equations. , Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New York-Berlin, 1971. Translated from the French by S. K. Mitter. MR 0271512
Igor Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, J. Comput. Appl. Math. 55 (1994), no. 3, 275–288. MR 1329875, DOI 10.1016/0377-0427(94)90034-5
Joseph M. Maubach, Local bisection refinement for $n$-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), no. 1, 210–227. MR 1311687, DOI 10.1137/0916014
William F. Mitchell, A comparison of adaptive refinement techniques for elliptic problems, ACM Trans. Math. Software 15 (1989), no. 4, 326–347 (1990). MR 1062496, DOI 10.1145/76909.76912
Pedro Morin, Ricardo H. Nochetto, and Kunibert G. Siebert, Convergence of adaptive finite element methods, SIAM Rev. 44 (2002), no. 4, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488 (electronic); MR1770058 (2001g:65157)]. MR 1980447, DOI 10.1137/S0036144502409093
Pedro Morin, Kunibert G. Siebert, and Andreas Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 707–737. MR 2413035, DOI 10.1142/S0218202508002838
Ricardo H. Nochetto, Kunibert G. Siebert, and Andreas Veeser, Theory of adaptive finite element methods: an introduction, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 409–542. MR 2648380, DOI 10.1007/978-3-642-03413-8_{1}2
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Kunibert G. Siebert, A convergence proof for adaptive finite elements without lower bound, IMA J. Numer. Anal. 31 (2011), no. 3, 947–970. MR 2832786, DOI 10.1093/imanum/drq001
Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X
C. T. Traxler, An algorithm for adaptive mesh refinement in $n$ dimensions, Computing 59 (1997), no. 2, 115–137. MR 1475530, DOI 10.1007/BF02684475
Jianli Xie and Jun Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal. 43 (2005), no. 4, 1504–1535. MR 2182138, DOI 10.1137/030602551
R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, New York, Stuttgart, 1996.
Nicholas Zabaras and Shinill Kang, On the solution of an ill-posed design solidification problem using minimization techniques in finite- and infinite-dimensional function spaces, Internat. J. Numer. Methods Engrg. 36 (1993), no. 23, 3973–3990. MR 1247810, DOI 10.1002/nme.1620362304
N. Zabaras and J. Liu, An analysis of two-dimensional linear inverse heat transfer problems using an integral method, Numer. Heat Transfer, 13 (1988), 527-533.