Analysis of an adaptive finite element method for recovering the robin coefficient

SIAM Journal on Control and Optimization

Volumn 53, Issue 2, 2015, Pages 622-644

  Xu, Yifeng ^a   Zou, Jun ^b  

^a SHANGHAI NORMAL UNIVERSITY   (China)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

A posteriori error estimates; Adaptive finite element method; Robin inverse problem

Indexed keywords

ADAPTIVE ALGORITHMS; ALGORITHMS; ERROR ANALYSIS; INVERSE PROBLEMS;

A-POSTERIORI ERROR ESTIMATES; ADAPTIVE FINITE ELEMENT METHODS; BOUNDARY MEASUREMENTS; LEAST-SQUARES FORMULATION; NON-LINEAR INVERSE PROBLEM; POSTERIORI ERROR ESTIMATOR; ROBIN INVERSE PROBLEMS; TIKHONOV REGULARIZATION;

FINITE ELEMENT METHOD;

EID: 84928952268     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/130941742     Document Type: Article

References (47)

1. M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math., Wiley-Interscience, New York, 2000.
2. W. Bangerth and A. Joshi, Adaptive finite element methods for the solution of inverse problems in optical tomography, Inverse Problems, 24 (2008), pp. 1-22.
3. R. Becker, H. Kapp, and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113-132.
4. R. Becker and B. Vexler, A posteriori error estimation for finite element discretization of parameter identification problems, Numer. Math., 96 (2004), pp. 435-459.
5. L. Beilina and M. 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), 045012.
6. L. Belenki, L. Diening, and C. Kreuzer, Optimality of an adaptive finite element method for the p-Laplacian equation, IMA J. Numer. Anal., 32 (2012), pp. 484-510.
7. P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math., 97 (2004), pp. 219-268.
8. S. Busenberg and W. Fang, Identification of semiconductor contact resistivity, Quart. J. Appl. Math., 49 (1991), pp. 639-649.
9. J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46 (2008), pp. 2524-2550.
10. S. Chaabane, C. Elhechmi, and M. Jaoua, A stable recovery method for the Robin inverse problem, Math. Comput. Simul., 66 (2004), pp. 367-383.
11. S. Chaabane, I. Fellah, M. Jaoua, and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems, 20 (2004), pp. 47-59.
12. S. Chaabane, J. Ferchichi, and K. Kunisch, Differentiability properties of the L1-tracking functional and application to the Robin inverse problem, Inverse Problems, 20 (2004), pp. 1083-1097.
13. S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems, 15 (1999), pp. 1425-1438.
14. S. Chantasiriwan, Inverse determination of steady-state heat transfer coefficient, Int. Comm. Heat Mass Transfer, 27 (2000), pp. 1155-1164.
15. P. G. Ciarlet, Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
16. L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the p-Laplacian equation, SIAM J. Numer. Anal., 46 (2008), pp. 614-638.
17. L. E. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
18. W. Fang and E. Cumberbatch, Inverse problems for metal oxide semiconductor field-effect transistor contact resistivity, SIAM J. Appl. Math., 52 (1992), pp. 699-709.
19. W. Fang and M. Lu, A fast collocation method for an inverse boundary value problem, Internat. J. Numer. Methods Engrg., 59 (2004), pp. 1563-1585.
20. D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical results and numerical methods, Inverse Problems, 15 (1999), pp. 41-48.
21. T. Feng, N. Yan, and W. Liu, Adaptive finite element methods for the identification of distributed parameters in elliptic equation, Adv. Comput. Math., 29 (2008), pp. 27-53.
22. E. M. Garau and P. Morin, Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA J. Numer. Anal., 31 (2011), pp. 914-946.
23. E. M. Garau, P. Morin, and C. Zuppa, Convergence of adaptive finite element methods for eigenvalue problems, Math. Models Methods Appl. Sci., 19 (2009), pp. 721-747.
24. E. M. Garau, P. Morin, and C. Zuppa, Convergence of an adaptive Kačanov FEM for quasi-linear problems, Appl. Numer. Math., 61 (2011), pp. 512-529.
25. E. M. Garau, P. Morin, and C. Zuppa, Quasi-optimal convergence rate of an AFEM for quasi-linear problems of monotone type, Numer. Math. Theory Methods Appl., 5 (2012), pp. 131-156.
26. A. Gaevskaya, R. Hoppe, Y. Iliash, and M. Kieweg, Convergence analysis of an adaptive finite element method for distributed control problems with control constraints, in Proceedings of the Conference on Optimal Control for PDEs, G. Leugering et al., eds., Oberwolfach, Germany, Birkhäuser, Basel, 2007.
27. M. Hintermüller and R. Hoppe, Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations, SIAM J. Control Optim., 48 (2010), pp. 5468-5487.
28. M. Hintermüller, R. Hoppe, Y. Iliash, and M. 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), pp. 540-560.
29. G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), pp. 977-994.
30. B. Jin and J. Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control. Optim., 48 (2009), pp. 1977-2002.
31. B. Jin and J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), pp. 677-701.
32. P. G. Kaup and F. Santosa, Nondestructive evaluation of corrosion damage using electrostatic measurements, J. Nondestr. Eval., 14 (1995), pp. 127-136.
33. J. Li, J. Xie, and J. Zou, An adaptive finite element reconstruction of distributed fluxes, Inverse Problems, 27 (2011), 075009 (25pp).
34. R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), pp. 1321-1349.
35. F. Lin and W. Fang, A linear integral equation approach to the Robin inverse problem, Inverse Problems, 21 (2005), pp. 1757-1772.
36. W. Liu and N. Yan, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math., 15 (2001), pp. 285-309.
37. T. J. Martin and G. S. Dulikravich, Inverse determination of steady heat convection coefficient distribution, J. Heat Transfer, 120 (1998), pp. 328-334.
38. P. Morin, K. G. Siebert, and A. Veeser, A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 18 (2008), pp. 707-737.
39. R. H. Nochetto, K. G. Siebert, and A. Veeser, Theory of adaptive finite element methods: An introduction, in Multiscale, Nonlinear and Adaptive Approximation, R. A. DeVore and A. Kunoth, eds., Springer, New York, 2009, pp. 409-542.
40. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
41. K. G. Siebert, A convergence proof for adaptive finite elements without lower bounds, IMA J. Numer. Anal., 31 (2011), pp. 947-970.
42. R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7 (2007), pp. 245-269.
43. R. Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp., 77 (2008), pp. 227-241.
44. C. Traxler, An algorithm for adaptive mesh refinement in n dimensions, Computing, 59 (1997), pp. 115-137.
45. R. Verfürth, A Review of A Posteriori Estimation and Adaptive Mesh-Refinement Techniques, Wiley, New York, 1996.
46. F. M. White, Heat and Mass Transfer, Addison-Wesley, Reading, MA, 1988.
47. Y. Xu and J. Zou, Convergence of an adaptive finite element method for distributed flux reconstruction, Math. Comp., to appear; preprint available as arXiv:1309.2101.