×

Global superconvergence of finite element methods for parabolic inverse problems. (English) Zbl 1212.35498

Summary: In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators.

MSC:

35R30 Inverse problems for PDEs
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
76R50 Diffusion
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML Link

References:

[1] C. Alvarez, C. Conca, L. Friz, O. Kavian, J. H. Ortega: Identification of immersed obstacles via boundary measurements. Inverse Probl. 21 (2005), 1531–1552. · Zbl 1088.35080
[2] H. Azari, W. Allegretto, Y. Lin, S. Zhang: Numerical procedures for recovering a time dependent coefficient in a parabolic differential equation. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 11 (2004), 181–199. · Zbl 1055.35132
[3] H. Azari, Ch. Li, Y. Nie, S. Zhang: Determination of an unknown coefficient in a parabolic inverse problem. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 11 (2004), 665–674. · Zbl 1059.35161
[4] H. Azari, S. Zhang: Identifying a time dependent unknown coefficient in a parabolic inverse problem. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms. Suppl. 12b (2005), 32–43.
[5] J. R. Cannon, H.-M. Yin: A class of nonlinear non-classical parabolic equations. J. Differ. Equations 79 (1989), 266–288. · Zbl 0702.35120
[6] J. R. Cannon, H.-M. Yin: Numerical solutions of some parabolic inverse problems. Numer. Methods Partial Differ. Equations 6 (1990), 177–191. · Zbl 0709.65105
[7] B. Canuto, O. Kavian: Determining coefficients in a class of heat equations via boundary measurements. SIAM J. Math. Anal. 32 (2001), 963–986 (electronic). · Zbl 0981.35096
[8] J. Douglas, Jr., B. F. Jones, Jr.: The determination of a coefficient in a parabolic differential equation. II. Numerical approximation. J. Math. Mech. 11 (1962), 919–926. · Zbl 0112.32603
[9] B. F. Jones, Jr.: The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness. J. Math. Mech. 11 (1962), 907–918. · Zbl 0112.32602
[10] Y. L. Keung, J. Zou: Numerical identification of parameters in parabolic systems. Inverse Probl. 14 (1998), 83–100. · Zbl 0894.35127
[11] R. A. Khachfe, Y. Jarny: Numerical solution of 2-D nonlinear inverse heat conduction problems using finite-element techniques. Numer. Heat Transfer, Part B: Fundamentals 37 (2000), 45–67.
[12] Q. Lin, N. Yan: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers, Baoding, 1996. (In Chinese.)
[13] Q. Lin, Q. Zhu: The Preprocessing and Postprocessing for the Finite Element Method. Shanghai Scientific & Technical Publishers, Shanghai, 1994. (In Chinese.)
[14] A. I. Prilepko, D. G. Orlovskii: Determination of the parameter of an evolution equation and inverse problems of mathematical physics I. Differ. Equations 21 (1985), 96–104. · Zbl 0571.35052
[15] A. G. Ramm: An inverse problem for the heat equation. J. Math. Anal. Appl. 264 (2001), 691–697. · Zbl 0987.35164
[16] A. G. Ramm: A non-overdetermined inverse problem of finding the potential from the spectral function. Int. J. Differ. Equ. Appl. 3 (2001), 15–29. · Zbl 1048.35137
[17] A. G. Ramm: Inverse problems for parabolic equations applications. Aust. J. Math. Anal. Appl. 2 (2005, Art. 10 (electronic)). · Zbl 1162.35384
[18] A. G. Ramm, S. V. Koshkin: An inverse problem for an abstract evolution equation. Appl. Anal. 79 (2001), 475–482. · Zbl 1020.35120
[19] X. T. Xiong, C. L. Fu, H. F. Li: Central difference schemes in time and error estimate on a non-standard inverse heat conduction problem. Appl. Math. Comput. 157 (2004), 77–91. · Zbl 1068.65117
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.