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Error Estimates for the Time Discretization of a Semilinear Integrodifferential Parabolic Problem with Unknown Memory Kernel

Part of: Equations and inequalities involving nonlinear operators Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  20 February 2017

Marijke Grimmonprez ,
Karel Van Bockstal  and
Marián Slodička
Marijke Grimmonprez*
Affiliation:
Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Ghent, Belgium
Karel Van Bockstal*
Affiliation:
Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Ghent, Belgium
Marián Slodička*
Affiliation:
Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Ghent, Belgium
*
*Corresponding author. Email addresses:[email protected] (Marijke Grimmonprez), [email protected] (Karel Van Bockstal), [email protected] (Marián Slodička)
*Corresponding author. Email addresses:[email protected] (Marijke Grimmonprez), [email protected] (Karel Van Bockstal), [email protected] (Marián Slodička)
*Corresponding author. Email addresses:[email protected] (Marijke Grimmonprez), [email protected] (Karel Van Bockstal), [email protected] (Marián Slodička)
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Abstract

This paper is devoted to the study of an inverse problem containing a semilinear integrodifferential parabolic equation with an unknown memory kernel. This equation is accompanied by a Robin boundary condition. The missing kernel can be recovered from an additional global measurement in integral form. In this contribution, an error analysis is performed for a time-discrete numerical scheme based on Backward Euler's Method. The theoretical results are supported by some numerical experiments.

Keywords

Parabolic inverse problemconvolution kernelerror estimates

MSC classification

Secondary: 47J35: Nonlinear evolution equations 65M12: Stability and convergence of numerical methods 65M32: Inverse problems
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 1 , February 2017 , pp. 116 - 144
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Lorenzi, A. and Sinestrari, E.. An inverse problem in the theory of materials with memory. Nonlinear Anal.-Theor., 12(12):1317–1335, 1988.Google Scholar
[2] Lorenzi, A.. Identification problems for integrodifferential equations. In Ill-posed problems in natural sciences. Proceedings of the international conference held in Moscow (Russia), August 19-25, 1991, pages 342–366. Utrecht: VSP; Moscow: TVP Science Publishers, 1992.Google Scholar
[3] Lorenzi, A. and Paparoni, E.. Direct and inverse problems in the theory of materials with memory. Rendiconti del Seminario Matematico della Università di Padova, 87:105–138, 1992.Google Scholar
[4] Grasselli, M.. An inverse problem in three-dimensional linear thermoviscoelasticity of boltzmann type. In Ill-posed problems in natural sciences. Proceedings of the international conference held in Moscow (Russia), August 19-25, 1991, pages 284–299. Utrecht: VSP; Moscow: TVP Science Publishers, 1992.Google Scholar
[5] Grasselli, M. and Lorenzi, A.. An inverse problem for an abstract nonlinear parabolic integrodifferential equation. Differ. Integral Equ., 6(1):63–81, 1993.Google Scholar
[6] von Wolfersdorf, L.. Inverse problems for memory kernels in heat flow and viscoelasticity. J. Inverse Ill-Posed Probl., 4(4):341–354, 1996.Google Scholar
[7] Janno, J., Kiss, E.M., and von Wolfersdorf, L.. On tikhonov regularization for identifying memory kernels in heat conduction and viscoelasticity. ZAMM - J. Appl. Math. Mech. / Z. Angew. Math. Me., 80(4):259–272, 2000.Google Scholar
[8] Janno, J. and von Wolfersdorf, L.. Inverse problems for memory kernels by laplace transform methods. Z. Anal. Anwend., 19(2):489–510, 2000.Google Scholar
[9] Bukhgeǐm, A. L. and Dyatlov, G. V.. Inverse problems for equations with memory. In Inverse problems and related topics. Proceedings of a seminar, Kobe Institute, Kobe, Japan, February 6–10, 1998, pages 19–35. Boca Raton, FL: Chapman & Hall/CRC, 2000.Google Scholar
[10] Colombo, F., Guidetti, D., and Lorenzi, A.. On applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type. Goldstein, Ruiz, Gisèle, (ed.) et al., Evolution equations. Proceedings of the conference, Blaubeuren, Germany, June 11–17, 2001 in honor of the 60th birthdays of Philippe Bénilan, Goldstein, Jerome A. and Nagel, Rainer. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 234, 77-89, 2003.Google Scholar
[11] Colombo, F., Guidetti, D., and Vespri, V.. Some global in time results for integrodifferential parabolic inverse problems. Favini, , Angelo, (ed.) et al., Differential equations. Inverse and direct problems. Papers of the meeting, Cortona, Italy, June 21–25, 2004. Boca Raton, FL: CRC Press. Lecture Notes in Pure and Applied Mathematics 251, 35-58, 2006.Google Scholar
[12] Colombo, F. and Guidetti, D.. A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces. Math. Models Methods Appl. Sci., 17(4):537–565, 2007.Google Scholar
[13] Guidetti, D.. Convergence to a stationary state for solutions to parabolic inverse problems of reconstruction of convolution kernels. Differ. Integral Equ., 20(9):961–990, 2007.Google Scholar
[14] Colombo, F. and Guidetti, D.. Some results on the identification of memory kernels. In Ruzhansky, M. and Wirth, J., editors, Modern aspects of the theory of partial differential equations. Including mainly selected papers based on the presentations at the 7th international ISAAC congress, London, UK, July 13–18, 2009., pages 121–138. Oper. Theor. 216. Basel: Birkhäuser, 2011.Google Scholar
[15] De Staelen, R.H. and Slodička, M.. Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement. Nonlinear Anal.-Theor., 112(0):43–57, 2015.Google Scholar
[16] De Staelen, R.H., Van Bockstal, K., and Slodička, M.. Error analysis in the reconstruction of a convolution kernel in a semilinear parabolic problem with integral overdetermination. J. of Comput. Appl. Math., 275(0):382–391, 2015.CrossRefGoogle Scholar
[17] Delleur, J. W.. The Handbook of Groundwater Engineering. CRS Press, 1999.Google Scholar
[18] Van Bockstal, K., De Staelen, R.H., and Slodička, M.. Identification of a memory kernel in a semilinear integrodifferential parabolic problem with applications in heat conduction with memory. J. Comput. Appl. Math., 289(0):196–207, 2015.Google Scholar
[19] Kačur, J.. Method of Rothe in evolution equations, volume 80 of Teub. Text. M. Teubner, Leipzig, 1985.Google Scholar
[20] Guenther, R. B. and Lee, L. W.. Partial Differential Equations of Mathematical Physics and Integral Equations. Dover Publications, 1988.Google Scholar
[21] Nečas, J.. Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967.Google Scholar
[22] Nochetto, R.H. and Verdi, C.. An efficient scheme to approximate parabolic free boundary problems: error estimates and implementation. Math. Comput., 51(183):27–53, 1988.CrossRefGoogle Scholar
[23] Slodička, M. and Dehilis, S.. A nonlinear parabolic equation with a nonlocal boundary term. J. Comput. Appl. Math., 233(12):3130–3138, 2010.Google Scholar
[24] Kufner, A., John, O., and Fučík, S.. Function Spaces. Monographs and textbooks on mechanics of solids and fluids. Noordhoff International Publishing, Leyden, 1977.Google Scholar
[25] Logg, Anders and Wells, Garth N.. Dolfin: Automated finite element computing. ACM T. Math. Software, 37(2), 2010.Google Scholar
[26] Logg, Anders, Wells, Garth N., and Hake, Johan. DOLFIN: a C++/Python Finite Element Library, chapter 10. Springer, 2012.Google Scholar
[27] Logg, A., Mardal, K-A., Wells, G.N., et al. Automated Solution of Differential Equations by the Finite Element Method. Springer, 2012.Google Scholar