Error Analysis of the Semi-Discretized Doubly Nonlinear Non- Local Thermistor Problem
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Authors: I. DAHI AND M. R. S. AMMI
DOI: 10.46793/KgJMat2610.1641D
Abstract:
In this paper, we study a doubly nonlinear parabolic equation obtained from the reduction of the wellknown nonlocal thermistor problem. Therefore, we focus our study on proving the existence of the solution to the semi-discrete problem. We also establish the stability and error estimates for a family of time discretization schemes. We investigate a time discretization of the continuous problem by the backward Euler scheme.
Keywords:
Existence, nonlinear parabolic equation, nonlocal, semi-discretization, Thermistor problem, backward Euler method.
References:
[1] P. Agarwal, M. R. Sidi Ammi and J. Asad, Existence and uniqueness results on time scales for fractional nonlocal thermistor problem in the conformable sense, Adv. Differ. Equ. 2021(1) (2021), 1–11. https://doi.org/10.1186/s13662-021-03319-7
[2] S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness uniqueness, blowup, SIAM J. Math. Anal. 25(4) (1994), 1128–1156. https://doi.org/10.1137/S0036141092233482
[3] K. Bartosz, T. Janiczko and P. Szafraniec, Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction, Appl. Anal. 97(8) (2018), 1432–1453. https://doi.org/10.1080/00036811.2017.1403586
[4] F. Benzekri and A. El Hachimi, Doubly nonlinear parabolic equations related to the p-Laplacian operator: Semi-discretization, Electron. J. Differ. Equ. 2003, Paper ID 2003.
[5] L. Boccardo and L. Orsina, An elliptic system related to the stationary thermistor problem, SIAM J. Appl. Math. 53(6) (2021), 6910–6931. https://doi.org/10.1137/21M1420058
[6] H. Brézis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan 25(4) (1973), 565–590. https://doi.org/10.2969/jmsj/02540565
[7] X. Chen, Existence and regularity of solutions of a nonlinear nonuniformly elliptic system arising from a thermistor problem, Ph.D. thesis, New York University, 1992.
[8] G. Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Q. Appl. Math. 47(1) (1989), 117–121. https://doi.org/10.1090/qam/987900
[9] I. Dahi and M. R. Sidi Ammi, Existence of capacity solution for a nonlocal thermistor problem in Musielak-Orlicz-Sobolev spaces, Ann. Funct. Anal. 14(1) (2023), 1–33. https://doi.org/10.1007/s43034-022-00237-x
[10] I. Dahi and M. R. Sidi Ammi, Existence of renormalized solutions for nonlocal thermistor problem via weak convergence of truncations, Rend. Circ. Mat. Palermo (2022). https://doi.org/10.1007/s12215-022-00837-5.
[11] I. Dahi, M. R. Sidi Ammi and M. Hichmani, A finite volume method for a nonlocal thermistor problem, Appl. Numer. Math. (2024). https://doi.org/10.1016/j.apnum.2024.08.016.
[12] I. Dahi and M. R. Sidi Ammi, Existence and uniqueness result of a solution with numeric simulation for nonlocal thermistor problem with the presence of memory term, J. Math. Sci. (2024). https://doi.org/10.1007/s10958-024-07124-x.
[13] A. Eden, B. Michaux and J. M. Rakotoson, Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis, Indiana Univ. Math. J. 39 (1990), 737–783.
[14] A. El Hachimi and M. R. Sidi Ammi, Thermistor problem: a nonlocal parabolic problem, Electron. J. Differ. Equ. 2004 (2004), 117–128.
[15] H. Gao, W. Sun and C. Wu, Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations, IMA J. Numer. Anal. 41(4) (2021), 3175–3200. https://doi.org/10.1093/imanum/draa063
[16] A. Glitzky, M. Liero and G. Nika, Analysis of a bulk-surface thermistor model for large-area organic LEDs, Port. Math. 78(2) (2021), 187–210.
[17] A. Glitzky, M. Liero and G. Nika, Dimension reduction of thermistor models for large-area organic light-emitting diodes, Discrete Contin. Dyn. Syst. Ser. S 14(11) (2021), 3953–3977.
[18] S. Harikrishnan, K. Kanagarajan and S. Sivasundaram, On the study of dynamic analysis of thermistor problem involving Ψ-Hilfer fractional derivative, Math. Eng. 10(1) (2019).
[19] S. D. Howison, J. F. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl. 174(2) (1993), 573–588. https://doi.org/10.1006/jmaa.1993.1142
[20] N. I. Kavallaris and T. Nadzieja, On the blow-up of the non-local thermistor problem, Proc. Edinb. Math. Soc. 50(2) (2007), 389–409. https://doi.org/10.1017/S001309150500101X
[21] M. Khuddush and K. R. Prasad, Existence, uniqueness and stability analysis of a tempered fractional order thermistor boundary value problems, J. Anal. (2022), 1–23. https://doi.org/10.1007/s41478-022-00438-6
[22] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some special cases, Eur. J. Appl. Math. 6(2) (1995), 127–144. https://doi.org/10.1017/S095679250000173X
[23] A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway, Eur. J. Appl. Math. 6(3) (1995), 201–224. https://doi.org/10.1017/S0956792500001807
[24] J. Liu, Z. Chai and B. Shi, A lattice Boltzmann model for the nonlinear thermistor equations, Int. J. Mod. Phys. C 31(3) (2020), Article ID 2050043. https://doi.org/10.1142/S0129183120500436
[25] A. A. Nanwate and S. P. Bhairat, On well-posedness of generalized thermistor-type problem, AIP Conf. Proc. 2435(1) (2022), Article ID 020018.
[26] C. V. Nikolopoulos and G. E. Zouraris, Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method, in: Progress in Industrial Mathematics at ECMI 2006, Springer, 2008, 827–832. https://doi.org/10.1007/978-3-540-71992-2_143
[27] M. R. Sidi Ammi, I. Dahi, A. El Hachimi and D. F. M. Torres, Existence result of the global attractor for a triply nonlinear thermistor problem, Moroc. J. Pure Appl. Anal. 9(1) (2023), 27–47.
[28] M. R. Sidi Ammi and O. Mul, Error estimates for the Chernoff scheme to approximate a nonlocal parabolic problem, Proc. Estonian Acad. Sci. Phys. Math. 456(4) (2007), 359–372.
[29] F. D. Thélin, Résultats d’existence et de non-existence pour la solution positive et bornée d’une EDP elliptique non linéaire, Ann. Fac. Sci. Toulouse Math. 8(3) (1986), 375–389.
[30] R. A. Van Gorder, A. Kamilova and R. G. Birkeland, Locating the baking isotherm in a Soderberg electrode: analysis of a moving thermistor model, SIAM J. Appl. Math. 81(4) (2021), 1691–1716. https://doi.org/10.1137/20M1314276
[31] H. Xie and W. Allegretto, Solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SIAM J. Math. Anal. 22(6) (1991), 1491–1499. https://doi.org/10.1137/0522096
[32] H. Yang, H. Liang, Y. Song, F. Sun, S. Liu and X. Wang, A linearization scheme of thermistor temperature sensor, Proc. SPIE 12252 (2022), Article ID 1225207. https://doi.org/10.1117/12.2640052
