Abstract
A generalized thermistor model is discretized thanks to a fully implicit vertex-centered finite volume scheme on simplicial meshes. An assumption on the stiffness coefficients is mandatory to prove a discrete maximum principle on the electric potential. This property is fundamental to handle the stability issues related to the Joule heating term. Then the convergence to a weak solution is established. Finally, numerical results are presented to show the efficiency of the methodology and to illustrate the behavior of the temperature together with the electric potential within the medium.
Acknowledgements
The authors acknowledge the anonymous referees for their comments that helped improve the presentation of this paper.
References
[1] M. Afif and B. Amaziane, Convergence of finite volume schemes for a degenerate convection–diffusion equation arising in flow in porous media, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 46, 5265–5286. 10.1016/S0045-7825(02)00458-9Search in Google Scholar
[2] G. Akrivis and S. Larsson, Linearly implicit finite element methods for the time-dependent Joule heating problem, BIT 45 (2005), no. 3, 429–442. 10.1007/s10543-005-0008-1Search in Google Scholar
[3] W. Allegretto, Y. Liu and A. Zhou, A box scheme for coupled systems resulting from microsensor thermistor problems, Dyn. Contin. Discrete Impuls. Syst. 5 (1999), no. 1–4, 209–223. Search in Google Scholar
[4] W. Allegretto and H. Xie, Existence of solutions for the time-dependent thermistor equations, IMA J. Appl. Math. 48 (1992), no. 3, 271–281. 10.1093/imamat/48.3.271Search in Google Scholar
[5] B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, J. Hyperbolic Differ. Equ. 7 (2010), no. 01, 1–67. 10.1142/S0219891610002062Search in Google Scholar
[6] O. Angelini, K. Brenner and D. Hilhorst, A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation, Numer. Math. 123 (2013), no. 2, 219–257. 10.1007/s00211-012-0485-5Search in Google Scholar
[7] S. Antontsev and M. Chipot, The thermistor problem: Existence, smoothness uniqueness, blowup, SIAM J. Math. Anal. 25 (1994), no. 4, 1128–1156. 10.1137/S0036141092233482Search in Google Scholar
[8] A. Bradji and R. Herbin, Discretization of coupled heat and electrical diffusion problems by finite-element and finite-volume methods, IMA J. Numer. Anal. 28 (2008), no. 3, 469–495. 10.1093/imanum/drm030Search in Google Scholar
[9] K. Brenner and R. Masson, Convergence of a vertex centred discretization of two-phase darcy flows on general meshes, Int. J. Finite Vol. 10 (2013), 1–37. Search in Google Scholar
[10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010. 10.1007/978-0-387-70914-7Search in Google Scholar
[11] P. Chatzipantelidis, R. Lazarov and V. Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. Methods Partial Differential Equations 20 (2004), no. 5, 650–674. 10.1002/num.20006Search in Google Scholar
[12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar
[13] G. Cimatti, A bound for the temperature in the thermistor problem, IMA J. Appl. Math. 40 (1988), no. 1, 15–22. 10.1093/imamat/40.1.15Search in Google Scholar
[14] G. Cimatti, On the stability of the solution of the thermistor problem, Appl. Anal. 73 (1999), no. 3–4, 407–423. 10.1080/00036819908840788Search in Google Scholar
[15] J. Droniou, Finite volume schemes for diffusion equations: Introduction to and review of modern methods, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1575–1619. 10.1142/S0218202514400041Search in Google Scholar
[16] J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The Gradient Discretisation Method, Springer, Cham, 2018. 10.1007/978-3-319-79042-8Search in Google Scholar
[17] C. M. Elliott and S. Larsson, A finite element model for the time-dependent Joule heating problem, Math. Comp. 64 (1995), no. 212, 1433–1453. 10.1090/S0025-5718-1995-1308451-4Search in Google Scholar
[18] R. E. Ewing, T. Lin and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39 (2002), no. 6, 1865–1888. 10.1137/S0036142900368873Search in Google Scholar
[19] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of Numerical Analysis. Vol. 7, Elsevier, Amsterdam (2000), 713–1018. 10.1016/S1570-8659(00)07005-8Search in Google Scholar
[20] R. Eymard and F. Sonier, Mathematical and numerical properties of control-volumel finite-element scheme for reservoir simulation, SPE Reservoir Eng. 9 (1994), no. 04, 283–289. 10.2118/25267-PASearch in Google Scholar
[21] F. O. Gallego and M. T. G. Montesinos, Existence of a capacity solution to a coupled nonlinear parabolic–elliptic system, Commun. Pure Appl. Anal. 6 (2007), no. 1, 23–42. 10.3934/cpaa.2007.6.23Search in Google Scholar
[22] H. Gao, Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations, J. Sci. Comput. 58 (2014), no. 3, 627–647. 10.1007/s10915-013-9746-4Search in Google Scholar
[23] H. Gao, B. Li and W. Sun, Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon, Numer. Math. 136 (2017), no. 2, 383–409. 10.1007/s00211-016-0843-9Search in Google Scholar
[24] M. Ghilani, E. H. Quenjel and M. Saad, Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media, Comput. Geosci. 23 (2019), no. 1, 55–79. 10.1007/s10596-018-9783-zSearch in Google Scholar
[25] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, Finite Volumes for Complex Applications. Vol. V, Wiley, New York (2008), 659–692. Search in Google Scholar
[26] M. Ibrahim and M. Saad, On the efficacy of a control volume finite element method for the capture of patterns for a volume-filling chemotaxis model, Comput. Math. Appl. 68 (2014), no. 9, 1032–1051. 10.1016/j.camwa.2014.03.010Search in Google Scholar
[27] B. Li, H. Gao and W. Sun, Unconditionally optimal error estimates of a Crank–Nicolson Galerkin method for the nonlinear thermistor equations, SIAM J. Numer. Anal. 52 (2014), no. 2, 933–954. 10.1137/120892465Search in Google Scholar
[28] B. Li and W. Sun, Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations, Int. J. Numer. Anal. Model. 10 (2013), no. 3, 622–633. Search in Google Scholar
[29] B. Li and C. Yang, Uniform bmo estimate of parabolic equations and global well-posedness of the thermistor problem, Forum Math. Sigma 3 (2015), Paper no. e26. 10.1017/fms.2015.29Search in Google Scholar
[30] J. L. Lions, Quelques méthodes de résolution des problemes aux limites non linéaires, Dunod, Paris, 1969. Search in Google Scholar
[31] M. T. G. Montesinos and F. O. Gallego, The evolution thermistor problem with degenerate thermal conductivity, Commun. Pure Appl. Anal. 1 (2002), no. 3, 313–325. 10.3934/cpaa.2002.1.313Search in Google Scholar
[32] M. T. G. Montesinos and F. O. Gallego, On the existence of solutions for a strongly degenerate system, C. R. Acad. Sci. Paris Ser. I 343 (2006), 119–123. 10.1016/j.crma.2006.06.002Search in Google Scholar
[33] H. Moussa, F. O. Gallego and M. Rhoudaf, Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Paper No. 14. 10.1007/s00030-018-0505-ySearch in Google Scholar
[34] B. Saad and M. Saad, Study of full implicit petroleum engineering finite-volume scheme for compressible two-phase flow in porous media, SIAM J. Numer. Anal. 51 (2013), no. 1, 716–741. 10.1137/120869092Search in Google Scholar
[35] D. Shi and H. Yang, Superconvergence analysis of nonconforming FEM for nonlinear time-dependent thermistor problem, Appl. Math. Comput. 347 (2019), 210–224. 10.1016/j.amc.2018.10.018Search in Google Scholar
[36] Y. Xing-Ye, Numerical analysis of nonstationary thermistor problem, J. Comput. Math. 12 (1994), no. 3, 213–223. Search in Google Scholar
[37] X. Xu, A strongly degenerate system involving an equation of parabolic type and an equation of elliptic type, Comm. Partial Differential Equations 18 (1993), no. 1–2, 199–213. 10.1080/03605309308820927Search in Google Scholar
[38] X. Xu, Local and global existence of continuous temperature in the electrical heating of conductors, Houston J. Math. 22 (1996), no. 2, 435–455. Search in Google Scholar
[39] G. Yuan, Regularity of solutions of the thermistor problem, Appl. Anal. 53 (1994), no. 3–4, 149–156. 10.1080/00036819408840253Search in Google Scholar
[40] G. Yuan and Z. Liu, Existence and uniqueness of the c^α solution for the thermistor problem with mixed boundary value, SIAM J. Math. Anal. 25 (1994), no. 4, 1157–1166. 10.1137/S0036141092237893Search in Google Scholar
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