Numerische Mathematik

, Volume 141, Issue 4, pp 881–916 | Cite as

Uniform-in-time bounds for approximate solutions of the drift–diffusion system

  • M. Bessemoulin-ChatardEmail author
  • C. Chainais-Hillairet


In this paper, we consider a numerical approximation of the Van Roosbroeck’s drift–diffusion system given by a backward Euler in time and finite volume in space discretization, with Scharfetter–Gummel fluxes. We first propose a proof of existence of a solution to the scheme which does not require any assumption on the time step. The result relies on the application of a topological degree argument which is based on the positivity and on uniform-in-time upper bounds of the approximate densities. Secondly, we establish uniform-in-time lower bounds satisfied by the approximate densities. These uniform-in-time upper and lower bounds ensure the exponential decay of the scheme towards the thermal equilibrium as shown in Bessemoulin-Chatard (Numer Math 25(3):147–168, 2016).

Mathematics Subject Classification

65M08 82D37 



  1. 1.
    Alikakos, N.D.: \(L^{p}\) bounds of solutions of reaction–diffusion equations. Comm. Partial Differ. Equ. 4(8), 827–868 (1979)CrossRefzbMATHGoogle Scholar
  2. 2.
    Angermann, L.: A mass-lumping semidiscretization of the semiconductor device equations. I. Properties of the semidiscrete problem. COMPEL 8(2), 65–105 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bessemoulin-Chatard, M., Chainais-Hillairet, C.: Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems. J. Numer. Math. 25(3), 147–168 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bessemoulin-Chatard, M., Chainais-Hillairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes. IMA J. Numer. Anal. 35(3), 1125–1149 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bessemoulin-Chatard, M., Chainais-Hillairet, C., Jüngel, A.: Uniform \(L^{\infty }\) estimates for approximate solutions of the bipolar drift–diffusion system. In: Finite Volumes for Complex Applications VIII, Springer Proceedings in Mathematics. Springer, Berlin (2017)Google Scholar
  6. 6.
    Bessemoulin-Chatard, M., Chainais-Hillairet, C., Vignal, M.-H.: Study of a finite volume scheme for the drift–diffusion system. Asymptotic behavior in the quasi-neutral limit. SIAM J. Numer. Anal. 52(4), 1666–1691 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezzi, F., Marini, L.D., Pietra, P.: Two-dimensional exponential fitting and applications to drift–diffusion models. SIAM J. Numer. Anal. 26(6), 1342–1355 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chainais-Hillairet, C., Filbet, F.: Asymptotic behavior of a finite volume scheme for the transient drift–diffusion model. IMA J. Numer. Anal. 27(4), 689–716 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chainais-Hillairet, C., Herda, M.: Large-time behavior of a family of finite volume schemes for boundary-driven convection–diffusion equations. (2018). arXiv:1810.01087
  10. 10.
    Chainais-Hillairet, C., Liu, J.-G., Peng, Y.-J.: Finite volume scheme for multi-dimensional drift–diffusion equations and convergence analysis. M2AN Math. Model. Numer. Anal. 37(2), 319–338 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chatard, M.: Asymptotic behavior of the Scharfetter–Gummel scheme for the drift–diffusion model. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F (eds.) Finite Volumes for Complex Applications VI Problems and Perspectives, volume 4 of Springer Proceedings in Mathematics, pp. 235–243. Springer, Berlin (2011)Google Scholar
  12. 12.
    Chen, Z., Cockburn, B.: Analysis of a finite element method for the drift–diffusion semiconductor device equations: the multidimensional case. Numer. Math. 71(1), 1–28 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Coughran Jr., W.M., Jerome, J.W.: Modular algorithms for transient semiconductor device simulation. I. Analysis of the outer iteration. In: Computational aspects of VLSI design with an emphasis on semiconductor device simulation (Minneapolis, MN, 1987), volume 25 of Lectures in Appl. Math., pp. 107–149. Am. Math. Soc., Providence, RI (1990)Google Scholar
  14. 14.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  15. 15.
    Di Francesco, M., Fellner, K., Markowich, P.A.: The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 464(2100), 3273–3300 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000)Google Scholar
  17. 17.
    Fiebach, A., Glitzky, A., Linke, A.: Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction–diffusion problems. Numer. Math. 128, 31–72 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Filbet, F., Herda, M.: A finite volume scheme for boundary-driven convection-diffusion equations with relative entropy structure. Numer. Math. 137(3), 535–577 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gajewski, H., Gärtner, K.: On the discretization of Van Roosbroeck’s equations with magnetic field. Z. Angew. Math. Mech. 76(5), 247–264 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gajewski, H., Gröger, K.: On the basic equations for carrier transport in semiconductors. J. Math. Anal. Appl. 113, 12–35 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gajewski, H., Gröger, K.: Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics. Math. Nachr. 140, 7–36 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gajewski, H., Gröger, K.: Initial-boundary value problems modelling heterogeneous semiconductor devices. In: Surveys on Analysis, Geometry and Mathematical Physics, volume 117 of Teubner-Texte Math., pp. 4–53. Teubner, Leipzig (1990)Google Scholar
  23. 23.
    Gajewski, H., Gröger, K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177, 109–130 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gärtner, K.: Existence of bounded discrete steady-state solutions of the Van Roosbroeck system on boundary conforming delaunay grids. SIAM J. Sci. Comput. 31(2), 1347–1362 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd ed. Springer, Berlin (1983)Google Scholar
  26. 26.
    Il’In, A.M.: A difference scheme for a differential equation with a small parameter multiplying the highest derivative. Math. Zametki 6, 237–248 (1969)MathSciNetGoogle Scholar
  27. 27.
    Kowalczyk, R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305(2), 566–588 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lazarov, R.D., Mishev, I.D., Vassilevski, P.S.: Finite volume methods for convection–diffusion problems. SIAM J. Numer. Anal. 33(1), 31–55 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Springer, Vienna (1990)CrossRefzbMATHGoogle Scholar
  30. 30.
    Mock, M.S.: An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5, 597–612 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Moser, J.: A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13(3), 457–468 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sacco, R., Saleri, F.: Mixed finite volume methods for semiconductor device simulation. Numer. Methods Partial Differ. Equ. 13(3), 215–236 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Scharfetter, D.L., Gummel, H.K.: Large signal analysis of a silicon Read diode. IEEE Trans. Electr. Dev. 16, 64–77 (1969)CrossRefGoogle Scholar
  34. 34.
    Van Roosbroeck, W.: Theory of the flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29, 560–607 (1950)CrossRefzbMATHGoogle Scholar
  35. 35.
    Wu, H., Jiang, J.: Global solution to the drift–diffusion–Poisson system for semiconductors with nonlinear recombination–generation rate. Asymptot. Anal. 85(1–2), 75–105 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Wu, H., Markowich, P.A., Zheng, S.: Global existence and asymptotic behavior for a semiconductor drift–diffusion–Poisson model. Math. Models Methods Appl. Sci. 18(3), 443–487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Université de Nantes - CNRS, Laboratoire de Mathématiques Jean LerayNantesFrance
  2. 2.Université de Lille - CNRS, Laboratoire Paul PainlevéLilleFrance

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