Numerische Mathematik

, Volume 132, Issue 4, pp 721–766 | Cite as

Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations

  • Jérôme DroniouEmail author
  • Robert Eymard


Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming finite element, mixed finite element and finite volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards’, Stefan’s and Leray–Lions’ models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli–Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin–Simon theorem. The model’s degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.

Mathematics Subject Classification

65M12 35K65 46N40 



The authors would like to thank Clément Cancès for fruitful discussions on discrete compensated compactness theorems.


  1. 1.
    Aavatsmark, I., Barkve, T., Boe, O., Mannseth, T.: Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. 127(1), 2–14 (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Akrivis, G., Makridakis, C., Nochetto, R.H.: Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods. Numer. Math. 114(1), 133–160 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Akrivis, G., Makridakis, C., Nochetto, R.H.: Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118(3), 429–456 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amann, H.: Compact embeddings of vector-valued Sobolev and Besov spaces. Glas. Mat. Ser. III 35(55), 161–177 (2000). (dedicated to the memory of Branko Najman)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Andreianov, B., Boyer, F., Hubert, F.: Discrete duality finite volume schemes for Leray–Lions-type elliptic problems on general 2D meshes. Numer. Methods Partial Differ. Equ. 23(1), 145–195 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Andreianov, B., Cancès C., Moussa, A.: A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs. HAL: hal-01142499 (2015) (submitted)Google Scholar
  7. 7.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces, vol 6. MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Mathematical Programming Society (MPS), Philadelphia, Philadelphia (2006)Google Scholar
  8. 8.
    Bertsch, M., De Mottoni, P., Peletier, L.: The Stefan problem with heating: appearance and disappearance of a mushy region. Trans. Am. Math. Soc 293, 677–691 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, X., Jüngel, A., Liu, J.-G.: A note on Aubin–Lions–Dubinskiĭ lemmas. Acta Appl. Math. 133, 33–43 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ciarlet, P.: The finite element method. In: Ciarlet, P.G., Lions, J.-L. (eds.) Part I, Handbook of Numerical Analysis. III. North-Holland, Amsterdam (1991)Google Scholar
  13. 13.
    Coudière, Y., Hubert, F.: A 3d discrete duality finite volume method for nonlinear elliptic equations. SIAM J. Sci. Comput. 33(4), 1739–1764 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Crouzeix, M., Raviart, P.-A.: onforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  16. 16.
    Diaz, J., de Thelin, F.: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25(4), 1085–1111 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dreher, M., Jüngel, A.: Compact families of piecewise constant functions in \(L^p(0, T;B)\). Nonlinear Anal. 75(6), 3072–3077 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Droniou, J.: Intégration et espaces de sobolev à valeurs vectorielles. Polycopiés de l’Ecole Doctorale de Mathématiques-Informatique de Marseille. (2001). Accessed 15 Jan 2015
  19. 19.
    Droniou, J.: Finite volume schemes for fully non-linear elliptic equations in divergence form. ESAIM Math. Model. Numer. Anal. 40(6), 1069 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Droniou, J., Eymard, R.: A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math. 105(1), 35–71 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: Gradient schemes for elliptic and parabolic problems (2015) (in preparation)Google Scholar
  22. 22.
    Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20(2), 265–295 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS) 23(13), 2395–2432 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Droniou, J., Eymard, R., Guichard, C.: Uniform-in-time convergence of numerical schemes for Richards’ and Stefan’s models. In: Finite Volumes for Complex Applications VII, Springer (2014)Google Scholar
  25. 25.
    Edwards, M.G., Rogers, C.F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2(4), 259–290 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ekeland, I., Témam, R.: Convex Analysis and Variational Problems, vol. 28. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (1999) (english edition, translated from the French)Google Scholar
  27. 27.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  28. 28.
    Eymard, R., Feron, P., Gallouët, T., Herbin, R., Guichard, C.: Gradient schemes for the Stefan problem. Int. J. Finite Vol. 10s (2013)Google Scholar
  29. 29.
    Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Eymard, R., Gallouët, T., Hilhorst, D., Naït Slimane, Y.: Finite volumes and nonlinear diffusion equations. RAIRO Modél. Math. Anal. Numér. 32(6):747–761 (1998)Google Scholar
  31. 31.
    Eymard, R., Guichard, C., Herbin, R.: Small-stencil 3d schemes for diffusive flows in porous media. M2AN 46, 265–290 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Eymard, R., Guichard, C., Herbin, R., Masson, R.: Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. ZAMM Z. Angew. Math. Mech. 94(7–8), 560–585 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Eymard, R., Gutnic, M., Hilhorst, D.: The finite volume method for Richards equation. Comput. Geosci. 3(3–4), 259–294 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Eymard, R., Herbin, R.: Gradient scheme approximations for diffusion problems. In: Finite Volumes for Complex Applications VI Problems and Perspectives, pp. 439–447 (2011)Google Scholar
  35. 35.
    Eymard, R., Herbin, R., Michel, A.: Mathematical study of a petroleum-engineering scheme. M2AN Math. Model. Numer. Anal. 37(6), 937–972 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gallouët, T., Latché, J.-C.: Compactness of discrete approximate solutions to parabolic PDEs–application to a turbulence model. Commun. Pure Appl. Anal. 11(6), 2371–2391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Glowinski, R., Rappaz, J.: Approximation of a nonlinear elliptic problem arising in a non-newtonian fluid flow model in glaciology. M2AN Math. Model. Numer. Anal. 37(1), 175–186 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    González, C., Ostermann, A., Palencia, C., Thalhammer, M.: Backward Euler discretization of fully nonlinear parabolic problems. Math. Comput. 71(237), 125–145 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Gwinner, J., Thalhammer, M.: Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods. Found. Comput. Math. 14(5), 913–949 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hermeline, F.: Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Eng. 192(16), 1939–1959 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kazhikhov, A.V.: Recent developments in the global theory of two-dimensional compressible Navier–Stokes equations. Seminar on Mathematical Sciences, vol. 25. Keio University,Department of Mathematics, Yokohama (1998)Google Scholar
  42. 42.
    Lubich, C., Ostermann, A.: Runge-Kutta methods for parabolic equations and convolution quadrature. Math. Comput. 60(201), 105–131 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lubich, C., Ostermann, A.: Linearly implicit time discretization of non-linear parabolic equations. IMA J. Numer. Anal. 15(4), 555–583 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lubich, C., Ostermann, A.: Runge-Kutta approximation of quasi-linear parabolic equations. Math. Comput. 64(210), 601–627 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Lubich, C., Ostermann, A.: Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour. Appl. Numer. Math. 22(1–3):279–292 (1996) (special issue celebrating the centenary of Runge-Kutta methods)Google Scholar
  46. 46.
    Maitre, E.: Numerical analysis of nonlinear elliptic-parabolic equations. M2AN Math. Model. Numer. Anal. 36(1), 143–153 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Minty, G.: On a monotonicity method for the solution of non-linear equations in Banach spaces. Proc. Natl. Acad. Sci. USA 50(6), 1038 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25(4), 784–814 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ostermann, A., Thalhammer, M.: Convergence of Runge-Kutta methods for nonlinear parabolic equations. Appl. Numer. Math. In: Ninth Seminar on Numerical Solution of Differential and Differential-Algebraic Equations (Halle, 2000) 42(1–3):367–380 (2002)Google Scholar
  50. 50.
    Ostermann, A., Thalhammer, M., Kirlinger, G.: Stability of linear multistep methods and applications to nonlinear parabolic problems. Appl. Numer. Math. In: Workshop on Innovative Time Integrators for PDEs 48(3–4):389–407 (2004)Google Scholar
  51. 51.
    Pop, I.S.: Numerical schemes for degenerate parabolic problems. In: Progress in Industrial Mathematics at ECMI 2004, vol. 8. Math. Ind., pp. 513–517. Springer, Berlin (2006)Google Scholar
  52. 52.
    Rulla, J., Walkington, N.J.: Optimal rates of convergence for degenerate parabolic problems in two dimensions. SIAM J. Numer. Anal. 33(1), 56–67 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityVictoriaAustralia
  2. 2.Laboratoire d’Analyse et de Mathématiques Appliquées, UMR 8050Université Paris-EstMarne-la-Vallée Cedex 2France

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