Afrika Matematika

, Volume 30, Issue 1–2, pp 53–90 | Cite as

Fully discrete approximation of general nonlinear Sobolev equations

  • F. Bekkouche
  • W. Chikouche
  • S. NicaiseEmail author


We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.


Fully discrete scheme Sobolev equations Error estimates 

Mathematics Subject Classification

65N30 65M12 65M15 35G30 


  1. 1.
  2. 2.
    Arnold, D.N., Douglas Jr., J., Thomée, V.: Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comput. 36(153), 53–63 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Avilez-Valente, P., Seabra-Santos, F.J.: A Petrov-Galerkin finite element scheme for the regularized long wave equation. Comput. Mech. 34(4), 256–270 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Barenblatt, G.I., Entov, V.M., Ryzhik, V.M.: Theory of Fluid Flow Through Natural Rocks. Kluwer, Dordrecht (1990)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bernardi, C.: Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26(5), 1212–1240 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2(4), 556–581 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Carroll, R.W., Showalter, R .E.: Singular and Degenerate Cauchy problems, vol. 127. Academic, New York (1976). Mathematics in Science and EngineeringGoogle Scholar
  8. 8.
    Chatzipantelidis, P.: Explicit multistep methods for nonstiff partial differential equations. Appl. Numer. Math. 27(1), 13–31 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chen, P.J., Gurtin, M.E.: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19, 614–627 (1968)zbMATHCrossRefGoogle Scholar
  10. 10.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  11. 11.
    Coleman, B.D., Noll, W.: An approximation theorem for functionals, with applications in continuum mechanics. Arch. Ratl. Mech. Anal. 6(355–370), 1960 (1960)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cuesta, C., van Duijn, C.J., Hulshof, J.: Infiltration in porous media with dynamic capillary pressure: travelling waves. Eur. J. Appl. Math. 11(4), 381–397 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains—Smoothness and Asymptotics of solutions, Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)zbMATHCrossRefGoogle Scholar
  14. 14.
    Dogan, A.: Numerical solution of regularized long wave equation using Petrov–Galerkin method. Commun. Numer. Methods Eng. 17(7), 485–494 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Ewing, R.E.: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal. 15(6), 1125–1150 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Gajewski, H., Zacharias, K.: Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum. Math. Nachr. 57, 127–140 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gao, F., Wang, X.: A modified weak Galerkin finite element method for Sobolev equation. J. Comput. Math. 33(3), 307–322 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)zbMATHGoogle Scholar
  19. 19.
    Hairer, E., Nø rsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, Volume 8 of Springer Series in Computational Mathematics. Springer, Berlin (1993)Google Scholar
  20. 20.
    Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 858–879 (1993)CrossRefGoogle Scholar
  21. 21.
    Hell, T., Ostermann, A., Sandbichler, M.: Modification of dimension-splitting methods–overcoming the order reduction due to corner singularities. IMA J. Numer. Anal. 35(3), 1078–1091 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kadlec, J.: The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czech. Math. J. 14(89), 386–393 (1964)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)Google Scholar
  24. 24.
    Kellogg, R.B.: Singularities in interface problems. In: Hubbard, B. (ed.) Numerical Solution of Partial Differential Equations II, pp. 351–400. Academic, New York (1971)CrossRefGoogle Scholar
  25. 25.
    Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4, 101–129 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lemrabet, K.: Régularité de la solution d’un problème de transmission. J. Math. Pures Appl. 56, 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lions, J.-L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer, Berlin (1961)Google Scholar
  28. 28.
    Liu, T., Lin, Y.-P., Rao, M., Cannon, J.R.: Finite element methods for Sobolev equations. J. Comput. Math. 20(6), 627–642 (2002)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nicaise, S.: Polygonal interface problems, Methoden und Verfahren der Mathematischen Physik, vol. 39. Verlag Peter D. Lang, Frankfurt am Main (1993)Google Scholar
  30. 30.
    Nicaise, S., Sändig, A.-M.: General interface problems. I, II. Math. Methods Appl. Sci. 17(6):395–429, 431–450 (1994)Google Scholar
  31. 31.
    Ohm, M.R., Lee, H.Y.: \(L^2\)-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations. Bull. Korean Math. Soc. 48(5), 897–915 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Ohm, M.R., Lee, H.Y., Shin, J.Y.: \(L^2\)-error estimates of the extrapolated Crank-Nicolson discontinuous Galerkin approximations for nonlinear Sobolev equations. J. Inequal. Appl., pages Art. ID 895187, 17 (2010)Google Scholar
  33. 33.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)zbMATHCrossRefGoogle Scholar
  34. 34.
    Ptashnyk, M.: Pseudoparabolic equations with convection. IMA J. Appl. Math. 72(6), 912–922 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Quarteroni, A.: Fourier spectral methods for pseudoparabolic equations. SIAM J. Numer. Anal. 24(2), 323–335 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Raugel, G.: Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A-B 286(18), A791–A794 (1978)zbMATHGoogle Scholar
  37. 37.
    Schwartz, L.: Mathematics for the Physical Sciences. Hermann; Addison-Wesley Publishing Co., Paris, Reading (1966)zbMATHGoogle Scholar
  38. 38.
    Showalter, R.E.: The Sobolev equation. I. Appl. Anal. 5(1), 15–22 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Showalter, R.E.: The Sobolev equation. II. Appl. Anal. 5(2), 81–99 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations, Monographs and Studies in Mathematics, vol. 1. Pitman, London (1977)Google Scholar
  41. 41.
    Ting, T.W.: Certain non-steady flows of second-order fluids. Arch. Ratl. Mech. Anal. 14, 1–26 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Ting, T.W.: A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45, 23–31 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Wahlbin, L.: Error estimates for a Galerkin method for a class of model equations for long waves. Numer. Math. 23, 289–303 (1975)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Univ. Valenciennes, EA 4015 - LAMAV - FR CNRS 2956ValenciennesFrance
  2. 2.Université de JijelLaboratoire LMPAJijelAlgeria

Personalised recommendations