Frontiers of Mathematics in China

, Volume 8, Issue 4, pp 825–836 | Cite as

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

  • Siriguleng He
  • Hong LiEmail author
  • Yang Liu
Research Article


This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L 2(H 1) and L 2(L 2) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k n ch 2, which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.


Nonlinear Sobolev equation time discontinuous Galerkin spacetime finite element method optimal error estimate 


65M12 65M60 


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  1. 1.
    Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 1994zbMATHCrossRefGoogle Scholar
  2. 2.
    Cockburn B, Lin S Y. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J Comput Phys, 1989, 84(1): 90–113MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cockburn B, Shu C W. The local discontinuous Galerkin method for time-depended convection diffusion systems. SIAM J Numer Anal, 1998, 35: 2440–2463MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Eriksson K, Johnson C. Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J Numer Anal, 1991, 28(1): 43–77MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Eriksson K, Johnson C. Adaptive finite element methods for parabolic problems II: optimal error estimates in L L 2 and L L . SIAM J Numer Anal, 1995, 32(3): 706–740MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ewing R E. Numerical solution of Sobolev partial differential equations. SIAM J Numer Anal, 1975, 12: 345–363MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ewing R E. Time-stepping Galerkin method for nonlinear Sobolev partial differential equations. SIAM J Numer Anal, 1978, 15: 1125–1150MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Karakashian O, Makridakis Ch. A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin mehtod. Math Comp, 1998, 67: 479–499MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Larsson S, Thomée V, Wahlbin L B. Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math Comp, 1998, 69(221): 45–71CrossRefGoogle Scholar
  10. 10.
    Li H, Liu R X. The space-time finite element methods for parabolic problems. Appl Math Mech, 2001, 22: 687–700zbMATHCrossRefGoogle Scholar
  11. 11.
    Liu Y, Li H, He S. Mixed time discontinuous space-time finite element method for convection diffusion equations. Appl Math Mech, 2008, 29(12): 1579–1586MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Reed N H, Hill T R. Triangle mesh methods for the Neutron transport equation. Report LA2 UR-73-479, Los Alamos Scientific Laboratory, 1973Google Scholar
  13. 13.
    Sudirham J J, Van Der Vegt J J W, Van Damme R M J. Space-time discontinuous Galerkin methods for advection-diffusion problems on time-dependent domains. Appl Numer Math, 2006, 56: 1491–1518MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sun T, Yang D. The finite element difference streamline-diffusion methods for Sobolev equation with convection dominated term. Appl Math Comput, 2002, 125: 325–345MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Sun T, Yang D. A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations. Appl Math Comput, 2008, 200: 147–159MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Sun T, Ma K. A space-time discontinuous Galerkin method for linear convectiondominated Sobolev equations. Appl Math Comput, 2009, 201: 490–503MathSciNetCrossRefGoogle Scholar
  17. 17.
    Thomée V. Galerkin Finite Element Methods for Parabolic Problems. Berlin: Springer-Verlag, 1997zbMATHCrossRefGoogle Scholar

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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina

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