A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

  • Yang Liu
  • Hong Li
  • Siriguleng He
  • Wei Gao
  • Sen Mu
Article

Abstract

A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear convection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.

Keywords

Sobolev equation nonlinear convection term variation of constants H1-Galerkin mixed method optimal error estimate 

MR Subject Classification

65M60 65N30 35Q10 

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References

  1. [1]
    S C Chen, H R Chen. New mixed element schemes for second order elliptic problem, Math Numer Sinica, 2010, 32(2): 213–218.MATHGoogle Scholar
  2. [2]
    Y P Chen, Y Q Huang. The superconvergence of mixed finite element methods for nonlinear hyperbolic equations, Comm Nonlinear Sci Numer Simul, 1998, 3(3): 155–158.MATHCrossRefGoogle Scholar
  3. [3]
    H B Chen, D Xue, X Q Liu. An H 1-Galerkin mixed finite element method for nonlinear parabolic partial integro-differential equations, Acta Math Appl Sinica, 2008, 31(4): 702–712.MATHGoogle Scholar
  4. [4]
    F Z Gao, H X Rui. Two splitting least-squares mixed element methods for linear Sobolev equations, Math Numer Sinica, 2008, 30(3): 269–282.MathSciNetMATHGoogle Scholar
  5. [5]
    L Guo, H Z Chen. H 1-Galerkin mixed finite element method for Sobolev equations, J Systems Sci Math Sci, 2006, 26(3): 301–314.MathSciNetMATHGoogle Scholar
  6. [6]
    Z W Jiang, H Z Chen. Error estimates for mixed finite element methods for Sobolev equation, Northeast Math J, 2001, 17(3): 301–314.MathSciNetMATHGoogle Scholar
  7. [7]
    C Johson, V Thomée. Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal Numer, 1981, 15: 41–78.MathSciNetGoogle Scholar
  8. [8]
    Y P Lin, T Zhang. Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions, J Math Anal Appl, 1992, 165: 180–191.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Y Liu, H Li. H 1-Galerkin mixed finite element methods for pseudo-hyperbolic equations, Appl Math Comput, 2009, 212: 446–457.MathSciNetMATHCrossRefGoogle Scholar
  10. 10]
    Y Liu, H Li, J F Wang. Error estimates of H 1-Galerkin mixed finite element method for Schrödinger equation, Appl Math J Chinese Univ, 2009, 24(1): 83–89.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Z D Luo. Mixed finite element estimate for second-order elliptic problem, ACTA Math Appl Sinica, 1993, 16(4): 473–476.MathSciNetMATHGoogle Scholar
  12. [12]
    Z D Luo, R X Liu. Mixed finite element analysis and numerical solitary solution for the RLW equation, SIAM J Numer Anal, 1998, 36(1): 89–104.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    A K Pani. An H 1-Galerkin mixed finite element method for parabolic partial equations, SIAM J Numer Anal, 1998, 35: 712–727.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    A K Pani, R K Sinha, A K Otta. An H 1-Galerkin mixed method for second order hyperbolic equations, Internat J Numer Anal Modeling, 2004, 1(2): 111–129.MathSciNetMATHGoogle Scholar
  15. [15]
    P A Raviart, J M Thomas. A mixed finite element methods for second order elliptic problems, In: Lecture Notes in Math, Vol 606, Springer, Berlin, 1977, 292–315.Google Scholar
  16. [16]
    D Y Shi, H H Wang. Nonconforming H 1-Galerkin mixed FEM for Sobolev equations on anisotropic meshes, Acta Math Appl Sinica (English Ser), 2009, 25(2): 335–344.MATHCrossRefGoogle Scholar
  17. [17]
    T J Sun, D P Yang. Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numer Methods Partial Differential Equations, 2008, 24: 879–896.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    R W Wang. Error estimates for H 1-Galerkin mixed finite element methods for hyperbolic type integro-differential equation, Math Numer Sinica, 2006, 28(1): 19–30.Google Scholar
  19. [19]
    F M Wheeler. A priori L 2 -error estimates for Galerkin approximations to parabolic differential equation, SIAM J Numer Anal, 1973, 10: 723–749.MathSciNetCrossRefGoogle Scholar
  20. [20]
    P X Zhao, H Z Chen. The characteristics-mixed finite element method for Sobolev equation, Math Appl, 2003, 16(4): 50–59.MathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Yang Liu
    • 1
  • Hong Li
    • 1
  • Siriguleng He
    • 1
  • Wei Gao
    • 1
  • Sen Mu
    • 2
  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina
  2. 2.College of Finance and StatisticsHunan UniversityChangshaChina

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