Advertisement

Applied Mathematics and Mechanics

, Volume 37, Issue 5, pp 647–658 | Cite as

Reduced-order finite element method based on POD for fractional Tricomi-type equation

  • Jincun Liu
  • Hong LiEmail author
  • Yang Liu
  • Zhichao Fang
Article

Abstract

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).

Key words

reduced-order finite element method (FEM) proper orthogonal decomposition (POD) fractional Tricomi-type equation unconditionally stable error estimate 

Chinese Library Classification

O242 

2010 Mathematics Subject Classification

39K99 65M60 65M12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Podlubny, I. Fractional Differential Equations, Academic Press, New York (1999)zbMATHGoogle Scholar
  2. [2]
    Liu, F., Anh, V., Turner, I., and Zhang, P. Time fractional advection dispersion equation. Journal of Applied Mathematics and Computing, 13(1), 233–245 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Yuste, S. B. and Acedo, L. An explicit finite difference method and a new von Numann-type stability analysis for fractional diffusion equation. SIAM Journal on Numerical Analysis, 42(5), 1862–1874 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Lin, Y. M. and Xu, C. J. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics, 225(2), 1533–1552 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Ervin, V. J. and Roop, J. P. Variational formulation for the stationary fractional advection dispersion equation. Numerical Methods for Partial Differential Equations, 22(3), 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Zhang, H., Liu, F., and Anh, V. Galerkin finite element approximations of symmetric spacefractional partial differnetial equations. Applied Mathematics and Computation, 217(6), 2534–2545 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Li, C. P., Zhao, Z. G., and Chen, Y. Q. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Computers and Mathematics with Applications, 62(3), 855–875 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Ford, N. J., Xiao, J. Y., and Yan, Y. B. A finite element method for time fractional partial differential equations. Fractional Calculus and Applied Analysis, 14(3), 454–474 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Zhang, X. D., Huang, P. Z., Feng, X. L., and Wei, L. L. Finite element method for two-dimensional time-fractional Tricomi-type equations. Numerical Methods for Partial Differential Equations, 29(4), 1081–1096 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Deng, W. H. Short memory principle and a predictor-corrector approach for fractional differential equations. Journal of Computational and Applied Mathematics, 206(1), 174–188 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Ford, N. J. and Simpson, A. C. The numerical solution of fractional differential equations: speed versus accuracy. Numerical Algorithms, 26(4), 333–346 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Holmes, P. J., Lumley, L., and Berkooz, G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  13. [13]
    Jolliffe, I. T. Principal Component Analysis, Springer-Verlag, Berlin (2002)zbMATHGoogle Scholar
  14. [14]
    Fukunaga, K. Introduction to Statistical Pattern Recognition, Academic Press, New York (1990)zbMATHGoogle Scholar
  15. [15]
    Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90(1), 117–148 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Kunisch, K. and Volkwein, S. Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM Journal on Numerical Analysis, 40(2), 492–515 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Sun, P., Luo, Z. D., and Zhou, Y. J. Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations. Applied Numerical Mathematics, 60, 154–164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Luo, Z. D., Chen, J., Sun, P., and Yang, X. Z. Finite element formulation based on proper orthogonal decomposition for parabolic equations. Science in China Series A: Mathematics, 52(3), 585–596 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Luo, Z. D., Li, H., Zhou, Y. J., and Huang, X. M. A reduced-order FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem. Journal of Mathematical Analysis and Applications, 385(1), 310–321 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Luo, Z. D. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete and Continuous Dynamical Systems Series B, 20(4), 1189–1212 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Liu, J. C., Li, H., Fang, Z. C., and Liu, Y. Application of low-dimensional finite element method to fractional diffusion equation. International Journal of Modeling, Simulation, and Scientific Computing, 5(4), 1450022 (2014)MathSciNetCrossRefGoogle Scholar
  22. [22]
    Adams, R. A. Sobolev Spaces, Academic Press, New York (1975)zbMATHGoogle Scholar
  23. [23]
    Thomée, V. Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotChina

Personalised recommendations