Skip to main content
Springer Nature Link
Log in

Numerical Approximations for the Fractional Fokker–Planck Equation with Two-Scale Diffusion

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Fractional Fokker–Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing numerical discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we first derive the fractional Fokker–Planck equation with two-scale diffusion from the Lévy process framework, and then the fully discrete scheme is built by using the \(L_{1}\) scheme for time discretization and finite element method for space. With the help of the sharp regularity estimate of the solution, we optimally get the spatial and temporal error estimates. Finally, we validate the effectiveness of the provided algorithm by extensive numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Acosta, G., Bersetche, F.M., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. Fract. Calc. Appl. Anal. 22, 767–794 (2019)

    Article  MathSciNet  Google Scholar 

  2. Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)

    Article  MathSciNet  Google Scholar 

  3. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)

    MATH  Google Scholar 

  4. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  5. Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid. Numer. Math. 131, 1–31 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of the integral fractional Laplacian. Numer. Math. 142, 235–278 (2019)

    Article  MathSciNet  Google Scholar 

  7. Bonito, A., Lei, W., Salgado, A.J.: Finite element approximation of an obstacle problem for a class of integro-differential operators. ESAIM M2AN 54, 229–253 (2020)

  8. Bonito, A., Pasciak, J.E.: Numerical approximation of fractional powers of regularly accretive operators. IMA J. Numer. Anal. 37, 1245–1273 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  10. Elliott, C.M., Larsson, S.: Error estimates with smooth and nonsmooth data for a finite element method for the Cahn–Hilliard equation. Math. Comp. 58, 603 (1992)

    Article  MathSciNet  Google Scholar 

  11. Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–211 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Eng. 346, 332–358 (2019)

    Article  MathSciNet  Google Scholar 

  13. Jin, B., Yan, Y., Zhou, Z.: Numerical approximation of stochastic time-fractional diffusion. ESAIM M2AN 53, 1245–1268 (2019)

  14. Lewin, L.: Polylogarithms and Associated Functions. North-Holland, New York (1981)

    MATH  Google Scholar 

  15. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  Google Scholar 

  16. Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65, 1–18 (1996)

    Article  MathSciNet  Google Scholar 

  17. Nie, D., Deng, W.: Numerical analysis for stochastic time-space fractional diffusion equation driven by fractional Gaussian noise. http://arxiv.org/pdf/2101.01963v1

  18. Nie, D., Sun, J., Deng, W.: Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states. Numer. Math. 146, 481–511 (2020)

    Article  MathSciNet  Google Scholar 

  19. Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)

    MATH  Google Scholar 

  20. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  21. Yan, Y., Khan, M., Ford, N.J.: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56, 210–227 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No. 12071195, AI and Big Data Funds under Grant No. 2019620005000775, Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2021-it26 and lzujbky-2021-kb15, and NSF of Gansu under Grant No. 21JR7RA537.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jing Sun, Weihua Deng & Daxin Nie

Authors
  1. Jing Sun
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Weihua Deng
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Daxin Nie
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Weihua Deng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, J., Deng, W. & Nie, D. Numerical Approximations for the Fractional Fokker–Planck Equation with Two-Scale Diffusion. J Sci Comput 91, 34 (2022). https://doi.org/10.1007/s10915-022-01812-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01812-z

Keywords