Skip to main content
Springer Nature Link
Log in

Error Estimate of Finite Element Approximation for Two-Sided Space-Fractional Evolution Equation with Variable Coefficient

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

Abstract

In this paper, we develop and analyze a finite element method (FEM) for a one-dimensional two-sided time-dependent space-fractional diffusion equation (sFDE) with variable diffusivity. We first prove the regularity of the solution to the steady-state sFDE equipped with variable diffusivity and then utilize the regularity estimate to analyze the property of an elliptic projection, which is of great importance to derive the error estimate for the time-dependent evolution problems. It has been well-known that the bilinear form corresponding to the finite element weak formulation for variable coefficient problems may not be coercive, and thus the well-posedness of the weak formulation can not be guaranteed. To perform the error estimate, we reformulate the steady-state equation to its equivalent form and then prove the weak coercivity of the corresponding bilinear form via the Gårding’s inequality, which leads to optimal-order error estimates of the FEM to the steady-state equation and thus to the interested evolution equation. Compared with some existing works on the FEM to variable-coefficient space-fractional problems, the main advantages of the developed numerical analysis techniques lie in the relaxed assumptions on the variable coefficient and its potential extensions to high-dimensional problems. Numerical experiments are performed to verify the theoretical findings.

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. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Elsevier, San Diego (2003)

    MATH  Google Scholar 

  2. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)

    MATH  Google Scholar 

  3. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36(6), 1403–1412 (2000)

    Article  Google Scholar 

  4. Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2007)

    Google Scholar 

  6. Crank, J.: The Mathematics of Diffusion. Oxford University Press, Oxford (1975)

    MATH  Google Scholar 

  7. Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  9. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ervin, V.J.: Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev spaces. J. Differ. Equ. 278, 294–325 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ervin, V.J., Heuer, N., Roop, J.P.: Regularity of the solution to 1-d fractional order diffusion equations. Math. Comput. 87, 2273–2294 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fu, H., Liu, H., Wang, H.: A finite volume method for two-dimensional Riemann–Liouville space-fractional diffusion equation and its efficient implementation. J. Comput. Phys. 388, 316–334 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hao, Z., Zang, H.: Optimal regularity and error estimates of a spectral Galerkin method for fractional advection–diffusion–reaction equations. SIAM J. Numer. Anal. 58, 211–233 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hao, Z., Park, M., Lin, G., Cai, Z.: Finite element method for two-sided fractional differential equations with variable coefficients: Galerkin approach. J. Sci. Comput. 79, 700–717 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jia, J., Zheng, X., Fu, H., Dai, P., Wang, H.: A fast method for variable-order space-fractional diffusion equations. Numer. Algorithms 85, 1519–1540 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, 383–406 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, M., Zhao, J., Huang, C., Chen, S.: Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data. J. Sci. Comput. 81(3), 1823–1859 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, F., Zhuang, P., Turner, I., Burrage, K., Anh, V.: A new fractional finite volume method for solving the fractional diffusion equation. Appl. Math. Model. 38, 3871–3878 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu, H., Cheng, A., Wang, H.: A parareal finite volume method for variable-order time-fractional diffusion equations. J. Sci. Comput. 85(1), 1–27 (2020)

    Article  MathSciNet  Google Scholar 

  24. Liu, J., Zhu, C., Chen, Y., Fu, H.: A Crank-Nicolson ADI quadratic spline collocation method for two-dimensional Riemann–Liouville space-fractional diffusion. Appl. Numer. Math. 160, 331–348 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44, 745–771 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Meerschaert, M.M., Sikorskii, A.: Stochastic models for fractional calculus. In: Studies in Mathematics, vol. 43. De Gruyter, Berlin (2012)

  27. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Metler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

  29. Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bulletin des Sciences Math́ematiques 136, 521–573 (2012)

  30. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  31. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  32. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

  33. Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  MATH  Google Scholar 

  34. Sun, H., Zhang, Y., Wei, S., Zhu, J., Chen, W.: A space fractional constitutive equation model for non-Newtonian fluid flow. Commun. Nonlinear Sci. Numer. Simul. 62, 409–417 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tang, T., Wang, L., Yuan, H., Zhou, T.: Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains. SIAM J. Sci. Comput. 42, A585–A611 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tang, T., Yuan, H., Zhou, T.: Hermite spectral collocation methods for fractional PDEs in unbounded domains. Commun. Comput. Phys. 24, 1143–1168 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51, 1088–1107 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, H., Yang, D., Zhu, S.: A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 290, 45–56 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, H., Yang, D., Zhu, S.: Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 52, 1292–1310 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, S., Chen, H., Wang, H.: Least-squared mixed variational formulation based on space decomposition for a kind of variable-coefficient fractional diffusion problems. J. Sci. Comput. 78(2), 687–709 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yao, W., Guo, Z., Sun, J., Wu, B., Gao, H.: Multiplicative noise removal for texture images based on adaptive anisotropic fractional diffusion equations. SIAM J. Imaging Sci. 12(2), 839–873 (2019)

    Article  MathSciNet  Google Scholar 

  42. Yin, B., Liu, Y., Li, H., Zhang, Z.: Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions. J. Sci. Comput. 84(1), 1–22 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. Yin, B., Liu, Y., Li, H., He, S.: Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions. J. Comput. Phys. 379, 351–372 (2019)

    Article  MathSciNet  Google Scholar 

  44. Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zheng, X., Ervin, V.J., Wang, H.: An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates. Commun. Appl. Math. Comput. 2, 147–162 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zheng, X., Ervin, V.J., Wang, H.: Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data. Comput. Meth. Appl. Math. 20, 573–589 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zheng, X., Ervin, V.J., Wang, H.: Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation. Appl. Numer. Math. 153, 234–247 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zheng, X., Ervin, V.J., Wang, H.: Optimal Petrov–Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval. J. Sci. Comput. 86, 1–22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was supported in part by the National Natural Science Foundation of China (Nos. 11971482, 12131014), by the National Science Foundation (No. DMS-1620194), by the China Postdoctoral Science Foundation (Nos. 2020M681136, 2021M700244, 2021TQ0017, 2021T140129), and by the International Postdoctoral Exchange Fellowship Program (Talent-Introduction Program) (No. YJ20210019).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Huan Liu

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Xiangcheng Zheng

  3. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

  4. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, Shandong, China

    Hongfei Fu

Authors
  1. Huan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiangcheng Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongfei Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongfei Fu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Liu, H., Zheng, X., Wang, H. et al. Error Estimate of Finite Element Approximation for Two-Sided Space-Fractional Evolution Equation with Variable Coefficient. J Sci Comput 90, 15 (2022). https://doi.org/10.1007/s10915-021-01698-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01698-3

Keywords