Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Optimal Petrov–Galerkin Spectral Approximation Method for the Fractional Diffusion, Advection, Reaction Equation on a Bounded Interval

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

Abstract

In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov–Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.

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
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington (1964)

  2. Acosta, G., Borthagaray, J.P., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for the (1-d)-fractional Laplacian. Math. Comput. 87, 1821–1857 (2018)

    Article  MathSciNet  Google Scholar 

  3. Babuška, I., Guo, B.: Direct and inverse approximation theorems for the \(p\)-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces. SIAM J. Numer. Anal. 39(5):1512–1538 (2001/02)

  4. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional-order governing equation of Lévy motion. Water Resour. Res. 36(6), 1413–1424 (2000)

    Article  Google Scholar 

  5. Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev World. Preprint IRMAR 07-14, Université de Rennes 1 (2007)

  6. Bernardi, C., Dauge, M., Maday, Y.: Polynomials in weighted Sobolev spaces: basics and trace liftings. Internal Report 92039, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris (1992)

  7. Buades, A., Coll, B., Morel, J.M.:Image denoising methods. A new nonlocal principle. SIAM Rev. 52(1):113–147 (2010). Reprint of “A review of image denoising algorithms, with a new one” [MR2162865]

  8. Chen, H., Wang, H.: Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation. J. Comput. Appl. Math. 296, 480–498 (2016)

    Article  MathSciNet  Google Scholar 

  9. Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comput. 85(300), 1603–1638 (2016)

    Article  MathSciNet  Google Scholar 

  10. Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792–7804 (2009)

    Article  MathSciNet  Google Scholar 

  11. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

    Book  Google Scholar 

  12. Ervin, V.J.: Regularity of the solution to fractional diffusion, advection, reaction equations. https://arxiv.org/abs/1911.03261 (2019)

  13. 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  Google Scholar 

  14. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)

    Article  MathSciNet  Google Scholar 

  15. Gatto, P., Hesthaven, J.S.: Numerical approximation of the fractional Laplacian via \(hp\)-finite elements, with an application to image denoising. J. Sci. Comput. 65(1), 249–270 (2015)

    Article  MathSciNet  Google Scholar 

  16. Ginting, V., Li, Y.: On the fractional diffusion–advection–reaction equation in \({\mathbb{R}}\). Fract. Calc. Appl. Anal. 22(4), 1039–1062 (2019)

    Article  MathSciNet  Google Scholar 

  17. Gui, W., Babuška, I.: The \(h,\;p\) and \(h\)-\(p\) versions of the finite element method in \(1\) dimension. II. The error analysis of the \(h\)- and \(h\)-\(p\) versions. Numer. Math. 49(6), 613–657 (1986)

    Article  MathSciNet  Google Scholar 

  18. Guo, B.-Y., Wang, L.-L.: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory 128(1), 1–41 (2004)

    Article  MathSciNet  Google Scholar 

  19. Hao, Z., Lin, G., Zhang, Z.: Error estimates of a spectral Petrov–Galerkin method for two-sided fractional reaction–diffusion equations. Appl. Math. Comput. 374, 125045 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Hao, Z., Zhang, Z.: Optimal regularity and error estimates of a spectral galerkin method for fractional advection–diffusion–reaction equations. SIAM J. Numer. Anal. 58(1), 211–233 (2020)

    Article  MathSciNet  Google Scholar 

  21. Jia, L., Chen, H., Ervin, V.J.: Existence and regularity of solutions to 1-D fractional order diffusion equations. Electron. J. Differ. Equ. 93, 1–21 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Jin, B., Lazarov, R., Lu, X., Zhou, Z.: A simple finite element method for boundary value problems with a Riemann–Liouville derivative. J. Comput. Appl. Math. 293, 94–111 (2016)

    Article  MathSciNet  Google Scholar 

  23. Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comput. 84(296), 2665–2700 (2015)

    Article  MathSciNet  Google Scholar 

  24. Jin, B., Lazarov, R., Zhou, Z.: A Petrov–Galerkin finite element method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 54(1), 481–503 (2016)

    Article  MathSciNet  Google Scholar 

  25. Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383–406 (2012)

    Article  MathSciNet  Google Scholar 

  26. Li, Y., Chen, H., Wang, H.: A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations. Math. Methods Appl. Sci. 40(14), 5018–5034 (2017)

    Article  MathSciNet  Google Scholar 

  27. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002), vol. 166, pp. 209–219 (2004)

  28. Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35(8), 4103–4116 (2011)

    Article  MathSciNet  Google Scholar 

  29. Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine. 1996), Volume 378 of CISM Courses and Lectures, pp. 291–348. Springer, Vienna (1997)

  30. Mao, Z., Chen, S., Shen, J.: Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Appl. Numer. Math. 106, 165–181 (2016)

    Article  MathSciNet  Google Scholar 

  31. Mao, Z., Em-Karniadakis, G.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56(1), 24–49 (2018)

    Article  MathSciNet  Google Scholar 

  32. Mao, Z., Shen, J.: Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)

    Article  MathSciNet  Google Scholar 

  33. Mao, Z., Shen, J.: Spectral element method with geometric mesh for two-sided fractional differential equations. Adv. Comput. Math. 44(3), 745–771 (2018)

    Article  MathSciNet  Google Scholar 

  34. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)

    Article  MathSciNet  Google Scholar 

  35. Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)

    Article  MathSciNet  Google Scholar 

  36. Szegő, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., 4th edition, 1975. American Mathematical Society, Colloquium Publications, vol. XXIII

  37. Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)

    Article  MathSciNet  Google Scholar 

  38. Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34(5), A2444–A2458 (2012)

    Article  MathSciNet  Google Scholar 

  39. Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51(2), 1088–1107 (2013)

    Article  MathSciNet  Google Scholar 

  40. Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)

    Article  MathSciNet  Google Scholar 

  41. Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E (3) 48(3), 1683–1694 (1993)

    Article  MathSciNet  Google Scholar 

  42. Zayernouri, M., Karniadakis, G.E.: Fractional Sturm–Liouville eigen-problems: theory and numerical approximation. J. Comput. Phys. 252, 495–517 (2013)

    Article  MathSciNet  Google Scholar 

  43. Zheng, X., Ervin, V.J., Wang, H.: Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension. Appl. Math. Comput. 361, 98–111 (2019)

    MathSciNet  MATH  Google Scholar 

  44. 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(1), 147–162 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially funded by the ARO MURI Grant W911NF-15-1-0562, by the National Science Foundation under Grants DMS-1620194 and DMS-2012291, and by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina. All data generated or analyzed during this study are included in this published article.

Author information

Authors and Affiliations

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

    Xiangcheng Zheng & Hong Wang

  2. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634-0975, USA

    V. J. Ervin

Authors
  1. Xiangcheng Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. J. Ervin
    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

Corresponding author

Correspondence to V. J. Ervin.

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

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, 29 (2021). https://doi.org/10.1007/s10915-020-01366-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01366-y

Keywords

Mathematics Subject Classification

Search

Navigation