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Spectral element method with geometric mesh for two-sided fractional differential equations

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Abstract

Solutions of two-sided fractional differential equations (FDEs) usually exhibit singularities at the both endpoints, so it can not be well approximated by a usual polynomial based method. Furthermore, the singular behaviors are usually not known a priori, making it difficult to construct special spectral methods tailored for given singularities. We construct a spectral element approximation with geometric mesh, describe its efficient implementation, and derive corresponding error estimates. We also present ample numerical examples to validate our error analysis.

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References

  1. Chen, M., Deng, W.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52(3), 1418–1438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, F., Xu, Q., Hesthaven, J.S.: A multi-domain spectral method for time-fractional differential equations. J. Comput. Phys. 293, 157–172 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Ervin, V.J., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in \(\mathbb {R}^{d}\). Numer. Methods Partial Differ. Equ. 23(2), 256–281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space–time fractional diffusion. Chem. Phys. 284(1), 521–541 (2002)

    Article  MATH  Google Scholar 

  6. Gorenflo, R., Vivoli, A., Mainardi, F.: Discrete and continuous random walk models for space-time fractional diffusion. Nonlinear Dyn. 38(1-4), 101–116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gui, W.-Z., Babuṡka, I.: The h, p and h-p versions of the finite element method in 1 dimension. I. The error analysis of the p-version. Numer Math. 49(6), 577–612 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gui, W.-Z., Babuṡka, 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  MATH  Google Scholar 

  9. Hilfer, R., Butzer, P.L., Westphal, U., Douglas, J., Schneider, W.R., Zaslavsky, G., Nonnemacher, T., Blumen, A., West, B.: Applications of fractional calculus in physics, volume 128. World Scientific (2000)

  10. Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: A finite element method for the fractional Sturm-liouville Problem. arXiv:1307.5114 (2013)

  11. Jin, B., Zhou, Z.: A finite element method with singularity reconstruction for fractional boundary value problems. ESAIM Math. Model. Numer Anal. 49(5), 1261–1283 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8(5), 1016–1051 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Liu, Q., Liu, F., Turner, I.W., Anh, V.V.: Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method. J. Comput. Phys. 222(1), 57–70 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Moffatt, H.K., Zaslavsky, G.M., Comte, P., Tabor, M.: Topological aspects of the dvnamics of fluidi and plasmas (1992)

  19. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic press, San Diego (1999)

    MATH  Google Scholar 

  20. Samko, S.G., Kilbas, A.A., Mariċev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publication (1993)

  21. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications. Springer (2011)

  22. Yanovsky, V.V., Chechkin, A.V., Schertzer, D., Tur, A.V.: Lévy anomalous diffusion and fractional fokker–planck equation. Phys. A Stat. Mech. Appl. 282(1), 13–34 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Zayernouri, M., Karniadakis, G.E.: Discontinuous spectral element methods for time- and space-fractional advection equations. SIAM J. Sci. Comput. 36(4), B684–B707 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zeng, F., Zhang, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations. SIAM J. Sci. Comput. 37(6), A2710–A2732 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zeng, F., Mao, Z., Karniadakis, G.E.: A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities. To be appeared on SIAM Journal Science Computer (2016)

  27. Zhao, X., Sun, Z.-Z., Hao, Z.-P.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36(6), A2865–A2886 (2014)

    Article  MATH  Google Scholar 

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Acknowledgments

Z.M. is supported in part by the MURI/ARO on “Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications” (W911NF-15-1-0562).

J.S is supported in part by AFOSR FA9550-16-1-0102 and NSF DMS-1620262, and by NSFC grants 11371298, 91630204 and 11421110001.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George St, Providence, RI, 02912, USA

    Zhiping Mao

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907-1957, USA

    Jie Shen

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  1. Zhiping Mao
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  2. Jie Shen
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Correspondence to Jie Shen.

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Communicated by: Martin Stynes

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Mao, Z., Shen, J. Spectral element method with geometric mesh for two-sided fractional differential equations. Adv Comput Math 44, 745–771 (2018). https://doi.org/10.1007/s10444-017-9561-9

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  • DOI: https://doi.org/10.1007/s10444-017-9561-9

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