Advertisement

Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

Abstract

This paper analyzes the convergence of a Petrov–Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Luchko, Y.: Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. Inter. J. Geomath. 1, 257–276 (2011)

  2. 2.

    Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Soliton Fract. 7(9), 1461–1477 (1996)

  3. 3.

    Mainardi, F.: Fractional diffusive waves. J. Comput. Acoust. 9(4), 1417–1436 (2001)

  4. 4.

    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)

  5. 5.

    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1998)

  6. 6.

    Bazhlekova, E.: Duhamel-type representation of the solutions of non-local boundary value problems for the fractional diffusion-wave equation. In: Proceedings of the 2nd International Workshop, Bulgarian Academy of Sciences, Sofia, pp. 32–40 (1998)

  7. 7.

    Bazhlekova, E.: Fractional Evolution Equations in Banach Spaces. PhD thesis, Eindhoven University of Technology (2001)

  8. 8.

    Sakamoto, K., Yamamoto, Y.: Initial value or boundary value problems for fractional diffusion wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)

  9. 9.

    Oldham, K., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

  10. 10.

    Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

  11. 11.

    Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci Comput. 38(1), A146–A170 (2016)

  12. 12.

    Li, B., Luo, H., Xie, X.: A time-spectral algorithm for fractional wave problems. J. Sci. Comput. 77(2), 1164–1184 (2018)

  13. 13.

    Li, B., Luo, H., Xie, X.: A space-time finite element method for fractional wave problems. (2018). preprint, arXiv:1803.03437

  14. 14.

    Lubich, C., Sloan, I., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65(213), 1–17 (1996)

  15. 15.

    Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75, 673–696 (2006)

  16. 16.

    Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51(1), 491–515 (2013)

  17. 17.

    Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56(1), 1–23 (2018)

  18. 18.

    Sheng, C., Shen, J.: A space-time Petrov–Galerkin spectral method for time fractional diffusion equation. Numer. Math. Theor. Meth. Appl. 11, 854–876 (2018)

  19. 19.

    Li, D., Wu, C., Zhang, Z.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with nonsmooth solutions in time direction. J. Sci. Comput. (2019). https://doi.org/10.1007/s10915-019-00943-0

  20. 20.

    Li, D., Zhang, J., Zhang, Z.: Unconditionally optimal error estimates of a linearized Galerkin method for nonlinear time fractional reaction-subdiffusion equations. J. Sci. Comput. 76, 848–866 (2018)

  21. 21.

    Lions, J., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972)

  22. 22.

    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Springer, Berlin (2007)

  23. 23.

    Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment. Springer, Berlin (1992)

  24. 24.

    Yamamoto, M.: Weak solutions to non-homogeneous boundary value problems for time-fractional diffusion equations. J. Math. Anal. Appl. 460(1), 365–381 (2018)

  25. 25.

    Gorenflo, R., Yamamoto, M.: Operator theoretic treatment of linear Abel integral equations of first kind. Jpn. J. Ind. Appl. Math. 16(1), 137–161 (1999)

  26. 26.

    Luchko, Y., Gorenflo, R., Yamamoto, M.: Time-fractional diffusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18(3), 799–820 (2015)

  27. 27.

    Lunardi, A.: Interpolation Theory. Edizioni della Normale, Pisa (2018)

  28. 28.

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

  29. 29.

    Li, B., Luo, H., Xie, X.: Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data. SIAM J. Numer. Anal. 57(2), 779–798 (2019)

  30. 30.

    Ervin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Meth. Part. D. E. 22(3), 558–576 (2006)

  31. 31.

    Demengel, F., Demengel, G.: Functional Spaces for the Theory of Elliptic Partial Differential Equations. Springer, London (2012)

  32. 32.

    Agranovich, M.: Sobolev Spaces. Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains. Springer, Cham (2015)

  33. 33.

    Babuška, I.: Error-bounds for finite element method. Numer. Math. 16(4), 322–333 (1971)

  34. 34.

    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

  35. 35.

    Clément, P.: Approximation by finite element functions using local regularization. RAIRO, Anal. Numer. 9, 77–84 (1975)

  36. 36.

    Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34(150), 441–463 (1980)

  37. 37.

    Li, B., Luo, H., Xie, X.: Error estimates of a discontinuous Galerkin method for time fractional diffusion problems with nonsmooth data. (2018). preprint, arXiv:1809.02015

  38. 38.

    Ke, R., Ng, M., Sun, H.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203–211 (2015)

Download references

Author information

Correspondence to Binjie Li.

Additional information

Publisher's Note

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

This work was supported in part by National Natural Science Foundation of China (11771312).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Luo, H., Li, B. & Xie, X. Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data. J Sci Comput 80, 957–992 (2019) doi:10.1007/s10915-019-00962-x

Download citation

Keywords

  • Fractional wave problem
  • Regularity
  • Petrov–Galerkin
  • Convergence analysis
  • Nonsmooth data