Advertisement

Advances in Computational Mathematics

, Volume 45, Issue 2, pp 1005–1029 | Cite as

Galerkin FEM for a time-fractional Oldroyd-B fluid problem

  • Mariam Al-Maskari
  • Samir KaraaEmail author
Article
  • 55 Downloads

Abstract

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results.

Keywords

Time-fractional Oldroyd-B fluid problem Finite element method Convolution quadrature Error estimate Nonsmooth data 

Mathematics Subject Classification (2010)

65M60 65M12 65M15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This research is supported by The Research Council of Oman under grant ORG/CBS/15/001.

References

  1. 1.
    Abdullah, M., Butt, A.R., Raza, N., Haque, E.U.: Semi-analytical technique for the solution of fractional Maxwell fluid. Can. J. Phys. 94, 472–478 (2017)CrossRefGoogle Scholar
  2. 2.
    Al-Maskari, M., Karaa, S.: The lumped mass FEM for a time-fractional cable equation. Appl. Numer. Math. 132, 73–90 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bazhlekova, E.: Subordination principle for a class of fractional order differential equations. Mathematics 3, 412–427 (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bazhlekova, E., Bazhlekov, I.: Viscoelastic flows with fractional derivative models: computational approach by convolutional calculus of Dimovski. Fract. Calc. Appl. Anal. 17, 954–976 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bazhlekova, E., Bazhlekov, I.: Peristaltic transport of viscoelastic bio-fluids with fractional derivative models. Biomath 5, 1605151 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bazhlekova, E., Bazhlekov, I.: On the Rayleigh-Stokes problem for generalized fractional Oldroyd-B fluids. AIP Conf. Proc. 1684, 080001–1–080001-12 (2015)Google Scholar
  7. 7.
    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 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, C.M., Liu, F., Anh, V.: Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Appl. Math. Comput. 204, 340–351 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, C.M., Liu, F., Anh, V.: A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. J. Comput. Appl. Math. 223, 777–789 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dehghan, M., Abbaszadeh, M.: A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng. Comput. 33, 587–605 (2017)CrossRefGoogle Scholar
  13. 13.
    Fetecau, C., Jamil, M., Fetecau, C., Vieru, D.: The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid. Z. Angew. Math. Phys. 60, 921–933 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fujita, H., Suzuki, T.: Evolution problems. Handbook of Numerical Analysis, vol. II, pp. 789–928, Handb. Numer. Anal., II. North-Holland, Amsterdam (1991)Google Scholar
  15. 15.
    Jamil, M., Rauf, A., Zafar, A.A., Khan, N.A.: New exact analytical solutions for Stokes’ first problem of Maxwell fluid with fractional derivative approach. Comput. Math. Appl. 62, 1013–1023 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khan, M., Ali, S.H., Hayat, T., Fetecau, C.: MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium. Int. J. Non Linear Mech. 43, 302–319 (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Khan, M., Anjum, A., Fetecau, C., Qi, H.: Exact solutions for some oscillating motions of a fractional Burgers’ fluid. Math. Comput. Model. 51, 682–692 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Khan, M., Anjum, A., Qi, H., Fetecau, C.: On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. Z. Angew. Math. Phys. 61, 133–145 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lin, Y., Jiang, W.: Numerical method for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. Numer. Methods Partial Differential Equations 27, 1599–1609 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lubich, C.: Convolution quadrature and discretized operational calculus-I. Numer. Math. 52, 129–145 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lubich, C.: Convolution quadrature revisited. BIT 44, 503–514 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65, 1–17 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation. IMA J. Numer. Anal. 30, 208–230 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mohebbi, A., Abbaszadeh, M., Dehghan, M.: Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Comput. Methods Appl. Mech. Eng. 264, 163–177 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Prüss, J.: Evolutionary Integral Equations and Applications Monographs in Mathematics, vol. 87. Basel, Birkhäuser Verlag (1993)CrossRefGoogle Scholar
  27. 27.
    Rasheed, A., Wahab, A., Shah, S.Q., Nawaz, R.: Finite difference-finite element approach for solving fractional Oldroyd-B equation. Adv. Difference Equ. 2016(236), 21 (2016)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)zbMATHGoogle Scholar
  29. 29.
    Tripathi, D., Pandey, S.K., Das, S.: Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Appl. Math Comput. 215, 3645–3654 (2010)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Tripathi, D.: Peristaltic transport of fractional Maxwell fluids in uniform tubes: applications in endoscopy. Comput. Math. Appl. 62, 1116–1126 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Vasileva, D., Bazhlekov, I., Bazhlekova, E.: Alternating direction implicit schemes for two-dimensional generalized fractional Oldroyd-B fluids. AIP Conf. Proc. 1684, 080014–1–080014-16 (2015)zbMATHGoogle Scholar
  32. 32.
    Wu, C.: Numerical solution for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. Appl. Numer. Math. 59, 2571–2583 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhao, C., Yang, C.: Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Appl. Math. Comput. 211, 502–509 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Zhu, P., Xie, S., Wang, X.: Nonsmooth data error estimates for FEM approximations of the time fractional cable equation. Appl. Numer. Math. 121, 170–184 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.FracDiff Research Group, Department of MathematicsSultan Qaboos UniversityMuscatOman

Personalised recommendations