Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients

Abstract

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates, which are optimal with respect to both approximation properties and regularity results, are derived for the semidiscrete problem with smooth as well as nonsmooth initial data. Specially, a priori error bounds for both primary and secondary variables in \(L^2\)-norm are established. Since the comparison between Fortin projection and the mixed Galerkin approximation of the secondary variable yields an improved rate of convergence, therefore, as a by-product, we derive \(L^p\)-estimates for the error in primary variable. Finally, some numerical experiments are conducted to confirm our theoretical findings.

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References

  1. 1.

    Bramble, J.H., Schatz, A.H., Thomée, V., Wahlbin, L.B.: Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14, 218–241 (1977)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, 2nd edn. Springer, New York (1991)

    Google Scholar 

  3. 3.

    Chen, H., Ewing, R., Lazarov, R.: Superconvergence of mixed finite element methods for for parabolic problems with nonsmooth initial data. Numer. Math. 78, 495–521 (1998)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Fujita, H., Suzuki, T.: Evolution problems. Handbook of Numerical Analysis, vol. II, pp. 789–928. North Holland, Amsterdam (1991)

    Google Scholar 

  5. 5.

    Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Goswami, D., Pani, A.K.: An alternate approach to optimal L2-error analysis of semidiscrete Galerkin methods for linear parabolic problems with nonsmooth initial data. Numer. Funct. Anal. Optim. 32, 946–982 (2011)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Goswami, D., Pani, A.K., Yadav, S.: Optimal error estimates of two mixed finite element methods for parabolic integro-differential equations with nonsmooth initial data. J. Sci. Comput. 56, 131–164 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Goswami, D., Pani, A.K., Yadav, S.: Optimal \(L^2\) estimates for semidiscrete Galerkin methods for parabolic integro-differential equations with nonsmooth data. ANZIAM J. 55, 245–266 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Huang, M., Thomée, V.: Some convergence estimates for semidiscrete Galerkin type schemes for time-dependent non-selfadjoint equations. Math. Comput. 37, 327–346 (1981)

    Article  Google Scholar 

  10. 10.

    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)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Jin, B., Lazarov, R., Pascal, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561–582 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jin, B., Lazarov, R., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jin, B., Li, B., Zhou, Z.: Subdiffusion with a time dependent coefficient: analysis and numerical solution. Math. Comput. 88, 2157–2186 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Johnson, C., Thomée, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 14, 41–78 (1981)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Karaa, S.: Semidiscrete finite element analysis time fractional parabolic problems: a unified approach. SIAM J. Numer. 56, 1673–1692 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Karaa, S., Mustapha, K., Pani, A.K.: Finite volume element method for two-dimensional fractional sub-diffusion problems. IMA J. Numer. Anal. 37, 945–964 (2017)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Karaa, S., Mustapha, K., Pani, A.K.: Optimal error analysis of a FEM for fractional diffusion problems by energy arguments. J. Sci. Comput. 74, 519–535 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Karaa, S., Pani, A.K.: Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data. ESAIM Math. Model. Numer. Anal. 52(2), 773–801 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    Google Scholar 

  20. 20.

    Le, K.N., McLean, W., Mustapha, K.: Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal. 54, 1763–1784 (2016)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Luskin, M., Rannacher, R.: On the smoothing property of the Galerkin method for parabolic equations. SIAM J. Numer. Anal. 19, 93–113 (1982)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mclean, W.: Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, 123–138 (2010)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Mustapha, K.: FEM for time-fractional diffusion equations, novel optimal error analyses. Math. Comput. 87, 2259–2272 (2018)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Mustapha, K., Schötzau, D.: Well-posedness of \(hp-\)version discontinuous Galerkin methods for fractional diffusion wave equations. IMA J. Numer. Anal. 34, 1226–1246 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Raviart, P., Thomas, J.A.: Mixed finite element method for second order elliptic problems. In: Mathematical Aspects of the Finite Element Method. In: Galligani, I., Magenes, E. (eds). Lecture Notes in Mathematics, vol. 606. Springer, Berlin (1977)

  27. 27.

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

    MathSciNet  Article  Google Scholar 

  28. 28.

    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)

    Google Scholar 

  29. 29.

    Zhao, Y., Chen, P., Bu, W., Liu, X., Tang, Y.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70, 407–428 (2017)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This research is supported by the Research Council of Oman grant ORG/CBS/15/001. The second author acknowledges the support from Institute Chair Professor’s fund and the support from SERB, Govt. India via MATRIX Grant No. MTR/201S/000309. Both the authors thank the referees for their valuable suggestions which help to improve the manuscript.

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Correspondence to Amiya K. Pani.

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Karaa, S., Pani, A.K. Mixed FEM for Time-Fractional Diffusion Problems with Time-Dependent Coefficients. J Sci Comput 83, 51 (2020). https://doi.org/10.1007/s10915-020-01236-7

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Keywords

  • Time-fractional diffusion equation
  • Time-dependent coefficients
  • Mixed finite element method
  • Semidiscrete method
  • Optimal error estimates
  • Smooth and nonsmooth initial data

Mathematics Subject Classification

  • 65M60
  • 65M12
  • 65M15
  • 65M70
  • 35S10