Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems


We derive optimal \(L^2\)-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order \(\alpha \in (0,1)\), for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results.

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  1. 1.

    Al-Maskari, M., Karaa, S.: Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data. SIAM J. Numer. Anal. 57, 1524–1544 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    MathSciNet  Article  Google Scholar 

  3. 3.

    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 

  4. 4.

    Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7, 33–76 (1973)

    MathSciNet  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chen, C., Thomée, V., Wahlbin, L.B.: Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 58, 587–602 (1992)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chen, Z.: Expanded mixed finite element methods for linear second-order elliptic problems, I. RAIRO Modél. Math. Anal. Numér. 32, 479–499 (1998)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Cockburn, B., Mustapha, K.: A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. 130, 293–314 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hecht, F., Pironneau, O., Le Hyaric, A.:

  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, A146–A170 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jin, B., Lazarov, R., Liu, Y., 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., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Eng. 346, 332–358 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Johnson, C., Larsson, S., Thomée, V., Wahlbin, L.B.: Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49, 331–357 (1987)

    MathSciNet  Article  Google Scholar 

  16. 16.

    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 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Karaa, S., Pani, A.K.: Mixed FEM for time-fractional diffusion problems with time-dependent coefficients. J Sci Comput. (2020).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    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, 773–801 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 21.

    Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of \(L^1\)-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86–103 (2017)

    MathSciNet  Google Scholar 

  22. 22.

    Li, D., Wang, J., Zhang, J.: Unconditionally convergent \(L^1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput. 39, A3067–A3088 (2017)

    Article  Google Scholar 

  23. 23.

    Li, X., Yang, X., Zhang, Y.: Error estimates of mixed finite element methods for time-fractional Navier–Stokes equations. J. Sci. Comput. 70, 500–515 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Lubich, C.: Convolution quadrature and discretized operational calculus-I. Numer. Math. 52, 129–145 (1988)

    MathSciNet  Article  Google Scholar 

  26. 26.

    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)

    MathSciNet  Article  Google Scholar 

  27. 27.

    McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equ. Appl. 22, 57–94 (2010)

    MathSciNet  Article  Google Scholar 

  28. 28.

    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)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Mustapha, K., Mustapha, H.: A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA J. Numer. Anal. 30, 555–578 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Nitsche, J.A.: Über ein Variationsprinzip zur Lösung yon Dirichlet-Problemen bei Verwendung von Teilrädumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)

    MathSciNet  Article  Google Scholar 

  31. 31.

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

    Google Scholar 

  32. 32.

    Thomée, V.: Galerkin finite element methods for parabolic problems. Springer, Berlin (1997)

    Google Scholar 

  33. 33.

    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 

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Correspondence to Samir Karaa.

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This research was supported by the Research Council of Oman Grant ORG/CBS/15/001.

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Karaa, S. Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems. J Sci Comput 83, 46 (2020).

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  • Semilinear fractional diffusion
  • Galerkin method
  • Nonconforming FE method
  • Mixed FE method
  • Convolution quadrature
  • Error estimate

Mathematics Subject Classification

  • 65M60
  • 65M12
  • 65M15