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# Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model

## Abstract

In this article, a time two-mesh (TT-M) algorithm combined with the H1-Galerkin mixed finite element (FE) method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model, which is faster than the H1-Galerkin mixed FE method. The Crank-Nicolson scheme with TT-M algorithm is used to discretize the temporal direction at time $$t_{n+\frac {1}{2}}$$, the FBN-đ formula is developed to approximate the distributed-order derivative, and the H1-Galerkin mixed FE method is used to approximate the spatial direction. TT-M mixed element algorithm mainly covers three steps: first, the mixed finite element solution of the nonlinear coupled system on the time coarse mesh ÎtC is calculated; next, based on the numerical solution obtained in the first step, the numerical solution of the nonlinear coupled system on time fine mesh ÎtF is obtained by using Lagrangeâs interpolation formula; finally, the numerical solution of the linearized system on time fine mesh ÎtF is solved by using the results in the second step. The existence and uniqueness of the solution for our numerical scheme are shown. Moreover, the stability and a priori error estimate are analyzed in detail. Furthermore, numerical examples with smooth and nonsmooth solutions are given to validate our method.

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## References

1. 1.

Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61(1), 132 (2000)

2. 2.

Yuste, S., Acedo, L., Lindenberg, K.: Reaction front in an $$A+B\rightarrow C$$ reaction-subdiffusion process. Phys. Rev. E 69(3), 036126 (2004)

3. 3.

Le Vot, F., Abad, E., Yuste, S.B.: Continuous-time random-walk model for anomalous diffusion in expanding media. Phys. Rev. E 96(3), 032117 (2017)

4. 4.

Raberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high-frequency financial data: an empirical study. Physica A: Stat. Mech. Appl. 314(1), 749â755 (2002)

5. 5.

Vong, S., Wang, Z.: A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions. J. Comput. Phys. 274, 268â282 (2014)

6. 6.

Atangana, A.: On the stability and convergence of the time-fractional variable order telegraph equation. J. Comput. Phys. 293, 104â114 (2015)

7. 7.

Daftardar-Gejji, V., Bhalekar, S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345(2), 754â765 (2008)

8. 8.

Zhou, J., Xu, D., Chen, H.B.: A weak Galerkin finite element method for multi-term time-fractional diffusion equations. East Asian J. Appl. Math. 8, 181â193 (2018)

9. 9.

YĂ©pez-MartĂ­nez, H., GĂłmez-Aguilar, J. F., Sosa, I.O., Reyes, J.M., Torres-JimĂ©nez, J: The Fengâs first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev. Mex. Fis. 62(4), 310â316 (2016)

10. 10.

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

11. 11.

Fan, W., Jiang, X., Liu, F., Anh, V.: The unstructured mesh finite element method for the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain. J. Sci. Comput. 77, 27â52 (2018)

12. 12.

Feng, L., Liu, F., Turner, I.: Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains. Commun. Nonlinear Sci. Numer. Simul. 70, 354â371 (2019)

13. 13.

Zeng, F., Zhang, Z., Karniadakis, G.E.: Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions. Comput. Meth. Appl. Mech. Eng. 327, 478â502 (2017)

14. 14.

Shi, Z.G., Zhao, Y.M., Liu, F., Tang, Y.F., Wang, F.L., Shi, Y.H.: High accuracy analysis of an H1-Galerkin mixed finite element method for two-dimensional time fractional diffusion equations. Comput. Math. Appl. 74(8), 1903â1914 (2017)

15. 15.

Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40(7-8), 4970â4985 (2016)

16. 16.

Liu, Y., Du, Y.W., Li, H., Liu, F.W., Wang, Y.J.: Some second-order đ schemes combined with finite element method for nonlinear fractional Cable equation. Numer. Algor. 80, 533â555 (2019). https://doi.org/10.1007/s11075-018-0496-0

17. 17.

Mainardi, F., Pagnini, G., Gorenflo, R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187(1), 295â305 (2007)

18. 18.

Kochubei, A.N.: Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340(1), 252â281 (2008)

19. 19.

Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96â104 (2009)

20. 20.

Ye, H., Liu, F., Anh, V., Turner, I.: Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains. IMA J. Appl. Math. 80(3), 825â838 (2015)

21. 21.

Gao, G.H., Alikhanov, A.A., Sun, Z.Z.: The temporal second order difference schemes based on the interpolation approximation foe solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73 (1), 93â121 (2017)

22. 22.

Li, J., Liu, F., Feng, L., Turner, I.: A novel finite volume method for the Riesz space distributed-order advection-diffusion equation. Appl. Math. Model. 46, 536â553 (2017)

23. 23.

Bu, W.P., Xiao, A.G., Zeng, W.: Finite difference/finite element methods for distributed-order time fractional diffusion equations. J. Sci. Comput. 72(1), 422â441 (2017)

24. 24.

Zhang, H., Liu, F., Jiang, X., Zeng, F., Turner, I.: A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation. Comput. Math. Appl. 76(10), 2460â2476 (2018)

25. 25.

Abbaszadeh, M., Dehghan, M.: Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed-order advection-diffusion equation. Eng. Comput., 1â17 (2019)

26. 26.

Karamali, G., Dehghan, M., Abbaszadeh, M.: Numerical solution of a time-fractional PDE in the electroanalytical chemistry by a local meshless method. Eng. Comput. 35(1), 87â100 (2019)

27. 27.

Guo, S., Mei, L., Zhang, Z., Jiang, Y.: Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction-diffusion equation. Appl. Math. Lett. 85, 157â163 (2018)

28. 28.

He, L., Ren, J.C.: High spatial accuracy analysis of linear triangular finite element for distributed order diffusion equations. Taiwan. J. Math. https://doi.org/10.11650/tjm/190803 (2019)

29. 29.

Li, X.L., Rui, H.X., Liu, Z.G.: Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation. Numer. Algor. 82(1), 321â347 (2019)

30. 30.

Aboelenen, T.: Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and SchrĂ¶dinger-type equations. Nonlinear Dyn. 92(2), 395â413 (2018)

31. 31.

Shi, Y.H., Liu, F., Zhao, Y.M., Wang, F.L., Turner, I.: An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain. Appl. Math. Model. 73, 615â636 (2019)

32. 32.

Fei, M.F., Huang, C.M.: Galerkin-Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation. Int. J. Comput Math. 97(6), 1183â1196 (2020)

33. 33.

Li, D., Wang, J., Zhang, J.: Unconditionally convergent L1-Galerkin FEMs for nonlinear time-fractional SchrĂ¶dinger equations. SIAM J. Sci. Comput. 39(6), A3067âA3088 (2017)

34. 34.

Hou, Y., Wen, C., Li, H., Liu, Y., Fang, Z.C., Yang, Y.N.: Some second-order Ï schemes combined with an H1-Galerkin MFE method for a nonlinear distributed-order sub-diffusion equation. Mathematics 8, 187 (2020)

35. 35.

Pani, A.K.: An H1-Galerkin mixed finite element methods for parabolic partial differential equations. SIAM J. Numer. Anal. 35(2), 712â727 (1998)

36. 36.

Shi, D.Y., Wang, J.J.: Superconvergence analysis of an H1-Galerkin mixed finite element method for Sobolev equations. Comput. Math. Appl. 72(6), 1590â1602 (2016)

37. 37.

Guo, L, Chen, H.Z.: H1-Galerkin mixed finite element method for the regularized long wave equation. Computing 77(2), 205â221 (2006)

38. 38.

Wang, J.F., Liu, T.Q., Li, H., Liu, Y., He, S.: Second-order approximation scheme combined with H1-Galerkin MFE method for nonlinear time fractional convection-diffusion equation. Comput. Math. Appl. 73, 1182â1196 (2017)

39. 39.

Yin, B.L., Liu, Y., Li, H., Zhang, Z.M.: Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions. J. Sci. Comput. 84(1), 2 (2020)

40. 40.

Yin, B.L., Liu, Y., Li, H., Zhang, Z.M.: Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations. arXiv:1906.01242v2 (2019)

41. 41.

Li, C., Ding, H.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38(15-16), 3802â3821 (2014)

42. 42.

Liu, Y., Yu, Z.D., Li, H., Liu, F.W., Wang, J.F.: Time two-mesh algorithm combined with finite element method for time fractional water wave model. Int. J. Heat Mass Transfer. 120, 1132â1145 (2018)

43. 43.

Yin, B.L., Liu, Y., Li, H., He, S.: Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions. J. Comput. Phys. 379, 351â372 (2019)

44. 44.

Zhao, M., He, S., Wang, H., Qin, G.: An integrated fractional partial differential equation and molecular dynamics model of anomalously diffusive transport in heterogeneous nano-pore structures. J. Comput. Phys. 373, 1000â1012 (2018)

45. 45.

Quarteroni, A., Sacco, R., Saleri, F.: Numerical mathematics. Springer Science and Business Media (2010)

46. 46.

Liu, Y., Yin, B.L., Li, H., Zhang, Z.M.: The unified theory of shifted convolution quadrature for fractional calculus. arXiv:1908.01136(2019)

47. 47.

Yin, B.L., Liu, Y., Li, H.: A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations. Appl. Math. Comput. 368, 124799 (2020)

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## Acknowledgments

The authors are grateful to the referees and editor for their valuable comments and good suggestions which greatly improved the presentation of the paper.

## Funding

This work is supported by the National Natural Science Foundation of China (12061053, 11661058, 11761053), Natural Science Foundation of Inner Mongolia (2020MS01003), and program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-17-A07).

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Correspondence to Yang Liu.

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Wen, C., Liu, Y., Yin, B. et al. Fast second-order time two-mesh mixed finite element method for a nonlinear distributed-order sub-diffusion model. Numer Algor 88, 523â553 (2021). https://doi.org/10.1007/s11075-020-01048-8

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### Keywords

• Time two-mesh algorithm
• H 1-Galerkin mixed finite element method
• Crank-Nicolson formula
• Nonlinear distributed-order sub-diffusion model
• Stability and a priori error analysis