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2023, Volume 31, Issue 12: 7207-7223. doi: 10.3934/era.2023365
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Research article

Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

  • 1.
    School of Foundation Studies, Zhejiang Pharmaceutical University, Ningbo 315500, China
  • 2.
    College of Science, Hunan University of Science and Engineering, Yongzhou 425199, China

  • Academic Editor: Alain Miranville
  • Received: 08 September 2023 Revised: 27 October 2023 Accepted: 07 November 2023 Published: 10 November 2023

  • In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.

    Citation: Jun Pan, Yuelong Tang. Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations[J]. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365

    Related Papers:

  • In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.


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    [1] I. Podlubny, Fractional differential equations, in Mathematics in Science and Engineering, Academic Press, San Diego, 1999.
    [2] Z. Sun, G. Gao, The Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015.
    [3] C. Li, F. Zeng, Numerical Methods for Fractional Calculas, Chapman and Hall/CRC Press, Boca Raton, 2015. https://doi.org/10.1201/b18503
    [4] F. Liu, P. Zhuang, Q. Liu, Numerical Methods for Fractional Partial Differential Equations and Their Applications, Science Press, Beijing, 2015.
    [5] Y. Lin, C. Xu, Finite difference/spectral approximation for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
    [6] M. Stynes, E. O'riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
    [7] X. Li, Y. Chen, C. Chen, An improved two-grid technique for the nonlinear time-fractional parabolic equation based on the block-centered finite difference method, J. Comput. Math., 40 (2022), 455–473. https://doi.org/10.4208/jcm.2011-m2020-0124 doi: 10.4208/jcm.2011-m2020-0124
    [8] X. Peng, D. Xu, W. Qiu, Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation, Math. Comput. Simulat., 208 (2023, ) 702–726. https://doi.org/10.1016/j.matcom.2023.02.004
    [9] H. Wang, Y. Chen, Y. Huang, W. Mao, A posteriori error estimates of the Galerkin spectral methods for space-time fractional diffusion equations, Adv. Appl. Math. Mech., 12 (2020), 87–100.
    [10] X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
    [11] F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871–3878. https://doi.org/10.1016/j.apm.2013.10.007 doi: 10.1016/j.apm.2013.10.007
    [12] C. Huang, M. Stynes, Superconvergence of a finite element method for the multi-term time-fractional diffusion problem, J. Sci. Comput., 82 (2020), 10. https://doi.org/10.1007/s10915-019-01115-w doi: 10.1007/s10915-019-01115-w
    [13] B. Tang, Y. Chen, X. Lin, A posteriori error error estimates of spectral Galerkin methods for multi-term time fractional diffusion equations, Appl. Math. Lett., 120 (2021), 107259. https://doi.org/10.1016/j.aml.2021.107259 doi: 10.1016/j.aml.2021.107259
    [14] H. Liu, X. Zheng, C. Chen, H. Wang, A characteristic finite element method for the time-fractional mobile/immobile advection diffusion model, Adv. Comput. Math., 47 (2021), 41. https://doi.org/10.1007/s10444-021-09867-6 doi: 10.1007/s10444-021-09867-6
    [15] S. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients, Appl. Numer. Math., 168 (2021), 1–12. https://doi.org/10.1016/j.apnum.2021.05.021 doi: 10.1016/j.apnum.2021.05.021
    [16] Y. Zhao, P. Chen, W. Bu, X. Liu, Y. Tang, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput., 70 (2017), 407–428. https://doi.org/10.1007/s10915-015-0152-y doi: 10.1007/s10915-015-0152-y
    [17] Z. Shi, Y. Zhao, F. Liu, Y. Tang, F. Wang, Y. Shi, High accuracy analysis of an $H^1$-Galerkin mixed finite element method for two-dimensional time fractional diffusion equations, Comput. Math. Appl., 74 (2017), 1903–1914. https://doi.org/10.1016/j.camwa.2017.06.057 doi: 10.1016/j.camwa.2017.06.057
    [18] M. Abbaszadeh, M. Dehghan, Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction-diffusion equation on nonrectangular domains, Comput. Math. Appl., 78 (2019), 1531–1547. https://doi.org/10.1016/j.camwa.2019.03.040 doi: 10.1016/j.camwa.2019.03.040
    [19] X. Li, Y. Tang, Interpolated coefficient mixed finite elements for semilinear time fractional diffusion equations, Fractal Fract., 7 (2023), 482. https://doi.org/10.3390/fractalfract7060482 doi: 10.3390/fractalfract7060482
    [20] M. Li, J. Zhao, C. Huang, S. Chen, Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data, J. Sci. Comput., 81 (2019), 1823–1859. https://doi.org/10.1007/s10915-019-01064-4 doi: 10.1007/s10915-019-01064-4
    [21] S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
    [22] J. Shen, Z. Sun, R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math., 8 (2018), 834–858.
    [23] X. Gu, H. Sun, Y. Zhang, Y. Zhao, Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian, Math. Methods Appl. Sci., 44 (2021), 441–463. https://doi.org/10.1002/mma.6746 doi: 10.1002/mma.6746
    [24] G. Gao, Z. Sun, H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33–50. https://doi.org/10.1016/j.jcp.2013.11.017 doi: 10.1016/j.jcp.2013.11.017
    [25] A. Alikhanov, C. Huang, A high-order $L2$ type difference scheme for the time-fractional diffusion equation, Appl. Meth. Comput., 411 (2021), 126545. https://doi.org/10.1016/j.amc.2021.126545 doi: 10.1016/j.amc.2021.126545
    [26] J. Ren, H. Liao, J. Zhang, Z. Zhang, Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems, J. Comput. Appl. Math., 389 (2021), 113352. https://doi.org/10.1016/j.cam.2020.113352 doi: 10.1016/j.cam.2020.113352
    [27] R. Feng, Y. Liu, Y. Hou, H. Li, Z. Fang, Mixed element algorithm based on a second-order time approximation scheme for a two-dimensional nonlinear time fractional coupled sub-diffusion model, Eng. Comput., 38 (2022), 51–68. https://doi.org/10.1007/s00366-020-01032-9 doi: 10.1007/s00366-020-01032-9
    [28] X. Zheng, H. Wang, A hidden-memory variable-order time-fractional optimal control model: Analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. https://doi.org/10.1137/20M1344962 doi: 10.1137/20M1344962
    [29] C. Li, Z. Zhao, Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855–875. https://doi.org/10.1016/j.camwa.2011.02.045 doi: 10.1016/j.camwa.2011.02.045
    [30] D. Li, C. Wu, Z. Zhang, Linearized Galerkin fems for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80 (2019), 403–419. https://doi.org/10.1007/s10915-019-00943-0 doi: 10.1007/s10915-019-00943-0
    [31] J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759–1777. https://doi.org/10.1137/S0036142992232949 doi: 10.1137/S0036142992232949
    [32] Q. Li, Y. Chen, Y. Huang, Y. Wang, Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math., 157 (2020), 38–54. https://doi.org/10.1016/j.apnum.2020.05.024 doi: 10.1016/j.apnum.2020.05.024
    [33] Y. Zeng, Z. Tan, Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations, Appl. Math. Comput., 434 (2022), 127408. https://doi.org/10.1016/j.amc.2022.127408 doi: 10.1016/j.amc.2022.127408
    [34] H. Fu, B. Zhang, X. Zheng, A high-order two-grid difference method for nonlinear time-fractional Biharmonic problems and its unconditional $\alpha$-robust error estimates, J. Sci. Comput., 96 (2023), 54. https://doi.org/10.1007/s10915-023-02282-7 doi: 10.1007/s10915-023-02282-7
    [35] W. Qiu, D. Xu, J. Guo, J. Zhou, A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model, Numer. Algor., 85 (2020), 39–58. https://doi.org/10.1007/s11075-019-00801-y doi: 10.1007/s11075-019-00801-y
    [36] A. Pehlivanov, G. Carey, R. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal., 31 (1994), 1368–1377. https://doi.org/10.1137/0731071 doi: 10.1137/0731071
    [37] A. Pani, An $H^1$-Galerkin mixed finite element methods for parabolic partial differential equations, SIAM J. Numer. Anal., 35 (1998), 712–727. https://doi.org/10.1137/S0036142995280808 doi: 10.1137/S0036142995280808
    [38] D. Yang, A splitting positive definite mixed finite element method for miscible displacement of compressible flow in porous media, Numer. Methods Partial Differ. Equation, 17 (2001), 229–249. https://doi.org/10.1002/num.3 doi: 10.1002/num.3
    [39] Y. Liu, Y. Du, H. Li, J. Wang, An $H^1$-Galerkin mixed finite element method for time fractional reaction-diffusion equation, J. Appl. Math. Comput., 47 (2015), 103–117. https://doi.org/10.1007/s12190-014-0764-7 doi: 10.1007/s12190-014-0764-7
    [40] J. Wang, T. Liu, H. Li, Y. Liu, S. He, Second-order approximation scheme combined with $H^1$-Galerkin MFE method for nonlinear time fractional convection-diffusion equation, Comput. Math. Appl., 73 (2017), 1182–1196. https://doi.org/10.1016/j.camwa.2016.07.037 doi: 10.1016/j.camwa.2016.07.037
    [41] T. Hou, C. Liu, C. Dai, L. Chen, Y. Yang, Two-grid algorithm of $H^1$-Galerkin mixed finite element methods for semilinear parabolic integro-differential equations, J. Comput. Math., 40 (2022), 667–685. https://doi.org/10.4208/jcm.2101-m2019-0159 doi: 10.4208/jcm.2101-m2019-0159
    [42] M. Tripathy, R. Sinha, Superconvergence of $H^1$-Galerkin mixed finite element methods for parabolic problems, Appl. Anal., 88 (2009), 1213–1231. https://doi.org/10.1080/00036810903208163 doi: 10.1080/00036810903208163
    [43] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. https:doi.org//10.1007/978-1-4612-3172-1
    [44] R. Ewing, M. Liu, J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J. Numer. Anal., 36 (1999), 772–787. https://doi.org/10.1137/S0036142997322801 doi: 10.1137/S0036142997322801
    [45] C. Huang, M. Stynes, Optimal $H^1$ sptial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation, Adv. Comput. Math., 46 (2020), 63. https://doi.org/10.1007/s10444-020-09805-y doi: 10.1007/s10444-020-09805-y
    [46] R. Li, W. Liu, The AFEPack Handbook, 2006. Available from: http://dsec.pku.edu.cn/rli/software.php
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