Taiwanese Journal of Mathematics

Superconvergence of FEM for Distributed Order Time Fractional Variable Coefficient Diffusion Equations

Yanhua Yang and Jincheng Ren

Full-text: Open access

Abstract

In this paper, a numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional variable coefficient diffusion equations is developed and a complete error analysis is provided. The weighted and shifted Grünwald formula is applied for the time-fractional derivative and finite element approach for the spatial discretization. The unconditional stability and the global superconvergence estimate of the fully discrete scheme are proved rigorously. Extensive numerical experiments are presented to illustrate the accuracy and efficiency of the scheme, and to verify the convergence theory.

Article information

Source
Taiwanese J. Math., Volume 22, Number 6 (2018), 1529-1545.

Dates
Received: 7 December 2017
Revised: 15 May 2018
Accepted: 19 June 2018
First available in Project Euclid: 12 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1531382428

Digital Object Identifier
doi:10.11650/tjm/180606

Mathematical Reviews number (MathSciNet)
MR3880239

Zentralblatt MATH identifier
07021703

Subjects
Primary: 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Keywords
distributed order diffusion equations finite element method fully discrete scheme superconvergence estimate

Citation

Yang, Yanhua; Ren, Jincheng. Superconvergence of FEM for Distributed Order Time Fractional Variable Coefficient Diffusion Equations. Taiwanese J. Math. 22 (2018), no. 6, 1529--1545. doi:10.11650/tjm/180606. https://projecteuclid.org/euclid.twjm/1531382428


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References

  • R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65, Academic Press, New York, 1975.
  • A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), 424–438.
  • A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002), no. 4, 046129.
  • A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Y. Gonchar, Distributed order time fractional diffusion equation, Fract. Calc. Appl. Anal. 6 (2003), no. 3, 259–279.
  • P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis II: Finite element methods, Amsterdam, North-Holland, 1991.
  • M. Cui, Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comput. Phys. 231 (2012), no. 6, 2621–2633.
  • K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math. 225 (2009), no. 1, 96–104.
  • H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes problem, Math. Comp. 80 (2011), no. 274, 697–722.
  • Q. Feng and F. Meng, Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method, Math. Methods Appl. Sci. 40 (2017), no. 10, 3676–3686.
  • G.-H. Gao and Z.-Z. Sun, Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput. 66 (2016), no. 3, 1281–1312.
  • G.-H. Gao, H.-W. Sun and Z.-Z. Sun, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys. 298 (2015), 337–359.
  • L. Guo, L. Liu and Y. Wu, Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions, Nonlinear Anal. Model. Control. 21 (2016), no. 5, 635–650.
  • M. Li and J. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput. 324 (2018), 254–265.
  • H.-L. Liao, P. Lyu, S. Vong and Y. Zhao, Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations, Numer. Algorithms 75 (2017), no. 4, 845–878.
  • H.-L. Liao, Y.-N. Zhang, Y. Zhao and H.-S. Shi, Stability and convergence of modified Du Fort-Frankel schemes for solving time-fractioal subdiffusion equations, J. Sci. Comput. 61 (2014), no. 3, 629–648.
  • Q. Lin and N. N. Yan, The Construction and Analysis of High Efficient Elements, Hebei University Press, China, 1996.
  • F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal. 16 (2013), no. 1, 9–25.
  • C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986), no. 3, 704–719.
  • M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65–77.
  • I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering 198, Academic Press, San Diego, CA, 1999.
  • J. Ren and Y. Ma, A superconvergent nonconforming mixed finite element method for the Navier-Stokes equations, Numer. Methods Partial Differential Equations 32 (2016), no. 2, 646–660.
  • J. Ren and Z.-Z. Sun, Efficient and stable numerical methods for multi-term time fractional sub-diffusion equations, East Asian J. Appl. Math. 4 (2014), no. 3, 242–266.
  • ––––, Efficient numerical solution of the multi-term time fractional diffusion-wave equation, East Asian J. Appl. Math. 5 (2015), no. 1, 1–28.
  • S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and applications, Gordon and Breach, London, 1993.
  • T. Shen, J. Xin and J. Huang, Time-space fractional stochastic Ginzburg-Landau equation driven by Gaussian white noise, Stoch. Anal. Appl. 36 (2018), no. 1, 103–113.
  • W. Tian, H. Zhou and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp. 84 (2015), no. 294, 1703–1727.
  • S. Vong and Z. Wang, A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, J. Comput. Phys. 274 (2014), 268–282.
  • Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), 1–15.
  • H. Ye, F. Liu, V. Anh and I. Turner, Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math. 80 (2015), no. 3, 825–838.
  • X. Zhang, L. Liu and Y. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling 55 (2012), no. 3-4, 1263–1274.
  • ––––, Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives, Appl. Math. Comput. 219 (2012), no. 4, 1420–1433.
  • J. Zhang, Z. Lou, Y. Ji and W. Shao, Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl. 462 (2018), no. 1, 57–83.
  • Y.-N. Zhang and Z.-Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys. 230 (2011), no. 24, 8713–8728.
  • Y.-N. Zhang, Z.-Z. Sun and H.-W. Wu, Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation, SIAM J. Numer. Anal. 49 (2011), no. 6, 2302–2322.
  • P. Zhuang, F. Liu, V. Anh and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 1079–1095.