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Finite element method for two-dimensional space-fractional advection–dispersion equations

Author

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  • Zhao, Yanmin
  • Bu, Weiping
  • Huang, Jianfei
  • Liu, Da-Yan
  • Tang, Yifa

Abstract

The backward Euler and Crank–Nicolson–Galerkin fully-discrete approximate schemes for two-dimensional space-fractional advection–dispersion equations are established. Firstly, we prove that the corresponding variational problem has a unique solution, and the proposed fully-discrete schemes are unconditionally stable, whose solutions are all unique. Secondly, the optimal error estimates are derived by use of properties of projection operator and fractional derivatives. Finally, numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.

Suggested Citation

  • Zhao, Yanmin & Bu, Weiping & Huang, Jianfei & Liu, Da-Yan & Tang, Yifa, 2015. "Finite element method for two-dimensional space-fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 553-565.
  • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:553-565
    DOI: 10.1016/j.amc.2015.01.016
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    Cited by:

    1. Yang, Hong & Lao, Cheng-Xue & She, Zi-Hang, 2023. "Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients," Applied Mathematics and Computation, Elsevier, vol. 445(C).
    2. Zieniuk, Eugeniusz, 2017. "Approximation of the derivatives of solutions in a normalized domain for 2D solids using the PIES methodAuthor-Name: Bołtuć, Agnieszka," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 138-155.
    3. Saffarian, Marziyeh & Mohebbi, Akbar, 2022. "Finite difference/spectral element method for one and two-dimensional Riesz space fractional advection–dispersion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 348-370.
    4. Qu, Wei & Li, Zhi, 2021. "Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    5. She, Zi-Hang & Qiu, Li-Min & Qu, Wei, 2023. "An unconditionally convergent RSCSCS iteration method for Riesz space fractional diffusion equations with variable coefficients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 633-646.
    6. Zhao, Jingjun & Zhao, Wenjiao & Xu, Yang, 2021. "Lagrange nodal discontinuous Galerkin method for fractional Navier-Stokes equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    7. Eyaya Fekadie Anley & Zhoushun Zheng, 2020. "Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term," Mathematics, MDPI, vol. 8(11), pages 1-27, October.
    8. Zhao, Jingjun & Zhao, Wenjiao & Xu, Yang, 2023. "Hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    9. Shi, Z.G. & Zhao, Y.M. & Liu, F. & Wang, F.L. & Tang, Y.F., 2018. "Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 290-304.
    10. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    11. Mohammed M. Al-Shomrani & Mohamed A. Abdelkawy & António M. Lopes, 2023. "Spectral Collocation Technique for Solving Two-Dimensional Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Model," Mathematics, MDPI, vol. 11(9), pages 1-14, April.
    12. Nyamoradi, Nemat & Rodríguez-López, Rosana, 2015. "On boundary value problems for impulsive fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 874-892.
    13. Jie Zhao & Hong Li & Zhichao Fang & Yang Liu, 2019. "A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids," Mathematics, MDPI, vol. 7(7), pages 1-18, July.

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