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Volume 41, Issue 3
Anisotropic $EQ_1^{rot}$ Finite Element Approximation for a Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation

Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi & Fawang Liu

DOI: 10.4208/jcm.2110-m2021-0180

J. Comp. Math., 41 (2023), pp. 458-481.

Published online: 2023-02

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  • Abstract

By employing $EQ_1^{rot}$ nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of $EQ_1^{rot}$ nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.

  • Keywords

Multi-term time-fractional mixed sub-diffusion and diffusion-wave equation, Nonconforming FEM, L1-CN scheme, Anisotropic meshes, Convergence and superconvergence.

  • AMS Subject Headings

65N15, 65N30, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

huijunfan@126.com (Huijun Fan)

zhaoym@lsec.cc.ac.cn (Yanmin Zhao)

mathwfl@163.com (Fenling Wang)

syhsdq@163.com (Yanhua Shi)

f.liu@qut.edu.au (Fawang Liu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-458, author = {Fan , HuijunZhao , YanminWang , FenlingShi , Yanhua and Liu , Fawang}, title = {Anisotropic $EQ_1^{rot}$ Finite Element Approximation for a Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {3}, pages = {458--481}, abstract = {

By employing $EQ_1^{rot}$ nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of $EQ_1^{rot}$ nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2110-m2021-0180}, url = {http://global-sci.org/intro/article_detail/jcm/21393.html} }
TY - JOUR T1 - Anisotropic $EQ_1^{rot}$ Finite Element Approximation for a Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation AU - Fan , Huijun AU - Zhao , Yanmin AU - Wang , Fenling AU - Shi , Yanhua AU - Liu , Fawang JO - Journal of Computational Mathematics VL - 3 SP - 458 EP - 481 PY - 2023 DA - 2023/02 SN - 41 DO - http://doi.org/10.4208/jcm.2110-m2021-0180 UR - https://global-sci.org/intro/article_detail/jcm/21393.html KW - Multi-term time-fractional mixed sub-diffusion and diffusion-wave equation, Nonconforming FEM, L1-CN scheme, Anisotropic meshes, Convergence and superconvergence. AB -

By employing $EQ_1^{rot}$ nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of $EQ_1^{rot}$ nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.

Fan , HuijunZhao , YanminWang , FenlingShi , Yanhua and Liu , Fawang. (2023). Anisotropic $EQ_1^{rot}$ Finite Element Approximation for a Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation. Journal of Computational Mathematics. 41 (3). 458-481. doi:10.4208/jcm.2110-m2021-0180
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