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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.

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