Volume 11, Issue 1
A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi & Yifa Tang

East Asian J. Appl. Math., 11 (2021), pp. 63-92.

Published online: 2020-11

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  • Abstract

An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical $L$1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable $u$ and the flux $\mathop{p} \limits ^{\rightarrow}=µ(\rm x)∇u$, respectively, in $H^1$- and $L^2$-norms is derived by using the relationship between the projection operator $R_h$ and the interpolation operator $I_h$. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

  • Keywords

Nonconforming mixed FEM, multi-term time-fractional mixed diffusion and diffusion-wave equations, $L$1 time-stepping method, Crank-Nicolson scheme, convergence and superconvergence.

  • AMS Subject Headings

65M10, 78A48

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-63, author = {Huijun and Fan and and 9651 and and Huijun Fan and Yanmin and Zhao and and 9653 and and Yanmin Zhao and Fenling and Wang and and 9654 and and Fenling Wang and Yanhua and Shi and and 9655 and and Yanhua Shi and Yifa and Tang and and 9656 and and Yifa Tang}, title = {A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients}, journal = {East Asian Journal on Applied Mathematics}, year = {2020}, volume = {11}, number = {1}, pages = {63--92}, abstract = {

An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical $L$1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable $u$ and the flux $\mathop{p} \limits ^{\rightarrow}=µ(\rm x)∇u$, respectively, in $H^1$- and $L^2$-norms is derived by using the relationship between the projection operator $R_h$ and the interpolation operator $I_h$. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.180420.200720}, url = {http://global-sci.org/intro/article_detail/eajam/18413.html} }
TY - JOUR T1 - A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients AU - Fan , Huijun AU - Zhao , Yanmin AU - Wang , Fenling AU - Shi , Yanhua AU - Tang , Yifa JO - East Asian Journal on Applied Mathematics VL - 1 SP - 63 EP - 92 PY - 2020 DA - 2020/11 SN - 11 DO - http://doi.org/10.4208/eajam.180420.200720 UR - https://global-sci.org/intro/article_detail/eajam/18413.html KW - Nonconforming mixed FEM, multi-term time-fractional mixed diffusion and diffusion-wave equations, $L$1 time-stepping method, Crank-Nicolson scheme, convergence and superconvergence. AB -

An unconditionally stable fully-discrete scheme on regular and anisotropic meshes for multi-term time-fractional mixed diffusion and diffusion-wave equations (TFMDDWEs) with variable coefficients is developed. The approach is based on a nonconforming mixed finite element method (FEM) in space and classical $L$1 time-stepping method combined with the Crank-Nicolson scheme in time. Then, the unconditionally stability analysis of the fully-discrete scheme is presented. The convergence for the original variable $u$ and the flux $\mathop{p} \limits ^{\rightarrow}=µ(\rm x)∇u$, respectively, in $H^1$- and $L^2$-norms is derived by using the relationship between the projection operator $R_h$ and the interpolation operator $I_h$. Interpolation postprocessing technique is used to establish superconvergence results. Finally, numerical tests are provided to demonstrate the theoretical analysis.

Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi & Yifa Tang. (2020). A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients. East Asian Journal on Applied Mathematics. 11 (1). 63-92. doi:10.4208/eajam.180420.200720
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