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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.

  • AMS Subject Headings

65M10, 78A48

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-11-63, author = {Fan , HuijunZhao , YanminWang , FenlingShi , Yanhua and Tang , Yifa}, 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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