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 - <p style="text-align: justify;">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&nbsp;$\mathop{p} \limits ^{\rightarrow}=µ(\rm x)∇u$, 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.</p>