A Superconvergent Nonconforming Mixed FEM for Multi-Term Time-Fractional Mixed Diffusion and Diffusion-Wave Equations with Variable Coefficients

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.