Nonconforming Mixed FEM Analysis for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation with Time-Space Coupled Derivative

Abstract

The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with spatial variable coefficient, which contains a time-space coupled derivative. The nonconforming $EQ_1^{rot}$ element and Raviart-Thomas element are employed for spatial discretization, and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization. Firstly, based on some significant lemmas, the unconditional stability analysis of the fully-discrete scheme is acquired. With the assistance of the interpolation operator $I_h$ and projection operator $R_h$, superclose and convergence results of the variable $u$ in $H^1$-norm and the flux $\vec{p}=\kappa_5(\textbf{x})\nabla u(\textbf{x},t)$ in $L^2$-norm are obtained, respectively. Furthermore, the global superconvergence results are derived by applying the interpolation postprocessing technique. Finally, the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.