Novel superconvergence analysis of anisotropic triangular FEM for a multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients

A fully discrete scheme is proposed for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation with variable coefficients on anisotropic meshes, where linear triangular finite elelment method (FEM) is used for the spatial discretization and modified L1 approximation coupled with Crank-Nicolson scheme is applied to temporal direction. The mixed equation concerned contains a time-space coupled derivative which is very different from the previous literature. The stability is firstly obtained. Based on the property of the projection operator, the special relation between the projection operator and the interpolation operator of linear triangular finite element, the optimal error estimation and the superclose result are deduced. Then the global superconvergence property is derived by the interpolated postprocessing technique. Some numerical experiments are carried out to confirm the theoretical analysis on anisotropic meshes.