Anisotropic linear triangle finite element approximation for multi-term time-fractional mixed diffusion and diffusion-wave equations with variable coefficient on 2D bounded domain

Abstract

A fully-discrete approximate scheme is established for the 2D multi-term time-fractional mixed diffusion and diffusion-wave equations with spatial variable coefficient by using linear triangle finite element method in space and classical L1 time-stepping method combined with Crank–Nicolson scheme in time. Then, the unconditionally stable analysis of the fully-discrete scheme is presented by employing some important lemmas. At the same time, both the spatial superclose property in H1-norm and convergence result in L2-norm are derived by skillfully dealing with numerical errors without any restrictions of time step τ and mesh size h. As a necessary way for obtaining the aimed numerical analysis, the relationship between the nonstandard projection operator Rh and the interpolation operator Ih of linear triangle finite element is introduced. Moreover, the global superconvergence is deduced by adopting interpolation postprocessing technique. Finally, numerical tests are provided to illustrate the efficiency and correctness of the theoretical results.