Nonconforming Element Approximation For Multi-term Time-fractional Mixed Diffusion And Diffusion-wave Equations

In this paper,basic theories about the Sobolev space and finite element methods are firstly introduced,then the definitions of several common kinds of fractional integrals and derivatives are presented.Secondly,based on the nonconforming mixed finite element method,the classical L1 method and the Crank-Nicolson scheme,a fully-discrete approximate scheme with anisotropic characteristic and unconditional stability is constructed for multi-term time-fractional mixed diffusion and diffusion-wave equation with variable coefficient.Furthermore,by using the relationship between the projection operator Rh and the interpolation operator Ih,combining with the properties of interpolation postprocessing operators,the convergence and superconvergence results for the original variable u and the(?)are obtained.At the same time,the Galerkin nonconforming finite element fully-discrete scheme is constructed for multi-term time-fractional mixed diffusion and diffusion-wave equation with time-space coupled derivative,and the stability of the fully-discrete scheme is proved.The convergence analysis and superconvergence result are given.Finally,several numerical examples verify the correctness of the theoretical analysis.