Nonconforming Mixed Element Approximation For Two Kinds Of Time Fractional Partial Differential Equations

Fractional partial differential equations（FPDEs）are widely used in various subject areas,which play an important role in solving many complex problems.Firstly,the relevant definitions of fractional derivatives and fractional integrals are given in this paper,and based on the Sobolev space,the related theories of the finite element methods（FEMs）and the mixed finite element methods（MFEMs）are expounded.Secondly,for multi-term time-fractional mixed sub-diffusion and diffusion wave equation with time-space coupled derivative,an unconditional stability fully discrete approximate scheme is established.The EQ₁^rot element and zero-order Raviart-Thomas（R-T）element are applied for spatial discretization.The classical L₁ time-stepping method combined with the CrankNicolson scheme are employed for temporal discretization,and based on some important lemmas,the unconditional stability analysis of the fully discrete scheme is obtained.Furthermore,by the relevant definitions and properties of the interpolation operator I_h and the projection operator Rh,the superclose and convergence results of the variable u and the flux （?）=κ₅（x）▽u（x,t） are derived,respectively,and the global superconvergence results are obtained by using the interpolation postprocessing technique.In addition,for two-dimensional distributed-order time fractional diffusion equations with variable coefficient,the Gauss integral is applied to approximate the distributed-order operator,and the original equation is transformed into a multi-term time-fractional differential equation,the stability of the fully discrete approximate scheme is received by using the nonconforming MFEM,then the superclose of correlated variables and the global superconvergence results are acquired.