| In this thesis, firstly, we review definitions of several kinds of fractional integral and derivative, Sobolev space, the basic theory of finite element and mixed finite element meth-ods. Secondly, based on bilinear finite element and R-T element, fully-discrete schemes of H1-Galerkin mixed finite elements for time-fractional diffusion equation and time-fractional Schrodinger equation are constructed, respectively. By the virtue of properties of bilinear interpolation operator and the classical L1 time stepping method, we provide some im-portant lemmas. And then, the superclose and superconvergence results for the original variable u in H1-norm and the superclose result for the flux p= ▽u in H(div, Ω)-norm are derived. Last but not least, based on a new mixed finite element method combined with L1 time stepping method, a fully-discrete scheme is established for a two-term time-fractional diffusion equation. At the same time, the spatial global superconvergence and temporal convergence order of O(h2+τ2-α) for both the original variable u in H1-norm and the flux p=▽u in L2-norm are derived by means of properties of bilinear element and interpolation postprocessing operator, respectively. |
