Mixed Finite Element Methods For Several Classes Of Time Fractional Partial Differential Equations

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.