Galerkin Finite Element Methods For Two Kinds Of Two-dimensional Fractional Diffusion Equations

In this thesis,we study the Galerkin mixed element method for the two-dimensional nonlinear Riemann-Liouville time-fractional fourth-order diffusion equation and the two-grid Galerkin method for the two-dimensional nonlinear space-time fractional diffusion equation.The specific research contents are as follows.In the first part,the Galerkin mixed finite element method combined with the time second-order discrete scheme is used to solve the nonlinear time-fractional diffusion equa-tion with the fourth derivative term.Firstly,we introduce an auxiliary variable ?=?u to change the fourth-order problem to a coupled system with two equations;further we propose a new second-order approximation formula for time-fractional derivatives,using a second-order discrete scheme at time t_n-??/2?;then we get a weak formulation and a fully dis-crete mixed element scheme;we detail the stability of the numerical scheme,the existence and uniqueness of the mixed element solution and derive a priori error estimates;finally,the validity of the proposed numerical scheme is numerically verified by a two-dimensional numerical example,and the correctness of the error theory results is also illustrated.In the second part,in order to improve the computational efficiency,a two-grid fi-nite element method is used to solve the two-dimensional nonlinear space-time fractional diffusion equation.Firstly,the second-order backward difference scheme is used to ap-proximate the time direction,and the time-fractional derivative is approximated by the WSGD operator.In order to reduce the computation time of the standard finite element method,a two-grid finite element algorithm is used for calculation.The specific algorith-m is to iteratively solve a nonlinear system on a coarse mesh and then to solve a linear system on a fine mesh;we prove the scheme stability of the two-grid algorithm and derive a priori error estimates with(O(?t²+?^r+1-?+H^2r+2-2?)convergence result based on fractional norm;finally,through a two-dimensional numerical calculation,the two-grid finite element algorithm can improve the computational efficiency and reduce the com-putation time.At the same time,the correctness of the theoretical results of fractional norm error estimation is also verified.