The Finite Element Methods For Fractional Differential Equations

In this dissertation, the finite element schemes for three types of fractional differ-ential equations with Caputo derivative have been established. The error estimates for the fully discrete schemes have been derived in details. The numerical results are in line with the theoretical analysis.The first work is to formulate a new discontinuous Galerkin numerical method for a type of nonlinear fractional Cauchy problem. The usual approximate method is to change the fractional differential equation into an integral equation, then to solve the equivalent integral equation. Our numerical method is different. By introducing an auxiliary function, we change the fractional differential equation into a system of equations. With the help of modified discontinuous Galerkin method, a new numerical scheme is derived. The existence and uniqueness of the solution to the discrete system are proved. The error estimates are also derived. Numerical results are in line with the theoretical results.The second work is to propose a fully discrete scheme for a type of fractional advection-diffusion equation. In the temporal direction we use the modified Crank-Xicolson method. and in the spatial direction we use the finite element method. The error estimates for the fully discrete scheme are derived in details. And the numerical examples are also included which agree with the theoretical analysis.The last work is to numerical study the time-space finite element algorithm for the nonlinear spatial fractional Fokker-Planck equation. where we adopt the modified Galerkin finite element method in spatial direction and a discontinuous Galerkin finite element approach in the time direction. The priori error estimation for this numeri-cal shceme is derived. Numerical examples for the nonlinear space-fractional Fokker-Planck equation are presented which confirm the theoretical convergence rates.