Fractional differential equations, particularly fractional partial differential equations (FPDEs) have many applications in areas such as diffusion processes, electromagnetics, electrochemistry, material science and turbulent flow. There are lots of work for the existence and uniqueness of the solutions for fractional partial differential equations. In recent years, people start to consider the numerical methods for solving fractional partial differential equation. The numerical methods include finite difference method, finite element method and the spectral method. In this dissertation, we mainly consider the finite element method, for the time fractional partial differential equation. We consider both time discretization and space discretization. We obtain the optimal error estimates both in time and space. The numerical examples demonstrate that the numerical results are consistent with the theoretical results
