Weak Galerkin Finite Element Method For The Time-fractional Diffusion Equation

The study of fractional differential equation has been applied to many fields including physics,chemistry,engineering,finance,medical research and so on,which has a wide application prospect,such as prevention and control of underground water pollution,vis-coelastic material and control system etc.This thesis uses the weak Galerkin finite element method to solve the time-fractional order diffusion equation and we present the discrete format,the analysis of stability and convergence.We consider adopting the Euler scheme to discretize the time direction and applying the L1-Galerkin method to approximate the fractional integral term.The weak Galerkin finite element method is utilized for spatial discretization,then we get the fully discrete format of the equation.Finally,by analysing the numerical examples,we obtain higher convergent results O(?2-? + h?+1),where ?,h and k represent time step,space step and the highest order of basis functions,we find that the numerical results are consistent with our theoretical results.This thesis consists of five chapters,which is organized as fol-lows:Chapter 1 introduces the background and current situation of fractional differential equation and weak finite element method.Chapter 2 presents the definitions and properties of the fractional derivative,gamma function,sobolev space and the basic knowledge of the weak Galerkin finite element method.Chapter 3 demon-strates a semi-discrete scheme of finite difference method in the time direction and a fully discrete scheme for weak Galerkin finite element method.Chapter 4 studies the stability and convergence of the fully discrete scheme.Chapter 5 gives the numerical examples,through numerical examples the validity and accuracy are proved when we use the proposed method to solve the time-fractional or-der diffusion equation.