Some Types Of Finite Element Methods For Fractional Partial Differential Equations

Since the fractional derivatives can describe the materials with memory, inheritance characters as well as physical process more precisely, fractional differential equations have been applied extensively and researched deeply in many areas. Unfortunately, it is usually difficult to solve out the analytic solutions of fractional differential equations. Even if the analytic solutions can be obtained, most of them often contain infinite series or special functions which are hard to be computed. Therefore the numerical methods for fractional differential equations have become more and more popular. Among the developed nu-merical methods, the finite element method has attracted much more attention because it is simple and universal, strongly adapted to the area at the same time it can flexibly generate meshes with low requirement for the solutions’smoothness. In this thesis, some types of finite element methods for fractional partial differential equations are researched. We mainly study the reduced-order finite element method for time fractional differential equations and the space-time finite element method for space fractional differential equa-tion and system of fractional differential equations. The main contributions of our work can be divided into three parts as follows.First, we study the classic finite element method for the time fractional Cable equa-tion in chapter 3. According to the idea of traditional finite element method, we apply finite difference discrete in time and finite element approximation in space, then obtain the fully discrete scheme of Crank-Nicolson type. Owing to the presented coefficients which are different from that of L1 algorithm in the discrete to time fractional derivative, the fully discrete scheme has good succession with respect to the number of time level n. The stability and optimal error estimate in L2 norm are also provided in detail. Moreover, two numerical examples of one-dimensional space and two-dimensional space are illustrated to confirm our theoretical analysis.Second, the time fractional diffusion equation, Tricomi type equation and Sobolev equation are analyzed by reduced-order finite element method which is based on proper orthogonal decomposition (POD) theory. These equations are all two-dimensional in space. The adopted reduced-order finite element method marches as follows:the snap-shots are obtained from the solutions in very short time interval which are solved by the general finite element formulation firstly. Then the optimal POD bases, the number of which is much less than that of classical finite element scheme, are computed from the snapshots in the least squares sense. After defining the space spanned by the POD bases as reduced-order finite element space, the approximate solutions are solved by the reduced-order finite element scheme. When finite element method is used to solve the time fractional differential equation, all the solutions on time levels which satisfy t< tn need to be saved and added up in order to get the solution on time tn. Since the degree of freedom on every time level is decreased, the total size of memory and calculation amount are reduced largely so that the computational burden is alleviated which is caused by the non-locality of fractional differential operators. We have given the fully discrete finite element scheme and reduced-order finite element scheme then proved their stability and convergence, respectively. The optimal error estimate in L2 norm of the two schemes are obtained. Numerical examples indicate that the reduced-order finite element method based on POD theory can largely reduce the size of memory to improve computing effi-ciency. Moreover, the reduced-order method can guarantee that the accuracy of numerical solutions is not lower than the one solved by classical finite element method. This part is arranged in chapter 4 to chapter 6. The algorithm process for the reduced-order finite element method is given in chapter 5 particularly.The third part includes the space-time finite element method for fractional differential equation and system of fractional differential equations. In chapter 7, the semilinear space fractional diffusion equation is considered by discontinuous space-time finite element method, which is discontinuous in time and continuous in space. By introducing suitable test functions in the approximate variational problem, we get the space-time finite element formulation which can be time stepping. For theoretical analysis, we make use of Radau integration formula to express the approximate solution as the linear combination of Lagrange interpolation polynomials which are based on the Radau points so that the existence and uniqueness for the weak solution are proved without any assumptions to the restriction of steps in space and time. After defining the elliptic projection operator, we present the estimation in L2 norm for it through Aubin-Nitsche trick. And then the optimal order error estimate in L∞(L2) norm for space-time finite element solution is obtained in detail. In chapter 8, the discontinuous space-time finite element method is applied to nonlinear fractional reaction-diffusion equations. The fully discrete scheme is obtained as well as its well-posedness and error estimate. Therefore, the discontinuous space-time finite element method is generalized to the system of fractional equations. Both the theoretical analysis and numerical simulation results show that the discontinuous space-time finite element method can reach high order accuracy both in time and in space. It holds true for fractional equations and this work lays the foundation for studying more complicated fractional equations.