Alternating Direction Implicit Finite Element Error Analysis For The Fractional Differential Equations

The fractional partial differential equation is a class of very important differential equations, which originated from many science fields, such as, heat conduction with memory, compression of porous viscoelastic media, atomic reaction, dynamics, life sciences, pharmacology, pathology, Newton fluidics, etc. So many mathematicians and researchers in the application fields are trying to use fractional differential equations to model. Fractional derivative is pseudo-differential operator and has the character of memory (nonlocal). Although it can describe practical problems beautifully, it can also bring con-siderable difficulties in numerical computation, especially for high-dimensional problems.Alternating direction implicit (ADI) finite element method is an effec-tive numerical methods to calculate high-dimensional partial differential equa-tions, which reduce a multidimensional problem to sets of independent one-dimensional problems. So ADI finite element method has less storage, low computation and the feature of high accuracy. The main objective of this thesis is to use ADI method to investigate the numerical solutions of the two-dimensional time fractional partial differential equations. Stable and efficient numerical schemes are proposed for these equations and the error estimates of our proposed numerical schemes are also established. The main work of this thesis contains the following three parts:In the first part, we use ADI finite difference method to solve the nu-merical solutions of the two dimensional fractional evolution equation. We first use the second-order difference quotient for the spatial discretization and the backward Euler for the time stepping combined with order one convolu-tion quadrature approximating the integral term and obtain the full discrete scheme. Then, applying the method of discrete energy method, we prove that the full discrete scheme is unconditionally stable and convergent. Finally, some numerical results and numerical simulations are presented to confirm the rates of convergence and the robustness of the numerical schemes. This method can effectively reduce the storage capacity which is caused by overall time-dependent and can calculate the long time solution.In the second part, ADI Galerkin schemes are formulated and analyzed for the two-dimensional time fractional evolution equation. We first use the Galerkin finite element for the spatial discretization and the backward Eu-ler, Grank-Nicolson for the time stepping combined with order one and two convolution quadrature and obtain the full discrete scheme. Then we rigorously prove the stability and error estimates of the full discrete scheme. Numerical examples is examined to the accuracy of the theoretical analysis. The method achieve arbitrary order accuracy in the spatial direction and effectively reduce the storage volume requirements caused by the overall time dependence.In the third part, we apply ADI finite element method to investigate the numerical solutions of the two-dimensional fractional diffusion-wave equation. We use Galerkin finite element method in space and Crank-Nicolson method in time and obtain the full discrete scheme. We prove the stability and error estimates of the scheme. Numerical examples verify the correctness of the conclusions.