Finite Element Methods For The Space Fractional Gray-Scott Model And Time Fractional Maxwell's System

In this thesis,finite element methods for the space fractional Gray-Scott model and time fractional Maxwell's system are developed.In Chapter 2,a time two-mesh?TT-M?finite element?FE?method for the two-dimensional space fractional Gray-Scott model is studied and discussed to get the numerical solutions effectively.The method mainly includes three steps:firstly,use an iterative method for solving the coupled nonlinear system on the time coarse grid;secondly,by an interpolation formula,ones can get any coarse values on the time fine mesh;finally,based on the computed coarser solutions,a linear FE system on time fine mesh can be constructed by using the two-variables Taylor's formula.Here,some theoretical results,which include stability and a priori error for the fully discrete scheme,are analyzed and proved.Furthermore,the computing data are given to verify the correctness of the theoretical results and the effectiveness of the current algorithm,and the numerical images are provided to observe the phenomenon of anomalous diffusion.In Chapter 3,mixed element algorithms with second-order time convergence results for the time fractional Maxwell's equations in the Cole-Cole dispersive medium are proposed.Fully discrete mixed element systems with parameters ? at time t=t_n-?,which are constructed by combining the generalized BDF2-? algorithms in temporal direction and mixed element methods in space direction,are formulated.For the two-dimensional case of the fractional Maxwell's system,the algorithm implementation process based on the rectangular subdivision is shown in detail.Finally,two numerical examples are provided to check the validity of our method and the influence of parameters ? and ?.