Finite Element Methods For Two Kinds Of Multidimensional Time Fractional Partial Differential Equations

In this thesis,the finite element methods for time-fractional diffusion wave equations and fourth-order time-fractional diffusion equations are studied.The numerical problem of multidimensional time-fractional wave equations is studied by combining the second-order Crank-Nicolson-WSGI time discrete format and finite element method.Firstly,the Caputo time fractional wave equation is transformed into a fractional integral equation by using an integral technique.The WSGI approximation formula is used to approximate the fractional integral,and then a second-order CrankNicolson finite element format is formed.Further,detailed stability analysis and prior error estimation are given,and numerical theoretical results are verified by two-dimensional and three-dimensional numerical examples.For the time-fractional fourth-order diffusion equation,we use the WSGD approximation formula of fractional derivatives to study the second-order Crank-Nicolson finite element method,and propose the stability and error analysis.Then,we use the Richardson extrapolation method in the temporal direction to construct the extrapolation solution,and obtain the third-order temporal precision.At the end,the effectiveness of the extrapolation algorithm is verified by comparing the numerical results before and after extrapolation.