Finite Element Methods Based On High-order Approximation Formulas For Time Fractional Partial Differential Equations

In this thesis,finite element methods combined with high-order approximation for-mulas for fractional derivatives are developed to solve three classes of time fractional partial differential equations.In Chapter 2,a time two-mesh(TT-M)algorithm combined with the~1-Galerkin mixed finite element(MFE)method is introduced to numerically solve the nonlinear distributed-order sub-diffusion model,which is faster than the~1-Galerkin MFE method.The TT-M Crank-Nicolson scheme is used to discretize the temporal direction,the FBN-formula with the composite trapezoid formula is developed to approximate the distributed-order derivative,and the~1-Galerkin MFE method is used to approximate the spatial direction.The existence and uniqueness of the solution for our numerical scheme are shown.Moreover,the stability and a priori error estimate are analyzed in detail.Further,numerical example with smooth solution is provided to validate the effectiveness and com-putational efficiency of the TT-M fast algorithm,and numerical results with nonsmooth solution are given to illustrate the corrected scheme can effectively solve the problem of missing precision caused by the weak singularity.In Chapter 3,a nonlinear distributed-order fourth order sub-diffusion model is solved by a kind of fully discrete mixed element method,where the high-order FBT-approx-imation formula is used to approximate the time direction,the TT-M fast algorithm is to speed up the calculation,and the space direction is discretized by a mixed element method.The theoretical error analysis is done in detail,and numerical examples with smooth or nonsmooth solutions are provided to test the correctness of the theory results,the computational efficiency of the fast algorithm,and the effectiveness of the correction scheme in improving calculation accuracy.In Chapter 4,the high-order approximation formula for the variable-order fractional derivative is structured and its second-order approximation accuracy is proved.The fully discrete finite element scheme with the proposed high-order approximation formula is developed for solving the variable-order fractional Allen-Cahn model,and numerical experiments are done to test the feasibility of the numerical method and the effect of the high-order approximation formula.