A high-order and extra-dof-free generalized finite element method for time-fractional diffusion equation

In this paper, a spatial high-order and extra-dof-free generalized finite element method (GFEM), which is based on partition of unity, is investigated for solving a class of fractional diffusion equations. An L1 scheme is used for time discretization. To resort to orthogonal projection, we first derive the error estimate of stationary problem by a local interpolation operator. Then we further study the well-posedness, stability and convergence of a discrete scheme of time-fractional diffusion problem. The offered numerical scheme demonstrates O(Δt1+δ-α+hk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(\Delta t^{1+\delta -\alpha }+h^{k})$$\end{document} convergence rates, where α∈0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \left( 0,1\right) $$\end{document}, 0≤δ≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \delta \le 1$$\end{document} and k is 3, 4, 5. Finally, ample numerical examples are given to illustrate our theoretical achievements.