Numerical Analysis of a Fast Finite Element Method for a Hidden-Memory Variable-Order Time-Fractional Diffusion Equation

摘要

Abstract We investigate a fast finite element scheme to a hidden-memory variable-order time-fractional diffusion equation. Different from the traditional L1 methods, a fast approximation to the hidden-memory variable-order fractional derivative is derived to reduce the computational cost of generating coefficients from O(N2)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(N^2)$$end{document} to O(NlogN)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(Nlog N)$$end{document}, where N refers to the number of time steps. We then develop different techniques from the analysis of L1 methods to prove error estimates for the corresponding fast fully-discrete finite element scheme. Furthermore, a fast divide and conquer algorithm is proposed to reduce the complexity of solving the linear systems from O(MN2)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(MN^2)$$end{document} to O(MNlog2N)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(MNlog ^2N)$$end{document} where M stands for the spatial degree of freedom. Numerical experiments are presented to substantiate the theoretical results.