Two-Grid Finite Element Method for the Time-Fractional Navier-Stokes Equation

Abstract

In this paper, we investigate a two-grid finite element method for solving the time-fractional Navier-Stokes system. In the first step, the fully nonlinear problem is spatially discretized on a coarse grid with the mesh size H and the time step k, incorporating Caputo fractional derivative approximations using the L1-scheme temporal discretization. In the second step, the problem is discretized on a fine grid with the mesh size h (preserving the same time-fractional discretization) and linearized around the velocity field

u_H

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u_H$$\end{document}

obtained from the coarse-grid solution. The proposed two-grid finite element strategy capitalizes on the temporal nonlocality inherent to Caputo derivatives. Through rigorous error decomposition in the

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

-norm framework, our analysis establishes that the global error admits the following decomposition:

\Vert u-u_h{\Vert }_{L^2}⩽C(h+H^2+k^{2-\alpha }).

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert u - u_h\Vert _{L^2} \leqslant C({h + H^2} +k^{2-\alpha }).$$\end{document}

Keywords

Error estimate / Time-fractional Navier-Stokes equation / Two-grid finite element method / 65M60 / 60N15 / 60N30

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Funding

National Natural Science Foundation of China(12101034)

RIGHTS & PERMISSIONS

Shanghai University