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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_H$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm framework, our analysis establishes that the global error admits the following decomposition: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Vert u - u_h\Vert _{L^2} \leqslant C({h + H^2} +k^{2-\alpha }).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> <mi>u</mi> <mo>-</mo> </mrow> <msub> <mi>u</mi> <mi>h</mi> </msub> <msub> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> </msub> <mo>⩽</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>h</mi> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <msup> <mi>k</mi> <mrow> <mn>2</mn> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

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

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发表日期：

2026年5月14日

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Wenyan Ma,
Xiaocui Li,
Zhiqi Liu,
Yinghao Xia

摘要

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 $u_H$ $u_{H}$ 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 $L^2$ $L^{2}$ -norm framework, our analysis establishes that the global error admits the following decomposition: $\Vert u - u_h\Vert _{L^2} \leqslant C({h + H^2} +k^{2-\alpha }).$ $‖ u - u_{h} ‖_{L^{2}} ⩽ C (h + H^{2} + k^{2 - α}) .$