Work

Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping

Year: 2025

Type: article

Abstract: We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (0<α<1) with mixed finite element methods (P1b–P1 and P2–P1) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term γ|u|r−2u, for r≥2. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models.

Source: Fractal and Fractional

DOI: 10.3390/fractalfract9070445

Authors: Shahid Hussain, Xinlong Feng, Arafat Hussain, Ahmed Bakhet

Institution: Xinjiang University

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,258

Funders: Natural Science Foundation of Xinjiang, Xinjiang Tianchi Doctoral Project