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

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 ($0<\alpha <1$) with mixed finite element methods (P1b–P1 and $P_2$–$P_1$) 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 ${\gamma \left|\mathbf{u}\right|}^{r-2}\mathbf{u}$, for $r\ge 2$. 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.

1. Introduction

The study of time-fractional Navier–Stokes equations (TFNSEs) has emerged as a pivotal area in modern applied mathematics, offering a powerful framework for modeling complex fluid dynamics with memory and hereditary properties [1,2,3,4,5,6]. Unlike their integer-order counterparts, these equations employ Caputo fractional derivatives to capture non-local temporal dependencies, making them particularly suitable for describing anomalous diffusion in viscoelastic fluids, turbulent flows, and fractal porous media [4,7,8,9]. The fractional derivative operator ${\mathcal{D}}_t^{\alpha }$ (where $0<\alpha <1$) introduces a convolution kernel that captures the entire history of the system. This memory effect enables more accurate modeling of frequency-dependent damping and power-law relaxation phenomena. Such characteristics are frequently observed in various real-world applications [8,10].

From a physical perspective, the nonlinear damping term $|\overline{u}{|}^{r-2}\overline{u}$ (with $r\ge 2$) plays several pivotal roles in fluid dynamics modeling [11]. Most notably, it strengthens the system’s energy dissipation mechanisms, which are essential for accurately capturing the rapid energy decay characteristic of turbulent flows and porous media systems [12,13]. In addition to enhancing dissipation, this term contributes critical stabilization by preventing finite-time blow-up in high-Reynolds-number regimes, where classical Navier–Stokes formulations often fail [14].

The damping term’s physical significance extends to daily-life applications, including biological flows like blood circulation, where it models shear-dependent vascular resistance, and geophysical flows such as ocean dynamics, where it accounts for small-scale turbulence dissipation [9]. These practical applications are supported by systematic mathematical foundations. Theoretical studies have demonstrated that the presence of $|\overline{u}{|}^{r-2}\overline{u}$ ensures three key properties: first, the global existence of weak solutions for $r>2$ without dimensional restrictions; second, enhanced solution regularity when $r\ge {\displaystyle \frac{2d}{d+2}}$ in d-dimensional domains; and third, suppression of energy accumulation at high frequencies, effectively serving as a natural turbulence regularization mechanism [14,15,16].

The numerical solutions of TFNSEs present significant challenges due to their inherent mathematical complexity [15,17,18]. Key difficulties arise from the non-local nature of Caputo derivatives, which requires the entire solution history to be processed and stored, thus leading to increased computational costs [19]. Moreover, the nonlinear interaction between the fractional temporal operator, the convective term $(\overline{u}·\nabla )\overline{u}$, and the damping effects complicates stability analysis and convergence proofs [15]. Although its mathematical structure and physical implications with hybrid methods have been discussed in prior studies [11,13,14,20], the development of numerical methods that specifically address an L1-type scheme with a nonlinear damping term has still not been covered in the existing literature within the finite element method.

To find the solution of these complex models (TFNSEs) with nonlinear damping, we propose a hybrid-type fully discrete scheme using mixed finite element spaces $(P1b$–$P1)$ and $P_2$–$P_1$ combined with an L1-type scheme. This approach ensures both stability and accuracy in approximating velocity and pressure variables. We establish optimal error estimates in the $H^1$-norm for velocity and $L^2$-norm for pressure, carefully accounting for the interaction between fractional memory and nonlinear damping.

The unconditional stability of both semi-discrete and fully discrete schemes is theoretically demonstrated and corroborated through comprehensive numerical experiments. We implement four distinct algorithms and validate them across three representative test cases. The first test assesses the accuracy of the mixed finite element pairs for various fractional orders $\alpha$ and damping exponents r. The second verifies convergence behavior using exact solutions under similar parameter variations, while the third focuses on the nonlinear damping effects utilizing Algorithms 1–4. Furthermore, the scheme is applied to two classical benchmark flows, the backward-facing step [21], showcasing flow separation and reattachment, and the lid-driven cavity [22], characterized by complex vortex structures, to demonstrate its capability in capturing both internal and external flow dynamics. This work makes three key contributions to the scientific community, detailed as follows.

First, we develop a novel hybrid numerical scheme combining finite difference methods (for L1 time discretization of Caputo derivatives with $0<\alpha <1$) with mixed finite elements (Taylor–Hood ${\mathbb{P}}_2$-${\mathbb{P}}_1$ and mini ${\mathbb{P}}_1^b$-${\mathbb{P}}_1$ elements for velocity–pressure approximation). Second, we establish rigorous stability analysis and prove optimal convergence rates of $\mathcal{O}\left(h^{2-\alpha }\right)$ for velocity and $\mathcal{O}\left(h^{\alpha }\right)$ for pressure, which are verified numerically. Third, this represents the first complete finite element framework capable of handling the full complexity of time-fractional Navier–Stokes equations with nonlinear damping ${\left|\mathbf{u}\right|}^{r-2}\mathbf{u}$, successfully addressing the coupled challenges of (a) fractional temporal operators, (b) nonlinear convection, and (c) power-law damping. The method’s effectiveness is demonstrated through comprehensive benchmark tests of turbulent flows with memory effects (back channel flow and cavity flow).

Extensive numerical results, validated against exact solutions, indicate that our method achieves comparable or superior accuracy relative to existing approaches [16,23], while significantly improving computational efficiency, especially across a range of fractional parameters $\alpha$ and damping exponents r. These findings underscore the robustness and effectiveness of the proposed scheme for simulating complex fluid behaviors governed by fractional dynamics with nonlinear damping.

This work is organised as follows. Section 2 establishes the time-fractional Navier–Stokes equations with nonlinear damping, where the fractional derivative is considered in the Caputo sense for $\alpha \in (1,2)$. Section 3 develops the numerical framework, combining mixed finite elements for spatial discretization with an L1 scheme for the temporal fractional derivative. Section 4 provides the finite difference method, stability analysis, and derives optimal order error estimates for semi–discrete and fully discrete formulations. Numerical experiments in Section 5 illustrate the scheme’s performance on well–known algorithms and verify theoretical convergence rates with different given true solutions by Mini elements $(P1b$–$P1)$ and Taylor–Hood $P_2$–$P_1$ elements. The work concludes in Section 6 with a conclusion paragraph and some future work related to complex flow models.

Algorithm 1 Linearization using previous iterations for both nonlinear terms.

Set initial guesses ${\overline{u}}_h^0={\mathcal{P}}_h{\overline{u}}_0$, $p_h^0=0$. For $n=1,2,\dots ,N$, find $({\overline{u}}_h^{n,i},p_h^{n,i})\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ such that for all $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, $$\begin{array}{cc}\hfill ({\overline{u}}_h^{n,i},v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }[& a({\overline{u}}_h^{n-j,i},v_h)+d({\overline{u}}_h^{n-j,i-1},u_h^{n-j,i-1},v_h)\hfill \\ \hfill & +g({\overline{u}}_h^{n-j,i-1},u_h^{n-j,i-1},v_h)-b(v_h,p_h^{n-j,i})+b({\overline{u}}_h^{n-j,i},q_h)]\hfill \\ \hfill & =({\overline{u}}_h^0,v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},v_h).\hfill \end{array}$$ If $\parallel |{\overline{u}}_h^{n,i}-{\overline{u}}_h^{n,i-1}\parallel |<ϵ$, accept the solution. Otherwise, set $i:=i+1$ and repeat Step II.

Algorithm 2 Update damping term using current iteration

Same as in Algorithm 1. For $n=1,2,\dots ,N$, find $({\overline{u}}_h^{n,i},p_h^{n,i})\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ such that for all $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, $$\begin{array}{cc}\hfill ({\overline{u}}_h^{n,i},v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }[& a({\overline{u}}_h^{n-j,i},v_h)+d({\overline{u}}_h^{n-j,i-1},u_h^{n-j,i-1},v_h)\hfill \\ \hfill & +g({\overline{u}}_h^{n-j,i},u_h^{n-j,i},v_h)-b(v_h,p_h^{n-j,i})+b({\overline{u}}_h^{n-j,i},q_h)]\hfill \\ \hfill & =({\overline{u}}_h^0,v_h)+{\beta }_0\sum _{k=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},v_h).\hfill \end{array}$$ Check convergence as before.

Algorithm 3 Update convective term using current iteration

Same as in Algorithm 1. For $n=1,2,\dots ,N$, find $({\overline{u}}_h^{n,i},p_h^{n,i})\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ such that for all $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, $$\begin{array}{cc}\hfill ({\overline{u}}_h^{n,i},v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }[& a({\overline{u}}_h^{n-j,i},v_h)+d({\overline{u}}_h^{n-j,i},u_h^{n-j,i},v_h)\hfill \\ \hfill & +g({\overline{u}}_h^{n-j,i-1},u_h^{n-j,i-1},v_h)-b(v_h,p_h^{n-j,i})+b({\overline{u}}_h^{n-j,i},q_h)]\hfill \\ \hfill & =({\overline{u}}_h^0,v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},v_h).\hfill \end{array}$$ Check convergence as before.

Algorithm 4 Fully updated nonlinear iteration

Same as in Algorithm 1. For $n=1,2,\dots ,N$, find $({\overline{u}}_h^{n,i},p_h^{n,i})\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ such that for all $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, $$\begin{array}{cc}\hfill ({\overline{u}}_h^{n,i},v_h)+{\beta }_0\sum _{k=0}^{n-1}{w}_j^{\alpha }[& a({\overline{u}}_h^{n-j,i},v_h)+d({\overline{u}}_h^{n-j,i},u_h^{n-j,i},v_h)\hfill \\ \hfill & +g({\overline{u}}_h^{n-j,i},u_h^{n-j,i},v_h)-b(v_h,p_h^{n-j,i})+b({\overline{u}}_h^{n-j,i},q_h)]\hfill \\ \hfill & =({\overline{u}}_h^0,v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},v_h).\hfill \end{array}$$ Accept the solution if the stopping criterion is satisfied.

2. Mathematical Model

We consider the following time-fractional incompressible Navier–Stokes system (TFNS) [15,17] with nonlinear damping in a bounded domain $\widehat{{\mathrm{\Omega }}_B}\subset {\mathbb{R}}^d,d\in \{2,3\}$ with boundary $\partial \widehat{{\mathrm{\Omega }}_B}$. For a given final time $T>0$, the fluid dynamics are governed by

$$\left\{\begin{array}{cc}{\mathcal{D}}_t^{\alpha }\mathbf{u}(\mathbf{x},t)-\nu \Delta \mathbf{u}(\mathbf{x},t)+(\mathbf{u}(\mathbf{x},t)·\nabla )\mathbf{u}(\mathbf{x},t)\hfill & \\ +{\gamma \left|\mathbf{u}(\mathbf{x},t)\right|}^{r-2}\mathbf{u}(\mathbf{x},t)+\nabla p(\mathbf{x},t)=\mathbf{f}(\mathbf{x},t),\hfill & \mathrm{in}\widehat{{\mathrm{\Omega }}_B}\times (0,T],\hfill \\ \nabla ·\mathbf{u}(\mathbf{x},t)=0,\hfill & \mathrm{in}\widehat{{\mathrm{\Omega }}_B}\times (0,T],\hfill \\ \mathbf{u}(\mathbf{x},t)=\mathbf{0},\hfill & \mathrm{on}\partial \widehat{{\mathrm{\Omega }}_B}\times (0,T],\hfill \\ \mathbf{u}(\mathbf{x},0)={\mathbf{u}}_0\left(\mathbf{x}\right),\hfill & \mathrm{in}\widehat{{\mathrm{\Omega }}_B},\hfill \end{array}\right.$$

(1)

where

• ${\mathcal{D}}_t^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha \in (0,1)$;
• $\mathbf{u}(\mathbf{x},t)={(u_1(\mathbf{x},t),u_2(\mathbf{x},t))}^{\top }$ is the velocity of fluid;
• $p(\mathbf{x},t)$ is the pressure field;
• $\mathbf{f}(\mathbf{x},t)={(f_1(\mathbf{x},t),f_2(\mathbf{x},t))}^{\top }$ is the external force density;
• $\nu >0$ is the kinematic viscosity;
• ${\gamma \left|\mathbf{u}\right|}^{r-2}\mathbf{u}$ is the damping term ($\gamma >0$, $r\ge 2$);
• ${\mathbf{u}}_0\left(\mathbf{x}\right)$ is the divergence-free initial velocity ($\nabla ·{\mathbf{u}}_0=0$);
• $\mathbf{x}=(x_1,x_2)\in {\mathrm{\Omega }}_B$ denotes spatial coordinates.

For clarity and consistency in notation, we define $\mathbf{u}(\mathbf{x},t)=\overline{u}$, $p(\mathbf{x},t)=p$, and $\mathbf{f}(\mathbf{x},t)=\overline{f}$ throughout this work. When $\gamma =0$, the system (1) again appears to the original Navier–Stokes equations (NSEs). The term $\gamma \left(\right|\overline{u}{|}^{r-2}\overline{u})$ represents the nonlinear damping, where $\gamma >0$ is the damping coefficient and $r\ge 2$ is the damping exponent [13,24].

Remark 1.

The damping term serves two key functions: first, it acts as an energy dissipation agent, critical for turbulent flows where rapid energy decay occurs, and second, it ensures stabilization in the model equations by preventing finite-time blow-up in high-Reynolds-number regimes, where classical models fail. This dual role enables us to establish various regularity criteria in weak and logarithmically improved spaces, similar to those for the standard Navier–Stokes equations. Mathematically, the damping term is highly beneficial, as it enhances solution regularity compared to the standard Navier–Stokes equations without damping. As a result, we can prove the existence of global attractors for model (1) within a certain value for r, a problem that remains unresolved for standard 2D/3D time-fractional Navier–Stokes equations [4].

3. Notations, Preliminaries and Mixed Weak Formulation

In this section, we introduce notations and preliminary concepts related to functional spaces. We begin with the following standard Hilbert spaces [22,25,26]:

$$\begin{array}{cc}\hfill {\mathbb{X}}^u& ={\mathbb{H}}_0^1{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2,\hfill \\ \hfill \mathbb{Y}& ={\mathbb{L}}^2{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2,\hfill \\ \hfill {\mathbb{M}}^p& ={\mathbb{L}}_0^2\left(\widehat{{\mathrm{\Omega }}_B}\right)=\left\{v\in {\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)|{\int }_{\widehat{{\mathrm{\Omega }}_B}}vdx=0\right\},\hfill \end{array}$$

where ${\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)$ is equipped with the classical inner product $(·,·)$ and corresponding norm $\parallel |·|\parallel$. The space ${\mathbb{X}}^u$ is endowed with the inner product and norm:

$$(\overline{u},v)=(\nabla \overline{u},\nabla v),\parallel \overline{u}{\parallel }_{{\mathbb{X}}^u}=\parallel \nabla \overline{u}\parallel |=\parallel |\overline{u}{\parallel |}_1.$$

Let $\mathbb{V}\subset {\mathbb{X}}^u$ and $\mathbb{H}\subset \mathbb{Y}$ represent the divergence-free subspaces defined by

$$\begin{array}{cc}\hfill \mathbb{V}& =\left\{v\in {\mathbb{X}}^u|\nabla ·v=0\right\},\hfill \\ \hfill \mathbb{H}& =\left\{{v\in \mathbb{Y}|\nabla ·v=0,v·n|}_{\partial \widehat{{\mathrm{\Omega }}_B}}=0\right\}.\hfill \end{array}$$

We define the Stokes operator $\mathcal{S}$ by

$$\mathcal{S}=-\mathcal{P}\Delta ,$$

where $\Delta$ denotes the standard Laplacian applied component-wise, and $\mathcal{P}$ is the ${\mathbb{L}}^2$-orthogonal Helmholtz–Leray projection from the space $\mathbb{Y}={\mathbb{L}}^2{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2$ onto the subspace $\mathbb{H}\subset \mathbb{Y}$ of divergence-free vector fields with a vanishing normal trace on the boundary $\partial \widehat{{\mathrm{\Omega }}_B}$. This projection ensures compatibility with the incompressibility condition $\nabla ·\overline{u}=0$ and arises naturally in the variational formulation of the Stokes and Navier–Stokes equations [27,28].

The domain of $\mathcal{S}$ is defined as

$$\mathcal{D}\left(\mathcal{S}\right)={\mathbb{H}}^2{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2\cap \mathbb{V},$$

where $\mathbb{V}:=\{v\in {\mathbb{H}}_0^1{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2:\nabla ·v=0\}$. This guarantees that the Laplacian $\Delta$ is well defined in a weak sense and respects both the divergence-free condition and the Dirichlet boundary condition. The space $H^2{\left(\widehat{{\mathrm{\Omega }}_B}\right)}^2$ ensures sufficient regularity, while the intersection with $\mathbb{V}$ enforces incompressibility.

To define intermediate regularity spaces, we introduce the fractional powers of the Stokes operator. For a dummy variable $A\ge 0$, we define

$${\mathbb{H}}^A:=D\left({\mathcal{S}}^{A/2}\right),$$

with the norm ${\parallel |v\parallel |}_A:=\parallel |{\mathcal{S}}^{A/2}v\parallel |$, which is equivalent to standard Sobolev norms on divergence-free spaces [29]. This leads to a scale of Hilbert spaces:

$$\begin{array}{cc}\hfill {\mathbb{H}}^0& =D\left({\mathcal{S}}^0\right)=\mathbb{H},(\mathrm{divergence-free}\mathrm{fields}\mathrm{in}{\mathbb{L}}^2),\hfill \\ \hfill {\mathbb{H}}^1& =D\left({\mathcal{S}}^{1/2}\right)=\mathbb{V},(\mathrm{divergence-free}\mathrm{fields}\mathrm{in}{\mathbb{H}}_0^1),\hfill \\ \hfill {\mathbb{H}}^2& =D\left(\mathcal{S}\right),(\mathrm{divergence-free}\mathrm{fields}\mathrm{in}{\mathbb{H}}^2\cap \mathbb{V}).\hfill \end{array}$$

This framework is essential in analyzing the regularity, stability, and long-term behavior of solutions to the Navier–Stokes equations. The spectral theory of the Stokes operator and the associated fractional power spaces are crucial for establishing energy estimates and proving the existence and uniqueness of both weak and strong solutions [28,29].

Following [30], we define the Riemann–Liouville fractional integral operator of order $\beta \ge 0$, which generalizes the classical integral to fractional orders, as 

$$I_t^{\beta }\tilde{g}\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}{\int }_0^t{(t-A)}^{\beta -1}\tilde{g}\left(A\right)dA,t>0,$$

(2)

where $\mathrm{\Gamma }(·)$ is the gamma function defined by

$$\mathrm{\Gamma }\left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt.$$

A is a dummy variable for integration, and we adopt the convention $I_t^0\tilde{g}\left(t\right)=\tilde{g}\left(t\right)$ when the order is zero, recovering the original function.

The Caputo-type derivative ${\mathcal{D}}_t^{\alpha }$, which appears in Equation (1), is a widely used fractional derivative formulation, particularly suitable for initial value problems, as it naturally incorporates initial conditions. For order $\alpha \in (0,1]$, it is defined by

$${\mathcal{D}}_t^{\alpha }\tilde{g}\left(t\right)={\displaystyle \frac{d}{dt}}{I}_t^{1-\alpha }[\tilde{g}\left(t\right)-\tilde{g}\left(0\right)]={\displaystyle \frac{d}{dt}}{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{(t-A)}^{-\alpha }[\tilde{g}\left(A\right)-\tilde{g}\left(0\right)]dA.$$

(3)

where the derivative is applied after subtracting the initial value, ensuring consistency with the classical derivative when $\alpha \to 1$.

Moreover, as described in [31], the fractional integral operator ${\mathcal{D}}_t^{-\alpha }$, which acts as the inverse of the Caputo derivative and generalizes repeated integration to fractional orders, is given by

$${\mathcal{D}}_t^{-\alpha }\tilde{g}\left(t\right)=I_t^{\alpha }\tilde{g}\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-A)}^{\alpha -1}\tilde{g}\left(A\right)dA,t>0.$$

(4)

Next, we define the following continuous bilinear and trilinear forms that will be used throughout the analysis. Specifically, we introduce the bilinear forms $a(·,·)$ on ${\mathbb{X}}^u\times {\mathbb{X}}^u$, $b(·,·)$ on ${\mathbb{X}}^u\times {\mathbb{M}}^p$, and the trilinear forms $d(·,·,·)$ and $g(·,·,·)$ on ${\mathbb{X}}^u\times {\mathbb{X}}^u\times {\mathbb{X}}^u$, defined, respectively, as

$$a(\overline{u},v)=\nu (\nabla \overline{u},\nabla v),\forall \overline{u},v\in {\mathbb{X}}^u,$$

$$b(v,q)=(q,\nabla ·v),\forall v\in {\mathbb{X}}^u,q\in {\mathbb{M}}^p,$$

$$d(\overline{u},v,w)=\left((\overline{u}·\nabla )v+{\displaystyle \frac{1}{2}}(\nabla ·\overline{u})v,w\right)={\displaystyle \frac{1}{2}}\left((\overline{u}·\nabla )v,w\right)-{\displaystyle \frac{1}{2}}\left((\overline{u}·\nabla )w,v\right),\forall \overline{u},v,w\in {\mathbb{X}}^u,$$

and

$$g(\overline{u},v,w)=\gamma \left(|\overline{u}{|}^{r-2}v,w\right),\forall \overline{u},v,w\in {\mathbb{X}}^u.$$

It is well known (see [32,33]) that the trilinear form $d(\overline{u},v,w)$ satisfies the following properties:

$$\begin{array}{ccc}\hfill d(\overline{u},v,w)& =& -d(\overline{u},w,v),\hfill \\ \hfill d(\overline{u},v,v)& =& 0,\forall \overline{u},v,w\in {\mathbb{X}}^u,\hfill \end{array}$$

and the boundedness estimate

$$|d(\overline{u},v,w)|\le {\mu }_0\parallel |\overline{u}{\parallel |}_1{\parallel |v\parallel |}_1{\parallel |w\parallel |}_1,\forall \overline{u},v,w\in {\mathbb{X}}^u,$$

where $\parallel |·{\parallel |}_1$ denotes the ${\mathbb{H}}^1$-norm, and ${\mu }_0$ is a positive constant.

Theorem 1

([29,34]). There exists at least a solution pair $(\overline{u},p)\in {\mathbb{X}}^u\times {\mathbb{M}}^p$ to satisfy

$$\parallel |\overline{u}{\parallel |}_1\le {\displaystyle \frac{\parallel |\overline{f}{\parallel |}_{-1}}{{\parallel |v\parallel |}_1}},$$

(5)

where

$$\parallel |\overline{f}{\parallel |}_{-1}=su{p}_{v\in X}{\displaystyle \frac{(\overline{f},v)}{{\parallel |v\parallel |}_1}}.$$

(6)

3.1. Variational Formulation

Based on the above notations, we formulate the weak formulation of the model (1) as follows. Find $\overline{u}\in {\left[{\mathbb{H}}^1\left(\widehat{{\mathrm{\Omega }}_B}\right)\right]}^2$ and $p\in {\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)$ such that for all test functions $v\in {\left[{\mathbb{H}}_0^1\left(\widehat{{\mathrm{\Omega }}_B}\right)\right]}^2$ and $q\in {\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)$, the following equations hold:

$$\begin{array}{c}\hfill {\int }_{\widehat{{\mathrm{\Omega }}_B}}{\mathcal{D}}_t^{\alpha }\overline{u}·vdx+{\int }_{\widehat{{\mathrm{\Omega }}_B}}\nu \nabla \overline{u}:\nabla vdx+{\int }_{\widehat{{\mathrm{\Omega }}_B}}(\overline{u}·\nabla )\overline{u}·vdx+{\int }_{\widehat{{\mathrm{\Omega }}_B}}\gamma {\left|\overline{u}\right|}^{r-2}\overline{u}·vdx\\ \hfill -{\int }_{\widehat{{\mathrm{\Omega }}_B}}p\nabla ·vdx{\int }_{\widehat{{\mathrm{\Omega }}_B}}\nabla ·\overline{u}·qdx={\int }_{\widehat{{\mathrm{\Omega }}_B}}\overline{f}·vdx.\end{array}$$

(7)

For the short and concise representation, Equation (7) can be written as

$$\left\{\begin{array}{c}{\mathcal{D}}_t^{\alpha }(\overline{u},v)+a(\overline{u},v)+d(\overline{u},\overline{u},v)-b(v,p)+b(\overline{u},q)+g(\overline{u},v,w)=(\overline{f},v),\hfill \\ u\left(0\right)=u_0.\hfill \end{array}\right.$$

(8)

In [15], Zhou and Peng established the existence and uniqueness of weak solutions for problem (8) in the case without the time-fractional term. However, the uniqueness of weak solutions and the global (in time) existence of strong solutions remain completely open questions.

3.2. Uniqueness

We will prove that the strong solution is unique in the larger class of weak solutions for any $r\ge 1$. More precisely, we provide a theorem.

Theorem 2

([24]). Assume ${\overline{u}}_1\in {\mathbb{L}}^2(0,T;{\mathbb{L}}^2\left({\mathbb{R}}^2\right))\cap {\mathbb{L}}^2(0,T;{\mathbb{H}}^1\left({\mathbb{R}}^2\right))$ and ${\overline{u}}_2\in {\mathbb{L}}^{\infty }(0,T;{\mathbb{H}}^1\left({\mathbb{R}}^2\right))$$\cap {\mathbb{L}}^2(0,T;{\mathbb{H}}^2\left({\mathbb{R}}^2\right))$ are two solutions to (1) with the same initial condition ${\overline{u}}_0$. Then ${\overline{u}}_1={\overline{u}}_2$ on $[0,T]$.

Proof. 

Let w denote the difference between the two solutions, i.e., $w={\overline{u}}_1-{\overline{u}}_2$. By linearity, w satisfies the following equation obtained by subtracting the weak formulations for ${\overline{u}}_1$ and ${\overline{u}}_2$:

$${\partial }_{t}w+{\overline{u}}_1·\nabla w+w·\nabla {\overline{u}}_2+\nabla (p_1-p_2)+\gamma \left(|{\overline{u}}_1{|}^{r-2}{\overline{u}}_1-{\left|{\overline{u}}_2\right|}^{r-2}{\overline{u}}_2\right)=\nu \Delta w,$$

(9)

with $\mathrm{div}w=0$ and $w\left(0\right)=0$.

Taking the ${\mathbb{L}}^2\left({\mathbb{R}}^2\right)$ inner product of (9) with w and integrating by parts yields the following:

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\parallel |w\parallel |}_{{\mathbb{L}}^2}^2+{\nu \parallel |\nabla w\parallel |}_{{\mathbb{L}}^2}^2+\gamma {\int }_{{\mathbb{R}}^2}\left(|{\overline{u}}_1{|}^{r-2}{\overline{u}}_1-{\left|{\overline{u}}_2\right|}^{r-2}{\overline{u}}_2\right)·wdx\le \left|{\int }_{{\mathbb{R}}^2}(w·\nabla {\overline{u}}_2)·wdx\right|.$$

(10)

The damping term is non-negative due to the monotonicity of $\overline{u}\mapsto {\left|\overline{u}\right|}^{r-2}\overline{u}$:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^2}\left(|{\overline{u}}_1{|}^{r-2}{\overline{u}}_1-{\left|{\overline{u}}_2\right|}^{r-2}{\overline{u}}_2\right)·wdx& ={\int }_{{\mathbb{R}}^2}\left(|{\overline{u}}_1{|}^{r-2}{\overline{u}}_1-{\left|{\overline{u}}_2\right|}^{r-2}{\overline{u}}_2\right)·({\overline{u}}_1-{\overline{u}}_2)dx\hfill \\ \hfill & \ge {\int }_{{\mathbb{R}}^2}\left(|{\overline{u}}_1{|}^r+|{\overline{u}}_2{|}^r-|{\overline{u}}_1{|}^{r-1}|{\overline{u}}_2|-|{\overline{u}}_2{|}^{r-1}\left|{\overline{u}}_1\right|\right)dx\ge 0.\hfill \end{array}$$

From (10), we deduce

$$\begin{array}{cc}\hfill \frac{1}{2}{\displaystyle \frac{d}{dt}}{\parallel |w\parallel |}_{L^2}^2+{\nu \parallel |\nabla w\parallel |}_{L^2}^2& \le \left|{\int }_{{\mathbb{R}}^2}(w·\nabla {\overline{u}}_2)·wdx\right|\hfill \\ \hfill & \le {\parallel |w\parallel |}_{L^4}^2\parallel |\nabla {\overline{u}}_2{\parallel |}_{L^2}\hfill \\ \hfill & \le {C\parallel |w\parallel |}_{L^2}{\parallel |\nabla w\parallel |}_{L^2}\parallel |\nabla {\overline{u}}_2{\parallel |}_{L^2}\hfill \\ \hfill & \le {\displaystyle \frac{\nu }{2}}{\parallel |\nabla w\parallel |}_{L^2}^2+{\displaystyle \frac{C}{2\nu }}{\parallel |w\parallel |}_{L^2}^2\parallel |\nabla {\overline{u}}_2{\parallel |}_{L^2}^2,\hfill \end{array}$$

(11)

where we used Hölder’s inequality, the 2D Gagliardo-Nirenberg inequality ${\parallel |w\parallel |}_{L^4}^2\le {C\parallel |w\parallel |}_{L^2}{\parallel |\nabla w\parallel |}_{L^2}$ (see the references for more utilizations [26,35]), and Young’s inequality.

Simplifying and applying Grönwall’s inequality [31,36] gives the following:

$${\parallel |w\left(t\right)\parallel |}_{{\mathbb{L}}^2}^2\le {\parallel |w\left(0\right)\parallel |}_{{\mathbb{L}}^2}^{2}exp\left({\displaystyle \frac{C}{\nu }}{\int }_0^t\parallel |\nabla {\overline{u}}_2\left(A\right){\parallel |}_{{\mathbb{L}}^2}^{2}ds\right)=0,$$

since $w\left(0\right)=0$ and ${\overline{u}}_2\in {\mathbb{L}}^2(0,T;{\mathbb{H}}^1\left({\mathbb{R}}^2\right))$. Thus, $w\equiv 0$ on $[0,T]$, the proof is completed.    □

4. Spatially Discrete Finite Element Formulation

Let ${\mathcal{T}}^h={\left\{K\right\}}_{K\in {\mathcal{T}}^h}$ be a shape-regular triangulation of the reference domain ${\widehat{\mathrm{\Omega }}}_B$, where each element K is a closed simplex with diameter $h_K=\mathrm{diam}\left(K\right)$ [37]. We define the local mesh size function $h:{\widehat{\mathrm{\Omega }}}_B\to {\mathbb{R}}^{+}$ piecewise as $h\left(x\right)=h_K$ for all $x\in K$ and the global mesh parameter as $h={max}_{K\in {\mathcal{T}}^h}{h}_K$, representing the largest element diameter in the triangulation [38]. The shape regularity condition requires the existence of a constant $\sigma >0$ such that $h_K/{\rho }_K\le \sigma$ for all $K\in {\mathcal{T}}^h$, where ${\rho }_K$ denotes the radius of element K ([39] [Section 3.1]).

We consider a mixed finite element approximation space $({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)\subset ({\mathbb{X}}^u,{\mathbb{M}}^p)$, where

${\mathbb{X}}_h^u\subset {\mathbb{X}}^u$ is the discrete velocity space; ${\mathbb{M}}_h^p\subset {\mathbb{M}}^p$ is the discrete pressure space. The corresponding divergence-free subspace ${\mathbb{V}}_h\subset {\mathbb{X}}_h^u$ is defined by

$${\mathbb{V}}_h=\{v_h\in {\mathbb{X}}_h^u\mid d(v_h,q_h)=0,\forall q_h\in {\mathbb{M}}_h^p\},$$

where $d(·,·)$ represents the bilinear form of divergence.

Let ${\mathcal{P}}_h:\mathbb{Y}\to {\mathbb{V}}_h$ denote the ${\mathbb{L}}^2$-orthogonal projection operator, uniquely determined by

$$({\mathcal{P}}_{h}v,v_h)=(v,v_h),\forall v\in \mathbb{Y},\forall v_h\in {\mathbb{V}}_h.$$

The following fundamental assumptions on the mixed finite element spaces will be essential for our analysis [32,40].

Assumption 1

([34,39]). Approximation Property: For any $(v,q)\in \mathcal{D}\left(S\right)\times ({\mathbb{M}}^p\cap {\mathbb{H}}^1\left(\widehat{{\mathrm{\Omega }}_B}\right))$, there exist approximations $({\pi }_{h}v,{\rho }_{h}q)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, and a positive constant $C>0$ independent of the mesh size h, such that

$$\begin{array}{cc}\hfill \parallel |v-{\pi }_{h}v{\parallel |}_{{\mathbb{H}}^1\left(\widehat{{\mathrm{\Omega }}_B}\right)}& \le {Ch\parallel |Av\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \parallel |q-{\rho }_{h}q{\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}& \le {Ch\parallel |q\parallel |}_{{\mathbb{H}}^1\left(\widehat{{\mathrm{\Omega }}_B}\right)}.\hfill \end{array}$$

(13)

Assumption 2

([41,42]). For any $(v,q)\in (X_h,M_h)$, the following relations hold:

$$\parallel |\nabla v\parallel |\le C{h}^{-1}\parallel |v\parallel |,\parallel |q\parallel |\le C{h}^{-1}{\parallel |q\parallel |}_{-1}.$$

(14)

Assumption 3

([34,39,42]). For all $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, there exists $\lambda >0$, such that

$$\underset{v_h\in {\mathbb{X}}_h^u}{sup}{\displaystyle \frac{(\nabla ·v_h,q_h)}{\parallel |v_h{\parallel |}_1}}\ge \lambda \parallel |q_h\parallel |.$$

(15)

Assumption 4

([39]). The ${\mathbb{L}}^2$-projection operator ${\mathbb{P}}_h$ satisfies the following approximation properties:

1. 

For all $v\in \mathcal{D}\left(S\right)$, the Stokes operator domain is

$$\parallel |v-P_h{v\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}+h\parallel |\nabla (v-P_{h}v){\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}\le C{h}^2{\parallel |Av\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}$$

(16)

2. 

For all $v\in {\mathbb{X}}^u$:

$$\parallel |v-P_h{v\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}\le Ch\parallel |\nabla (v-P_{h}v){\parallel |}_{{\mathbb{L}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)}$$

(17)

The semi-discrete FEM seeks $({\overline{u}}_h,p_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$, satisfying

$$\begin{array}{cc}\hfill & {\mathcal{D}}_t^{\alpha }({\overline{u}}_h,v_h)+a({\overline{u}}_h,v_h)+d({\overline{u}}_h,{\overline{u}}_h,v_h)-b(v_h,p_h)+b({\overline{u}}_h,q_h)+g({\overline{u}}_h,{\overline{u}}_h,v_h)=(\overline{f},v_h)\hfill \end{array}$$

(18)

with ${\overline{u}}_h\left(0\right)={\mathcal{P}}_h{\overline{u}}_0$. The discrete Stokes operator ${\mathcal{S}}_h=-{\mathcal{P}}_h{\Delta }_h$ satisfies $(-{\Delta }_h{\overline{u}}_h,v_h)=(\nabla {\overline{u}}_h,\nabla v_h)$. The trilinear form $d(·,·,·)$ is anti-symmetric, conservative ($d({\overline{u}}_h,v_h,v_h)=0$), and bounded [27].

Lemma 1

(Caputo Inequality). Let ${\overline{u}}_h\in C^1([0,T];{\mathbb{L}}^2\left(\mathrm{\Omega }\right))$ and $0<\alpha <1$. Then, the Caputo fractional derivative satisfies the coercivity identity

$$({\mathcal{D}}_t^{\alpha }{\overline{u}}_h\left(t\right),{\overline{u}}_h\left(t\right))\ge {\displaystyle \frac{1}{2}}{\mathcal{D}}_t^{\alpha }{\parallel {\overline{u}}_h\left(t\right)\parallel }^2,$$

where ${\mathcal{D}}_t^{\alpha }$ denotes the Caputo fractional derivative of order α.

Theorem 3

(Uniqueness and Stability). Let (${\overline{u}}_h={\overline{u}}_h\left(t\right)$) be the semi-discrete finite element approximation of the solution of the TFNS problem with nonlinear damping. Then, for any $t\in [0,T]$, the following estimate holds:

$$\parallel |{\overline{u}}_h{\left(t\right)\parallel |}^2+{\displaystyle \frac{\nu T^{\alpha -1}}{2\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t\parallel |{\overline{u}}_h{\left(A\right)\parallel |}_1^{2}dA\le \parallel |{\overline{u}}_{0h}{\parallel |}^2+{\displaystyle \frac{(1-{\alpha }_1)T^{1+\beta }}{2\nu (1+\beta )\mathrm{\Gamma }\left(\alpha \right)}}+{\displaystyle \frac{{\alpha }_{1}T}{2\nu \mathrm{\Gamma }\left(\alpha \right)}}\underset{A\in [0,T]}{max}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1},$$

(19)

where $\beta ={\displaystyle \frac{\alpha -1}{1-{\alpha }_1}}$ and ${\alpha }_1\in (0,1)$.

Proof. 

Choosing test functions $(v,q)=({\overline{u}}_h,p_h)$ in Equation (18), we can obtain the following:

$${\mathcal{D}}_t^{\alpha }({\overline{u}}_h,{\overline{u}}_h)+a({\overline{u}}_h,{\overline{u}}_h)+d({\overline{u}}_h,{\overline{u}}_h,{\overline{u}}_h)-b({\overline{u}}_h,p_h)+b({\overline{u}}_h,p_h)+g({\overline{u}}_h,{\overline{u}}_h,{\overline{u}}_h)=(\overline{f},{\overline{u}}_h),$$

(20)

We begin with the weak formulation tested by ${\overline{u}}_h\left(t\right)$ and estimate the time-fractional Caputo derivative using Lemma 1:

$$\left({\mathcal{D}}_t^{\alpha }{\overline{u}}_h\left(t\right),{\overline{u}}_h\left(t\right)\right)\ge {\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}\parallel |{\overline{u}}_h\left(t\right){\parallel |}^2.$$

The bilinear form $a(·,·)$ satisfies the following coercivity property:

$$a({\overline{u}}_h\left(t\right),{\overline{u}}_h\left(t\right))\ge \nu \parallel |{\overline{u}}_h\left(t\right){\parallel |}_1^2.$$

The right-hand-side forcing term is estimated using dual norms and Young’s inequality:

$$\left(\overline{f}\left(t\right),{\overline{u}}_h\left(t\right)\right)\le \parallel \overline{f}{\left(t\right)\parallel }_{-1}\parallel {\overline{u}}_h{\left(t\right)\parallel }_1\le {\displaystyle \frac{1}{2\nu }}\parallel \overline{f}{\left(t\right)\parallel }_{-1}^2+{\displaystyle \frac{\nu }{2}}{\parallel {\overline{u}}_h\left(t\right)\parallel }_1^2.$$

Substituting into the weak formulation and integrating over time with the fractional kernel, we can obtain the following:

$$\begin{array}{c}\hfill \parallel |{\overline{u}}_h{\left(t\right)\parallel |}^2+\frac{\nu }{2\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-A)}^{\alpha -1}\parallel |{\overline{u}}_h\left(A\right){\parallel |}_1^{2}dA\\ \hfill \le \parallel |{\overline{u}}_{0h}{\parallel |}^2+{\displaystyle \frac{1}{2\nu \mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-A)}^{\alpha -1}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2}dA.\end{array}$$

(21)

We now apply Hölder’s inequality with exponents ${\displaystyle \frac{1}{1-{\alpha }_1}}$ and ${\displaystyle \frac{1}{{\alpha }_1}}$ in order to estimate the integral:

$$\begin{array}{c}\hfill {\int }_0^t{(t-A)}^{\alpha -1}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2}dA\le {\left({\int }_0^t{(t-A)}^{{\displaystyle \frac{\alpha -1}{1-{\alpha }_1}}}dA\right)}^{1-{\alpha }_1}{\left({\int }_0^t\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1}dA\right)}^{{\alpha }_1}.\end{array}$$

(22)

The first integral is a standard power integral, giving

$$\begin{array}{c}\hfill {\int }_0^t{(t-A)}^{{\displaystyle \frac{\alpha -1}{1-{\alpha }_1}}}dA={\displaystyle \frac{t^{\beta +1}}{\beta +1}},\mathrm{where}\beta ={\displaystyle \frac{\alpha -1}{1-{\alpha }_1}}.\end{array}$$

(23)

Hence,

$$\begin{array}{c}\hfill {\int }_0^t{(t-A)}^{\alpha -1}\parallel |\overline{f}{\left(A\right)\parallel |}_{-1}^{2}dA\le {\displaystyle \frac{t^{1+\beta }}{1+\beta }}\underset{A\in [0,t]}{max}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1}.\end{array}$$

(24)

Substituting this Equation (24) back into inequality (21), we may obtain the following:

$$\begin{array}{c}\hfill \parallel |{\overline{u}}_h{\left(t\right)\parallel |}^2+{\displaystyle \frac{\nu }{2\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-A)}^{\alpha -1}\parallel |{\overline{u}}_h\left(A\right){\parallel |}_1^{2}dA\\ \hfill \le \parallel |{\overline{u}}_{0h}{\parallel |}^2+\frac{t^{1+\beta }}{2\nu (1+\beta )\mathrm{\Gamma }\left(\alpha \right)}\underset{A\in [0,t]}{max}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1}.\end{array}$$

(25)

To obtain a uniform-in-time bound, we observe that for all $A\in [0,t]\subseteq [0,T]$, we have ${(t-A)}^{\alpha -1}\ge T^{\alpha -1}$. Hence,

$$\begin{array}{c}\hfill {\int }_0^t{(t-A)}^{\alpha -1}\parallel |{\overline{u}}_h{\left(A\right)\parallel |}_1^{2}dA\ge T^{\alpha -1}{\int }_0^t\parallel |{\overline{u}}_h\left(A\right){\parallel |}_1^{2}dA.\end{array}$$

(26)

Substituting this relation into the left-hand side of the Equation (26), we obtain

$$\parallel |{\overline{u}}_h{\left(t\right)\parallel |}^2+{\displaystyle \frac{\nu T^{\alpha -1}}{2\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t\parallel |{\overline{u}}_h{\left(A\right)\parallel |}_1^{2}dA\le \parallel |{\overline{u}}_{0h}{\parallel |}^2+{\displaystyle \frac{t^{1+\beta }}{2\nu (1+\beta )\mathrm{\Gamma }\left(\alpha \right)}}\underset{A\in [0,t]}{max}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1}.$$

Taking $t=T$ to obtain a global-in-time bound and rewriting the last term using convex splitting, we have

$${\displaystyle \frac{T^{1+\beta }}{2\nu (1+\beta )\mathrm{\Gamma }\left(\alpha \right)}}={\displaystyle \frac{(1-{\alpha }_1)T^{1+\beta }}{2\nu (1+\beta )\mathrm{\Gamma }\left(\alpha \right)}},\mathrm{and}{\displaystyle \frac{{\alpha }_{1}T}{2\nu \mathrm{\Gamma }\left(\alpha \right)}}\underset{A\in [0,T]}{max}\parallel |\overline{f}\left(A\right){\parallel |}_{-1}^{2/{\alpha }_1},$$

we thus obtain the desired estimate.    □

This completes the proof of the uniqueness and stability of the discrete solutions. The theorem’s key features include complete handling of fractional derivatives, careful treatment of nonlinear terms via skew-symmetry and monotonicity properties, and validity across the full range of parameters $0<\alpha <1$, all while maintaining mathematical rigor with computational relevance. The bound holds for any discrete time $dt>0$, establishing unconditional stability.

Remark 2.

This shows that (Theorem 3) serves as the foundation for proving the convergence of numerical approximations, long–time stability properties, and optimal error estimates for the finite element method.

We now present the convergence analysis.

Theorem 4

(Convergence Analysis). Assume $(\overline{u},p)\in {\mathbb{H}}^2\left(\widehat{{\mathrm{\Omega }}_B}\right)\times {\mathbb{H}}^1\left(\widehat{{\mathrm{\Omega }}_B}\right)$ and suppose $({\overline{u}}_h,p_h)$ is the approximate solution of (18). For any $t\in [0,T]$ with $0<\alpha <1$, there exists a constant $C>0$ independent of h, such that

$$\parallel \overline{u}-{\overline{u}}_h\parallel \le C{h}^2,\parallel p-p_h\parallel \le Ch.$$

(27)

Proof. 

Let $({\mathbf{e}}_u,e_p)=(\overline{u}-{\overline{u}}_h,p-p_h)$. By subtracting the discrete formulation (18) from the continuous formulation (8), we obtain the following error equation:

$$\left\{\begin{array}{c}{\mathcal{D}}_t^{\alpha }({\mathbf{e}}_u,v)+a({\mathbf{e}}_u,v)+\mathcal{N}({\mathbf{e}}_u,{\overline{u}}_h,v)-b(v,e_p)+b({\mathbf{e}}_u,q)=0,\forall (v,q)\in V_h\times Q_h,\hfill \\ {\mathbf{e}}_u\left(0\right)={\overline{u}}_0-P_h{\overline{u}}_0,\hfill \end{array}\right.$$

(28)

where $\mathcal{N}(\overline{u},v,w):=d(\overline{u},v,w)+d(v,\overline{u},w)$.

Taking $v={\mathbf{e}}_u$ and $q=e_p$ in (28) and applying the definition of $a(·,·)$ and $d(·,·,·)$, we obtain the following energy estimate:

$${\mathcal{D}}_t^{\alpha }\parallel {\mathbf{e}}_u{\parallel }^2+\nu \parallel {\mathbf{e}}_u{\parallel }_1^2\le C\parallel {\mathbf{e}}_u\parallel \parallel {\mathbf{e}}_u{\parallel }_1{\parallel {\overline{u}}_h\parallel }_1.$$

Applying Young’s inequality with $ϵ=\nu /2$ gives

$${\mathcal{D}}_t^{\alpha }\parallel {\mathbf{e}}_u{\parallel }^2+{\displaystyle \frac{\nu }{2}}\parallel {\mathbf{e}}_u{\parallel }_1^2\le C\parallel {\mathbf{e}}_u{\parallel }^2{\parallel {\overline{u}}_h\parallel }_1^2.$$

(29)

Applying the fractional integral operator ${\mathcal{I}}^{\alpha }$ to both sides of (29), we obtain the following:

$$\parallel {\mathbf{e}}_u{\left(t\right)\parallel }^2\le {\parallel {\mathbf{e}}_u\left(0\right)\parallel }^2+C{\mathcal{I}}^{\alpha }\left(\parallel {\mathbf{e}}_u{\parallel }^2{\parallel {\overline{u}}_h\parallel }_1^2\right)\left(t\right).$$

(30)

Using the generalized Grönwall inequality for fractional differential inequalities (see e.g., [43]), we obtain

$$\parallel {\mathbf{e}}_u{\left(t\right)\parallel }^2\le C\parallel {\mathbf{e}}_u{\left(0\right)\parallel }^2\le C{\parallel {\overline{u}}_0-P_h{\overline{u}}_0\parallel }^2\le C{h}^4,$$

(31)

where the last inequality follows from the approximation property of the projection operator Assumption 4, i.e.,

$$\parallel {\overline{u}}_0-P_h{\overline{u}}_0\parallel \le C{h}^2{\parallel {\overline{u}}_0\parallel }_2.$$

Hence,

$$\parallel {\mathbf{e}}_u\left(t\right)\parallel \le C{h}^2.$$

(32)

To estimate the pressure error, we apply the inf-sup condition (cf. Assumption (15)) with $q=0$:

$$\begin{array}{cc}\hfill \beta \parallel e_p\parallel & \le \underset{v_h\in V_h}{sup}{\displaystyle \frac{\left|b\right(v_h,e_p\left)\right|}{\parallel v_h{\parallel }_1}}\hfill \\ \hfill & =\underset{v_h\in V_h}{sup}{\displaystyle \frac{|C{D}_t^{\alpha }({\mathbf{e}}_u,v_h)+a({\mathbf{e}}_u,v_h)+\mathcal{N}({\mathbf{e}}_u,{\overline{u}}_h,v_h)|}{\parallel v_h{\parallel }_1}}\hfill \\ \hfill & \le C\left(\parallel D_t^{\alpha }{\mathbf{e}}_u{\parallel }_{-1}+\parallel {\mathbf{e}}_u{\parallel }_1+\parallel {\mathbf{e}}_u{\parallel }_1{\parallel {\overline{u}}_h\parallel }_1\right).\hfill \end{array}$$

We estimate the dual norm of the fractional derivative using the following bound:

$$\parallel D_t^{\alpha }{\mathbf{e}}_u{\parallel }_{-1}\le C\parallel {\mathbf{e}}_u\parallel ,$$

which, together with the earlier bounds $\parallel {\mathbf{e}}_u\parallel \le C{h}^2$ and $\parallel {\mathbf{e}}_u{\parallel }_1\le Ch$, implies

$$\parallel e_p\parallel \le C(h^2+h+h·1)\le Ch.$$

(33)

Therefore, the velocity and pressure errors satisfy the estimates

$$\parallel \overline{u}-{\overline{u}}_h\parallel \le C{h}^2,\parallel p-p_h\parallel \le Ch,$$

which thus completes the proof.    □

4.1. Fractional Time Discretization via the Finite Difference Method

The numerical treatment of time-fractional derivatives has been well established through various discretization approaches [44,45,46,47,48]. For our temporal discretization, we consider a uniform partition of the time interval $[0,T]$ given by

$$t_n=n\tau ,\mathrm{for}n=0,1,\dots ,N,\mathrm{where}\tau =T/N$$

At each discrete time level $t_n$, we approximate the Riemann–Liouville fractional integral operator of order $\alpha \in (0,1)$ as follows:

$$I_t^{\alpha }g\left(t_n\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\sum _{j=1}^n{\int }_{t_{j-1}}^{t_j}{(t_n-A)}^{\alpha -1}g\left(A\right)dA$$

$$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\sum _{j=1}^n\left({\int }_{t_{k-1}}^{t_j}{(t_n-A)}^{\alpha -1}g\left(t_k\right)dA\right)+{\gamma }_{\alpha }^n$$

$$={\displaystyle \frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}\sum _{j=1}^{n}g\left(t_j\right)\left[{(n-j+1)}^{\alpha }-{(n-j)}^{\alpha }\right]+{\gamma }_{\alpha }^n$$

$$={\displaystyle \frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}\sum _{j=0}^n{w}_j^{\alpha }g\left(t_{n-j}\right)+{\gamma }_{\alpha }^n$$

(34)

where the fractional weights $w_j^{\alpha }$ are defined by the difference formula

$$w_j^{\alpha }={(j+1)}^{\alpha }-j^{\alpha },j\ge 0,$$

and where the local truncation error ${\gamma }_{\alpha }^n$ satisfies the estimate

$${\gamma }_{\alpha }^n={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\sum _{j=1}^n{\int }_{t_{j-1}}^{t_j}{(t_n-A)}^{\alpha -1}[g\left(A\right)-g\left(t_j\right)]dA$$

$$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}\sum _{j=1}^n{\int }_{t_{j-1}}^{t_j}{(t_n-A)}^{\alpha -1}{g}^{\prime }\left(\zeta \right)(A-t_j)dA,A<\zeta <t_j.$$

Therefore, we have

$$|{\gamma }_{\alpha }^n|\le {\displaystyle \frac{\tau }{\mathrm{\Gamma }\left(\alpha \right)}}\underset{0\le t\le t_j}{max}\left|g^{\prime }\left(t\right)\right|{\int }_{t_{j-1}}^{t_j}{(t_n-A)}^{\alpha -1}dA$$

$$\le {\displaystyle \frac{N^{\alpha }{\tau }^{\alpha +1}}{\mathrm{\Gamma }\alpha +1}}\underset{0\le t\le t_j}{max}\left|g^{\prime }\left(t\right)\right|.$$

Lemma 2

(Numerical approximation of fractional integrals). (See [43]) Let $g\in C^1\left([0,T]\right)$ be a continuously differentiable function. Then, for any $\alpha \in (0,1)$, the Riemann–Liouville fractional integral $I_t^{\alpha }g\left(t_n\right)$ at time $t_n$ admits the following numerical approximation:

$$I_t^{\alpha }g\left(t_n\right)={\displaystyle \frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}\sum _{j=0}^n{w}_j^{\alpha }g\left(t_{n-j}\right)+{\gamma }_{\alpha }^n,$$

(35)

where the weights $w_j^{\alpha }:={(j+1)}^{\alpha }-j^{\alpha }$ and the local truncation error ${\gamma }_{\alpha }^n$ satisfy the bound

$$|{\gamma }_{\alpha }^n|\le C{\tau }^{\alpha +1}\mathit{for}\mathit{all}n=0,1,\dots ,N.$$

(36)

Lemma 3.

(See [43]) Consider the temporal discretization $0=t_0<t_1<\cdots <t_N=T$ with $\alpha >0$, where the fractional weights ${\left\{w_j^{\alpha }\right\}}_{j=0}^{\infty }$ are generated by the Formula (34). These weights satisfy the following fundamental properties.

(i) 

Positivity and initialization: $w_0^{\alpha }=1$ and $w_j^{\alpha }>0$ for all $j=0,1,2,\dots$

(ii) 

Monotonicity: $w_j^{\alpha }>w_{j+1}^{\alpha }$ for all $j=0,1,2,\dots$

(iii) 

Summation bound: ${\sum }_{j=0}^{n-1}{w}_j^{\alpha }=n^{\alpha }\le N^{\alpha }$

Utilizing integral operator (4) for both sides of Equation (18) yields a weak formulation:

$$\begin{array}{cc}\hfill ({\overline{u}}_h,v_h)& +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-A)}^{\alpha -1}[a({\overline{u}}_h,v_h)+d({\overline{u}}_h,{\overline{u}}_h,v_h)\hfill \\ \hfill & -b(v_h,p_h)+b({\overline{u}}_h,q_h)+g({\overline{u}}_h,{\overline{u}}_h,v)]dA\hfill \\ \hfill & =({\overline{u}}_h,v_h)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-A)}^{\alpha -1}(\overline{f},v_h)dA.\hfill \end{array}$$

(37)

Fully Discrete Scheme

Let $({\overline{u}}_h^n,p_h^n)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ denote the discrete approximations of the exact solutions $({\overline{u}}_h\left(t_n\right),p_h\left(t_n\right))$ at time $t=t_n$. The fully discrete numerical scheme for problem (8) is derived through the combination of the fractional approximation (35) with the variational formulation (37), yielding the following system. For each $n=1,\dots ,N$, find $({\overline{u}}_h^n,p_h^n)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$ such that for all test functions $(v_h,q_h)\in ({\mathbb{X}}_h^u,{\mathbb{M}}_h^p)$,

$$\begin{array}{ccc}\hfill ({\overline{u}}_h^n,v_h)& +& {\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }[a({\overline{u}}_h^{n-j},v_h)+d({\overline{u}}_h^{n-j},{\overline{u}}_h^{n-j},v_h)\hfill \\ \hfill & -& b(v_h,p_h^{n-j})+b({\overline{u}}_h^{n-j},q_h)+g({\overline{u}}_h^{n-j},{\overline{u}}_h^{n-j},v_h)]\hfill \\ \hfill & =& ({\overline{u}}_h^0,v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},v_h),\hfill \end{array}$$

(38)

where the time-stepping coefficient ${\beta }_0={\tau }^{\alpha }/\mathrm{\Gamma }(\alpha +1)$ encapsulates the temporal discretization of the fractional derivative operator.

Now, we present the stability of the numerical scheme.

4.2. Stability Analysis of the Fully Discrete Scheme

Theorem 5.

For any $0<\tau <T$, the fully discrete scheme given in (38) is unconditionally stable, satisfying the following stability estimate:

$$\parallel |{\overline{u}}_h^n{\parallel |}^2+{\beta }_1\parallel |{\overline{u}}_h^n{\parallel |}_1^2\le C^{†}(\parallel |{\overline{u}}_h^0{\parallel |}^2+\sum _{j=0}^n\parallel |{\overline{f}}^j{\parallel |}_{-1}^2).$$

Proof. 

For the case $n=1$ and taking j = 0 to $n-1$ in Equation (38), we obtain

$$\begin{array}{c}\hfill ({\overline{u}}_h^1,v_h)+{\beta }_0{w}_0^{\alpha }\left[a({\overline{u}}_h^1,v_h)+d({\overline{u}}_h^0,{\overline{u}}_h^1,v_h)-b(v_h,p_h^1)+b({\overline{u}}_h^1,q_h)+g({\overline{u}}_h^0,{\overline{u}}_h^1,v_h)\right]\\ \hfill =({\overline{u}}_h^0,v_h)+{\beta }_0{w}_0^{\alpha }({\overline{f}}^1,v_h).\end{array}$$

(39)

where

$${\beta }_0=\frac{{\tau }^{\alpha }}{\mathrm{\Gamma }(\alpha +1)},w_0^{\alpha }=1^{\alpha }-0^{\alpha }=1.$$

Taking $v_h={\overline{u}}_h^1$ and $q_h=p_h^1$ in (39), the convective term vanishes for the test function equal to velocity, and the standard properties of the convective terms are ${\beta }_{0}d({\overline{u}}_h^0,{\overline{u}}_h^1,{\overline{u}}_h)=0$. We also assume regularity for the nonlinear term ${\beta }_{0}g({\overline{u}}_h^0,{\overline{u}}_h^1,{\overline{u}}_h)\ge 0$. Now, Equation (39) reduces to

$$({\overline{u}}_h^1,{\overline{u}}_h^1)+{\beta }_{0}a({\overline{u}}_h^1,{\overline{u}}_h^1)=({\overline{u}}_h^0,{\overline{u}}_h^1)+{\beta }_0({\overline{f}}^1,{\overline{u}}_h^1).$$

Using Cauchy–Schwarz and Young’s inequalities, we obtain

$$\parallel |{\overline{u}}_h^1{\parallel |}^2+{\beta }_0\nu \parallel |{\overline{u}}_h^1{\parallel |}^2\le {\displaystyle \frac{1}{2}}[\parallel |{\overline{u}}_h^0{\parallel |}^2+\parallel |{\overline{u}}_h^1{\parallel |}^2]+[{\displaystyle \frac{{\beta }_0}{2\nu }}\parallel |{\overline{f}}^1{\parallel |}_{-1}^2+{\displaystyle \frac{{\beta }_0\nu }{2}}\parallel |{\overline{u}}_h^1{\parallel |}_1^2],$$

that is,

$$\parallel |{\overline{u}}_h^1{\parallel |}^2+{\beta }_0\parallel |{\overline{u}}_h^1{\parallel |}_1^2\le {\displaystyle \frac{1}{2}}\parallel |{\overline{u}}_h^0{\parallel |}^2+{\displaystyle \frac{{\beta }_0}{2\nu }}\parallel |{\overline{f}}^1{\parallel |}_{-1}^2.$$

For the discrete solution ${\overline{u}}_h^j\in {\mathbb{X}}_h^u$ and $p_h^j\in {\mathbb{M}}_h^p$ with test functions $v_h={\overline{u}}_h^j$ and $q_h=p_h^j$, we establish the following uniform bound:

$$\parallel |{\overline{u}}_h^j{\parallel |}^2+{\beta }_1\parallel |{\overline{u}}_h^j{\parallel |}_1^2\le C(\parallel |{\overline{u}}_h^0{\parallel |}^2+\sum _{j=1}^j\parallel |{\overline{f}}^j{\parallel |}_{-1}^2),j=2,3,\dots ,n-1.$$

Setting $v_h={\overline{u}}_h^{n-j}$ and $q_h=p_h^{n-j}$ in (38), we obtain

$$({\overline{u}}_h^n,{\overline{u}}_h^{n-j})+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }a({\overline{u}}_h^{n-j},{\overline{u}}_h^{n-j})=({\overline{u}}_h^0,{\overline{u}}_h^{n-j})+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},{\overline{u}}_h^{n-j}).$$

By employing the elementary identity $uv={\displaystyle \frac{1}{2}}(u^2+v^2)-{\displaystyle \frac{1}{2}}{(u-v)}^2$, which is valid for any real numbers $u,v\in \mathbb{R}$ together with the application of Young’s inequality, we obtain

$${\displaystyle \frac{1}{2}}\left[\parallel |{\overline{u}}_h^n{\parallel |}^2+\parallel |{\overline{u}}_h^{n-j}{\parallel |}^2\right]+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }\parallel |{\overline{u}}_h^{n-j}{\parallel |}_1^2=({\overline{u}}_h^0,{\overline{u}}_h^{n-j})+{\displaystyle \frac{1}{2}}\parallel |{\overline{u}}_h^n,{\overline{u}}_h^{n-j}{\parallel |}^2+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }({\overline{f}}^{n-j},{\overline{u}}_h^{n-j})$$

$$\le {\displaystyle \frac{1}{2}}\left[\parallel |{\overline{u}}_h^0{\parallel |}^2+\parallel |{\overline{u}}_h^{n-j}{\parallel |}^2\right]+{\displaystyle \frac{1}{2}}\parallel |{\overline{u}}_h^n,{\overline{u}}_h^{n-j}{\parallel |}^2+\sum _{j=0}^{n-1}{w}_j^{\alpha }({\displaystyle \frac{{\beta }_0}{2\nu }}\parallel |{\overline{f}}^{n-j}{\parallel |}_{-1}^2+{\displaystyle \frac{{\beta }_{0\nu }}{2}}\parallel |{\overline{u}}_h^{n-j}{\parallel |}_1^2).$$

Together with Lemma 3, $\left(ii\right)$$(i.e.,0<w_{j+1}^{\alpha }<w_j^{\alpha }<1)$ and ${\displaystyle \frac{{\beta }_{0\nu }}{2}}{\sum }_{j=0}^{n-1}{w}_j^{\alpha }\parallel |{\overline{f}}^{n-j}{\parallel |}_{-1}^2\ge 0$, we obtain

$$\parallel |{\overline{u}}_h^n{\parallel |}^2+{\beta }_1\parallel |{\overline{u}}_h^n{\parallel |}_1^2\le \left[\parallel |{\overline{u}}_h^0{\parallel |}^2+\parallel |{\overline{u}}_h^{n-j}{\parallel |}^2\right]+{\displaystyle \frac{{\beta }_0}{2\nu }}\sum _{j=0}^{n-1}{w}_j^{\alpha }\parallel |{\overline{f}}^{n-j}{\parallel |}_{-1}^2$$

$$\le C^{†}(\parallel |{\overline{u}}_h^0\parallel |+\sum _{j=0}^n\parallel |{\overline{f}}^j{\parallel |}_{-1}^2),$$

(40)

where $C^{†}$ is a constant depending on ${\beta }_0$. The proof is completed.   □

Lemma 4.

Let $\nu >0$ be the viscosity coefficient and

$$\parallel |{\overline{u}}_h^n{\parallel |}^2+\sum _{k=0}^n\parallel |{\overline{f}}^j{\parallel |}_{-1}^2\le {\displaystyle \frac{{\beta }_1\nu }{C^{†}\mu }};$$

(41)

then, we want to prove

$$a(v_h,v_h)+d(v_h,{\overline{u}}_h^n,v_h)\ge 0,$$

(42)

Proof. 

Utilizing Equations (40) and (41), we obtain

$$\parallel |{\overline{u}}_h^n{\parallel |}_1^2\le {\displaystyle \frac{\nu }{\mu }}$$

Thus, we have:

$$\nu -\mu \parallel |{\overline{u}}_h^n{\parallel |}_1^2\ge 0.$$

By the trilinear property of $d(v_h,{\overline{u}}_n^h,v_h)$, it follows that

$$a(v_h,v_h)+d(v_h,{\overline{u}}_h^n,v_h)\ge (\nu -\mu \parallel |{\overline{u}}_h^n{\parallel |}_1^2)\parallel |v_h{\parallel |}_1^2\ge 0.$$

Thus, the  proof of the lemma is completed.   □

Theorem 6.

Let $\alpha \in (0,1)$ be any given fractional order. Suppose that $({\overline{u}}_h\left(t_n\right),p_h\left(t_n\right))$ denotes the solution to Equation (18), and $({\overline{u}}_h^n,p_h^n)$ denotes the solution to scheme (38). Then, there exists a constant $C=C(\alpha ,\overline{{\mathrm{\Omega }}_B},T)>0$, which is independent of the time-step size τ, such that the following error estimates hold:

$$\parallel |{\overline{u}}_h-{\overline{u}}_h^n\parallel |\le C{\tau }^{\alpha }+1,\parallel |p_h-p_h^n\parallel |\le C{\tau }^{\alpha }+1.$$

(43)

Proof. 

We begin by defining the error functions for the velocity and pressure at time level n:

$$\begin{array}{cc}\hfill e_u^n& :={\overline{u}}_h\left(t_n\right)-{\overline{u}}_h^n\in X_h,\hfill \\ \hfill e_p^n& :=p_h\left(t_n\right)-p_h^n\in M_h,\hfill \end{array}$$

with initial error $e_u^0=0$. From the semi-discrete formulation (35) and the fully discrete scheme (38), we derive the error equation:

$$\begin{array}{c}\hfill (e_u^n,v_h)+{\beta }_0\sum _{j=0}^{n-1}{w}_j^{\alpha }[\alpha (e_u^{n-j},v_h)+d(e_u^{n-j},{\overline{u}}_h^{n-j},v_h)-b(v_h,e_p^{n-j})\\ \hfill +b\left(e_u^{n-j}\right),q_h+g(e_u^{n-j},{\overline{u}}_h^{n-j},v_h)]=({\gamma }_{\alpha }^n,v_h),\end{array}$$

(44)

where ${\gamma }_{\alpha }^n$ is the truncation error for the velocity.

Setting $v_h=e_u^1$ and $q_h=e_p^1$ in (44) for $n=1$, we obtain

$$(e_u^1,e_u^1)+{\beta }_0[\alpha (e_u^1,e_u^1)+d(e_u^1,{\overline{u}}_h^1,e_u^1)+g(e_u^1,{\overline{u}}_h^1,e_u^1)]=({\gamma }_{\alpha }^1,e_u^1).$$

Using Cauchy–Schwarz inequality in combination with Lemma 4, we derive

$$\parallel |e_u^1\parallel |\le \parallel |{\gamma }_{\alpha }^1\parallel |\le C{\tau }^{\alpha +1}.$$

(45)

Suppose that $\parallel |e_u^m\parallel |\le C{\tau }^{\alpha +1}$ holds for $m=2,3,\dots ,n-1$. To verify that the first inequality in (43) is satisfied for $m=n$, we choose $v_h=e_u^{n-j}$ and $q_h=e_p^{n-j}$ in (44). Then,

$$(e_u^n,e_u^{n-j})+{\beta }_0\sum _{k=0}^{n-1}{w}_j^{\alpha }\left[\alpha (e_u^n,e_u^{n-j})+d(e_u^{n-j},{\overline{u}}_h^{n-j},e_u^{n-j})+C(e_u^{n-j},{\overline{u}}_h^{n-j},e_u^{n-j})\right]=({\gamma }_{\alpha }^n,e_u^{n-j}).$$

Using the identity $uv={\displaystyle \frac{1}{2}}(u^2+v^2)-{\displaystyle \frac{1}{2}}{(u-v)}^2$, Young’s inequality, and Lemma 4, we derive the following:

$${\displaystyle \frac{1}{2}}\left[\parallel |e_u^n{\parallel |}^2+\parallel |e_u^{n-j}{\parallel |}^2\right]+{\displaystyle \frac{1}{2}}[\parallel |e_u^n-e_u^{n-j}{\parallel |}^2]+{\displaystyle \frac{1}{2}}[\parallel |e_u^{n-j}{\parallel |}^2+\parallel |{\gamma }_{\alpha }^n{\parallel |}^2],$$

(46)

which implies that

$$\begin{array}{ccc}\hfill \parallel |e_u^n{\parallel |}^2& \le & C{\tau }^{2\alpha +2}.\hfill \end{array}$$

(47)

From the inverse estimate (14) together with above inequality (47),

$$\begin{array}{ccc}\hfill \parallel |e_u^n{\parallel |}_1& \le & C{h}^{-1}\parallel |e_u^n\parallel |\le C{\tau }^{\alpha +1}.\hfill \end{array}$$

(48)

For the pressure estimate, by choosing $v_h=e_u^1$ and $q_h=0$ with $n=1$ in (44) and applying the Cauchy–Schwarz inequality together with (12)–(15) and (48), we obtain

$$\begin{array}{ccc}\hfill \parallel |e_p^1\parallel |& \le & \underset{V_h}{sup}{\displaystyle \frac{\left|b\right(e_u^1,e_p^1\left)\right|}{\lambda \parallel |e_u^1{\parallel |}_1}}\hfill \\ \hfill & =& \underset{v_h\in V_h}{sup}{\displaystyle \frac{|\parallel |e_u^1{\parallel |}^2+{\beta }_0\left[\nu \parallel |e_u^1{\parallel |}_1^2+d(e_u^1,{\overline{u}}_h^1,e_u^1)+g(e_u^1,{\overline{u}}_h^1,e_u^1)\right]-({\gamma }_{\alpha }^1,e_u^1)|}{\lambda \parallel |e_u^1{\parallel |}_1}}\hfill \\ \hfill & \le & C{\tau }^{\alpha +1}.\hfill \end{array}$$

(49)

By assuming $\parallel |e_u^m\parallel |\le C{\tau }^{\alpha +1}$ for $m=2,3,\dots ,n-1$, selecting $v_h=e_u^{n-j}$ and $q_h=0$ in (44), and using (47) and (48), we similarly derive, as in (49), the result for $m=n$.

$$\begin{array}{cc}\hfill & \parallel |e_p^n\parallel |\le \underset{v_h\in V_h}{sup}{\displaystyle \frac{\left|b\right(e_u^n,e_p^n\left)\right|}{\lambda \parallel |e_u^n{\parallel |}_1}}\le \underset{v_h\in V_h}{sup}{\displaystyle \frac{\left|{\sum }_{j=0}^{n-1}{w}_j^{\alpha }d(e_u^{n-j},e_p^{n-j})\right|}{\lambda \parallel |e_u^n{\parallel |}_1}}\hfill \\ \hfill & =\underset{v_h\in V_h}{sup}{\displaystyle \frac{|(e_u^n,e_u^{n-j})+{\beta }_0{\sum }_{j=0}^{n-1}{w}_j^{\alpha }\left[\nu \parallel |e_u^{n-j}{\parallel |}_1^2+d(e_u^{n-j},{\overline{u}}_h^{n-j},e_u^{n-k})+g(e_u^{n-j},{\overline{u}}_h^{n-j},e_u^{n-j})\right]-({\gamma }_{\alpha }^n,e_u^{n-j})|}{\lambda \parallel |e_u^n{\parallel |}_1}}\hfill \\ \hfill & \le C{\tau }^{\alpha +1}.\hfill \end{array}$$

   □

This completes the proof.

Next, we provide the error estimate for the fully discrete scheme.

4.3. Error Analysis

Theorem 7

(Optimal Error). For $0<\alpha <1$, let $(\overline{u},p)$ be the solution of the continuous weak formulation (7) and $({\overline{u}}_h,p_h)$ be the numerical solution obtained from the discrete scheme (38). Then, there exists a positive constant $C=C(\alpha ,\widehat{{\mathrm{\Omega }}_B},T)$, independent of the spatial mesh size h and time-step τ, such that the following error estimates hold:

$$\parallel |u-{\overline{u}}_h\parallel |\le C(h^2+{\tau }^{\alpha +1}),\parallel |p-p_h\parallel |\le C(h+{\tau }^{\alpha +1}).$$

(50)

Proof. 

It is easy to show that Equation (50) follows from Theorems 4 and 7 via triangle inequality. The overall error estimates (50) are optimal because the spatial error rates match the theoretical properties of the mixed finite element spaces. The temporal error rate matches the theoretical convergence order of the fractional time-stepping scheme. Thus, the derived errors represent the best possible rates achievable under the given mixed finite elements ($P1b$–$P1$) or ($P_2$–$P_1$) and temporal discretization frameworks.   □

5. Numerical Example

In this section, all simulations are performed using FreeFem++ [49] by solver UMFPACK, with post-processing and visualization in MATLAB 2021a. For spatial discretization, we use Mini-elements ($P1b$–$P1$) for stability and Taylor–Hood elements ($P_2$–$P_1$) for higher-order accuracy [22,25]. We briefly restate the goals of the numerical validation, such as verifying convergence rates and comparing element types. Four iterative algorithms (Algorithm 1, Algorithm 2, Algorithm 3 and Algorithm 4) are used to solve the discretized problem, differing mainly in the treatment of nonlinear terms. The numerical examples include Example 1 on convergence verification (Table 1, Table 2,Table 3, Table 4 and Table 5, Figure 1 and Figure 2), Example 2 on performance with different exact solutions (Table 6, Table 7 and Table 8, Figure 3), and Example 3 focusing on Algorithm 4 (Table 7 and Table 8, Figure 4).

To handle the nonlinear terms in the discretized system, we propose the following iterative algorithms.

5.1. Iterative Algorithms for the Fully Discrete Scheme

This problem has two nonlinear terms: a trilinear convective term $d(·,·,·)$ and a quasilinear damping term $g(·,·,·)$. We consider the following four iterative algorithms for their treatment.

Remark 3.

The time discretization in Algorithm 1 is considered implicitly through the approximation of the Caputo fractional derivative by a discrete convolution sum over the solution history. Specifically, the weights $w_j^{\alpha }$, which are dependent on α and the time-step size $\Delta t$, represent the memory effects of built-in fractional-order models. The summation ${\sum }_{j=0}^{n-1}{w}_j^{\alpha }$ over previous time levels $n-j$ effectively reflects the full time evolution of the system up to the current time-step $t_n$. Consequently, the algorithm does not require an explicit finite difference quotient in time; instead, the fractional derivative’s nonlocal nature is preserved through this history-dependent formulation, ensuring both consistency and stability in the time integration.

5.2. Example 1 (Convergence Rate Verifications)

The main objective of this example is to demonstrate the numerical convergence order in order to check the theoretical findings and explain the effectiveness of the introduced method.

We consider a bounded square domain ($\widehat{{\mathrm{\Omega }}_B}$) = ${[0,1]}^2$ for all the numerical executions. To this end, we consider 2D problems with known analytical solutions of the $(\overline{u},p)=(({\overline{u}}_1,u_2),p)$ by utilizing homogeneous boundary conditions. The exact solutions for the right-hand-side values are given as follows [9]:

$$\begin{array}{ccc}\hfill {\overline{u}}_1& =& 2x^2{(x-1)}^{2}y(y-1)(2y-1)e^{-t},\hfill \\ \hfill {\overline{u}}_2& =& -2y^2(y-1)2x(x-1)(2x-1)e^{-t},\hfill \\ \hfill p& =& (x^2-y^2)e^{-t},\hfill \end{array}$$

From the mathematical model, we can find the right-hand side, which is known as the load function:

$$\overline{f}={\mathcal{D}}_t^{\alpha }\overline{u}-\nu \Delta \overline{u}+(\overline{u}·\nabla )\overline{u}+\gamma {\left|\overline{u}\right|}^{r-2}\overline{u}+\nabla p,$$

(51)

final form of $\overline{f}={({\overline{f}}_1,{\overline{f}}_2)}^T$ is

$$\begin{array}{cc}\hfill {\overline{f}}_1& =C{D}_t^{\alpha }{\overline{u}}_1-\nu \Delta {\overline{u}}_1+{\overline{u}}_1{\displaystyle \frac{\partial {\overline{u}}_1}{\partial x}}+{\overline{u}}_2{\displaystyle \frac{\partial {\overline{u}}_1}{\partial y}}+\gamma {({\overline{u}}_1^2+u_2^2)}^{(r-2)/2}{\overline{u}}_1+{\displaystyle \frac{\partial p}{\partial x}},\hfill \\ \hfill {\overline{f}}_2& =C{D}_t^{\alpha }{\overline{u}}_2-\nu \Delta u_2+u_1{\displaystyle \frac{\partial {\overline{u}}_2}{\partial x}}+{\overline{u}}_2{\displaystyle \frac{\partial {\overline{u}}_2}{\partial y}}+\gamma {({\overline{u}}_1^2+{\overline{u}}_2^2)}^{(r-2)/2}{\overline{u}}_2+{\displaystyle \frac{\partial p}{\partial y}}.\hfill \end{array}$$

The forcing term $\overline{f}$ is

$$\begin{array}{cc}\hfill {\overline{f}}_1& =2x^2{(x-1)}^{2}y(y-1)(2y-1)C{D}_t^{\alpha }{e}^{-t}\hfill \\ \hfill & -\nu \left[4y(y-1)(2y-1)(6x^2-6x+1)+2x^2{(x-1)}^2(12y-6)\right]e^{-t}\hfill \\ \hfill & +\left[8x^3{(x-1)}^3{y}^2{(y-1)}^2(2x-1)(2y-1)\right.\hfill \\ \hfill & \left.-4x^2{(x-1)}^2{y}^2{(y-1)}^2(2x-1)(6y^2-6y+1)\right]e^{-2t}\hfill \\ \hfill & +\gamma {\left[4x^4{(x-1)}^4{y}^2{(y-1)}^2{(2y-1)}^2+4y^4{(y-1)}^4{x}^2{(x-1)}^2{(2x-1)}^2\right]}^{(r-2)/2}\hfill \\ \hfill & \times 2x^2{(x-1)}^{2}y(y-1)(2y-1)e^{-rt/2}+2x{e}^{-t},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\overline{f}}_2& =-2y^2{(y-1)}^{2}x(x-1)(2x-1)C{D}_t^{\alpha }{e}^{-t}\hfill \\ \hfill & -\nu \left[-4x(x-1)(2x-1)(6y^2-6y+1)-2y^2{(y-1)}^2(12x-6)\right]e^{-t}\hfill \\ \hfill & +\left[-4x^2{(x-1)}^2{y}^2{(y-1)}^2(2y-1)(6x^2-6x+1)\right.\hfill \\ \hfill & \left.+8x^2{(x-1)}^2{y}^3{(y-1)}^3(2x-1)(2y-1)\right]e^{-2t}\hfill \\ \hfill & +\gamma {\left[4x^4{(x-1)}^4{y}^2{(y-1)}^2{(2y-1)}^2+4y^4{(y-1)}^4{x}^2{(x-1)}^2{(2x-1)}^2\right]}^{(r-2)/2}\hfill \\ \hfill & \times (-2y^2{(y-1)}^{2}x(x-1)(2x-1))e^{-rt/2}-2y{e}^{-t}.\hfill \end{array}$$

Table 1 presents ${\mathbb{L}}^2$ errors and convergence rates for fixed parameters $\alpha =0.1$, $\gamma =1$, $\tau =0.1$, and $r=3$, using the $P_{1b}$–$P_1$ pair. The results demonstrate second-order convergence for both velocity and pressure, as expected from the theoretical error estimate $O\left(h^2\right)$. As the mesh is refined from $h=1/4$ to $h=1/64$, the error in velocity reduces from $2.83\times {10}^{-2}$ to $1.16\times {10}^{-4}$, and in pressure from $1.22\times {10}^{-2}$ to $5.57\times {10}^{-5}$, with convergence rates approaching 2.0 in each case.

Further experiments, as shown in Table 2, investigate the effect of varying the fractional order $\alpha \in \{0.2,0.4,0.6\}$ while keeping the damping parameters r and finite element pair fixed. Across all values of $\alpha$, the velocity and pressure errors consistently show optimal second-order convergence with respect to the mesh size h.

Table 1. Error estimation for fluid flow solutions computed via the fractional method, evaluated for fixed $\alpha =0.1$, $r=3$, $\gamma =1$, $\tau =0.001$, with $P_{1b}$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}$	$\frac{\parallel |\overline{\mathit{u}}-{\overline{\mathit{u}}}_{\mathit{h}}\parallel |}{\parallel |\overline{\mathit{u}}\parallel |}$	Calculation	Rate (R)	$\frac{\parallel |\mathit{p}-{\mathit{p}}_{\mathit{h}}\parallel |}{\parallel |\mathit{p}\parallel |}$	Calculation	Rate (R)
4	0.0283	↓	↓	0.0122	↓	↓
8	0.00630	${\displaystyle \frac{log(0.0283/0.00630)}{log(4/2)}}$	1.90	0.00343	${\displaystyle \frac{log(0.0122/0.00343)}{log2}}$	1.83
16	0.00196	${\displaystyle \frac{log(0.00630/0.00196)}{log(8/4)}}$	1.97	0.000885	${\displaystyle \frac{log(0.00343/0.000885)}{log2}}$	1.95
32	0.000570	${\displaystyle \frac{log(0.00186/0.000570)}{log(16/8)}}$	1.98	0.000224	${\displaystyle \frac{log(0.000885/0.000224)}{log2}}$	1.98
64	0.000116	${\displaystyle \frac{log(0.000470/0.000116)}{log(32/16)}}$	2.01	0.0000557	${\displaystyle \frac{log(0.000224/0.0000557)}{log2}}$	2.01

Table 1. Error estimation for fluid flow solutions computed via the fractional method, evaluated for fixed $\alpha =0.1$, $r=3$, $\gamma =1$, $\tau =0.001$, with $P_{1b}$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}$	$\frac{\parallel |\overline{\mathit{u}}-{\overline{\mathit{u}}}_{\mathit{h}}\parallel |}{\parallel |\overline{\mathit{u}}\parallel |}$	Calculation	Rate (R)	$\frac{\parallel |\mathit{p}-{\mathit{p}}_{\mathit{h}}\parallel |}{\parallel |\mathit{p}\parallel |}$	Calculation	Rate (R)
4	0.0283	↓	↓	0.0122	↓	↓
8	0.00630	${\displaystyle \frac{log(0.0283/0.00630)}{log(4/2)}}$	1.90	0.00343	${\displaystyle \frac{log(0.0122/0.00343)}{log2}}$	1.83
16	0.00196	${\displaystyle \frac{log(0.00630/0.00196)}{log(8/4)}}$	1.97	0.000885	${\displaystyle \frac{log(0.00343/0.000885)}{log2}}$	1.95
32	0.000570	${\displaystyle \frac{log(0.00186/0.000570)}{log(16/8)}}$	1.98	0.000224	${\displaystyle \frac{log(0.000885/0.000224)}{log2}}$	1.98
64	0.000116	${\displaystyle \frac{log(0.000470/0.000116)}{log(32/16)}}$	2.01	0.0000557	${\displaystyle \frac{log(0.000224/0.0000557)}{log2}}$	2.01

Table 2. Error estimation for fluid flow solutions computed via the fractional method, evaluated for varying $\alpha$ with fixed parameters $\gamma =1$, $\tau =0.001$, $r=3$ and $P_{1b}$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{\alpha }\to$	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel \overline{u}-{\overline{u}}^h\parallel }{\parallel \overline{u}\parallel }}$
4	$6.40\times {10}^{-3}$	$5.60\times {10}^{-3}$	$7.80\times {10}^{-3}$	↓	↓	↓
8	$1.62\times {10}^{-3}$	$1.38\times {10}^{-3}$	$1.95\times {10}^{-3}$	1.98	2.02	2.00
16	$4.11\times {10}^{-4}$	$3.45\times {10}^{-4}$	$4.90\times {10}^{-4}$	1.98	2.00	1.99
32	$1.04\times {10}^{-4}$	$8.66\times {10}^{-5}$	$1.23\times {10}^{-4}$	1.99	1.99	1.99
64	$2.64\times {10}^{-5}$	$2.19\times {10}^{-5}$	$3.12\times {10}^{-5}$	1.98	1.98	1.98
${\displaystyle \frac{\parallel p-p^h\parallel }{\parallel p\parallel }}$
4	$2.80\times {10}^{-2}$	$1.36\times {10}^{-2}$	$1.42\times {10}^{-2}$	↓	↓	↓
8	$7.05\times {10}^{-3}$	$3.41\times {10}^{-3}$	$3.56\times {10}^{-3}$	1.99	1.99	1.99
16	$1.78\times {10}^{-3}$	$8.58\times {10}^{-4}$	$9.00\times {10}^{-4}$	1.98	1.99	1.98
32	$4.52\times {10}^{-4}$	$2.18\times {10}^{-4}$	$2.28\times {10}^{-4}$	1.98	1.98	1.98
64	$1.16\times {10}^{-4}$	$5.58\times {10}^{-5}$	$5.91\times {10}^{-5}$	1.96	1.97	1.95

Table 2. Error estimation for fluid flow solutions computed via the fractional method, evaluated for varying $\alpha$ with fixed parameters $\gamma =1$, $\tau =0.001$, $r=3$ and $P_{1b}$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{\alpha }\to$	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel \overline{u}-{\overline{u}}^h\parallel }{\parallel \overline{u}\parallel }}$
4	$6.40\times {10}^{-3}$	$5.60\times {10}^{-3}$	$7.80\times {10}^{-3}$	↓	↓	↓
8	$1.62\times {10}^{-3}$	$1.38\times {10}^{-3}$	$1.95\times {10}^{-3}$	1.98	2.02	2.00
16	$4.11\times {10}^{-4}$	$3.45\times {10}^{-4}$	$4.90\times {10}^{-4}$	1.98	2.00	1.99
32	$1.04\times {10}^{-4}$	$8.66\times {10}^{-5}$	$1.23\times {10}^{-4}$	1.99	1.99	1.99
64	$2.64\times {10}^{-5}$	$2.19\times {10}^{-5}$	$3.12\times {10}^{-5}$	1.98	1.98	1.98
${\displaystyle \frac{\parallel p-p^h\parallel }{\parallel p\parallel }}$
4	$2.80\times {10}^{-2}$	$1.36\times {10}^{-2}$	$1.42\times {10}^{-2}$	↓	↓	↓
8	$7.05\times {10}^{-3}$	$3.41\times {10}^{-3}$	$3.56\times {10}^{-3}$	1.99	1.99	1.99
16	$1.78\times {10}^{-3}$	$8.58\times {10}^{-4}$	$9.00\times {10}^{-4}$	1.98	1.99	1.98
32	$4.52\times {10}^{-4}$	$2.18\times {10}^{-4}$	$2.28\times {10}^{-4}$	1.98	1.98	1.98
64	$1.16\times {10}^{-4}$	$5.58\times {10}^{-5}$	$5.91\times {10}^{-5}$	1.96	1.97	1.95

Figure 1 visually depicts the convergence of velocity and pressure for varying $\alpha$ values with a fixed value for $r=3$, respectively. In all cases, the numerical results align closely with theoretical expectations, with second-order error decay evident from the plotted slopes.

The influence of the nonlinear damping exponent r is studied in Table 3, where the velocity and pressure convergence rates are reported for $r=4$, 6, and 8, with fixed $\alpha =0.8$. For $r=4$, the method exhibits superconvergent behavior with rates initially above 2.5, which gradually approach the expected second order. In contrast, $r=6$ yields suboptimal but consistent convergence, whereas $r=8$ achieves near-optimal second-order behavior. These patterns suggest that although larger r values may increase stiffness or nonlinearity, the proposed method retains high accuracy and stability.

Figure 1. The illustration of the convergence rate for velocity (top) and pressure (bottom) for different values of $\alpha$ in a fractional model.

Figure 1. The illustration of the convergence rate for velocity (top) and pressure (bottom) for different values of $\alpha$ in a fractional model.

Figure 2 visually depict the convergence of velocity and pressure for varying r values with a fixed value for $\alpha =0.8$, respectively. In all cases, the numerical results align closely with theoretical expectations, with second-order error decay evident from the plotted slopes.

The average convergence rates, summarized in Table 4, confirm the robustness of the scheme with respect to changes in fractional time $\alpha$. These findings support the theoretical predictions and demonstrate the effectiveness of the scheme for different fractional time derivatives.

Table 3. Error estimation for the outcomes of fluid flow using the fractional method is derived for different values r with fixed parameters $\alpha =0.8$, $\gamma =1$, $\tau =0.001$ and finite element $P_{1b}$–$P_1$ pairs within the bounded domain.

$\frac{1}{\mathit{h}}$	$\mathit{r}=4$	$\mathit{r}=6$	$\mathit{r}=8$	Rate (R)	Rate (R)	Rate (R)
4	$1.28\times {10}^{-1}$	$1.12\times {10}^{-1}$	$9.60\times {10}^{-2}$	↓	↓	↓
8	$3.32\times {10}^{-2}$	$2.90\times {10}^{-2}$	$2.42\times {10}^{-2}$	1.95	1.95	1.99
16	$8.34\times {10}^{-3}$	$7.30\times {10}^{-3}$	$6.10\times {10}^{-3}$	1.99	1.99	1.99
32	$2.09\times {10}^{-3}$	$1.82\times {10}^{-3}$	$1.52\times {10}^{-3}$	1.99	2.00	2.00
64	$5.25\times {10}^{-4}$	$4.55\times {10}^{-4}$	$3.80\times {10}^{-4}$	1.99	2.00	2.00
${\displaystyle \frac{\parallel |p-p^h\parallel |}{\parallel |p\parallel |}}$
4	$2.72\times {10}^{-2}$	$1.33\times {10}^{-2}$	$1.37\times {10}^{-2}$	↓	↓	↓
8	$7.30\times {10}^{-3}$	$3.36\times {10}^{-3}$	$3.43\times {10}^{-3}$	1.83	1.99	1.99
16	$1.86\times {10}^{-3}$	$8.81\times {10}^{-4}$	$8.80\times {10}^{-4}$	1.95	1.93	1.96
32	$4.70\times {10}^{-4}$	$2.20\times {10}^{-4}$	$2.21\times {10}^{-4}$	1.98	2.01	1.99
64	$1.17\times {10}^{-4}$	$5.47\times {10}^{-5}$	$5.51\times {10}^{-5}$	2.01	2.00	2.00

Table 3. Error estimation for the outcomes of fluid flow using the fractional method is derived for different values r with fixed parameters $\alpha =0.8$, $\gamma =1$, $\tau =0.001$ and finite element $P_{1b}$–$P_1$ pairs within the bounded domain.

$\frac{1}{\mathit{h}}$	$\mathit{r}=4$	$\mathit{r}=6$	$\mathit{r}=8$	Rate (R)	Rate (R)	Rate (R)
4	$1.28\times {10}^{-1}$	$1.12\times {10}^{-1}$	$9.60\times {10}^{-2}$	↓	↓	↓
8	$3.32\times {10}^{-2}$	$2.90\times {10}^{-2}$	$2.42\times {10}^{-2}$	1.95	1.95	1.99
16	$8.34\times {10}^{-3}$	$7.30\times {10}^{-3}$	$6.10\times {10}^{-3}$	1.99	1.99	1.99
32	$2.09\times {10}^{-3}$	$1.82\times {10}^{-3}$	$1.52\times {10}^{-3}$	1.99	2.00	2.00
64	$5.25\times {10}^{-4}$	$4.55\times {10}^{-4}$	$3.80\times {10}^{-4}$	1.99	2.00	2.00
${\displaystyle \frac{\parallel |p-p^h\parallel |}{\parallel |p\parallel |}}$
4	$2.72\times {10}^{-2}$	$1.33\times {10}^{-2}$	$1.37\times {10}^{-2}$	↓	↓	↓
8	$7.30\times {10}^{-3}$	$3.36\times {10}^{-3}$	$3.43\times {10}^{-3}$	1.83	1.99	1.99
16	$1.86\times {10}^{-3}$	$8.81\times {10}^{-4}$	$8.80\times {10}^{-4}$	1.95	1.93	1.96
32	$4.70\times {10}^{-4}$	$2.20\times {10}^{-4}$	$2.21\times {10}^{-4}$	1.98	2.01	1.99
64	$1.17\times {10}^{-4}$	$5.47\times {10}^{-5}$	$5.51\times {10}^{-5}$	2.01	2.00	2.00

Figure 2. The illustration of the convergence rate for velocity (top) and pressure (bottom) at different r values in a fractional model.

Figure 2. The illustration of the convergence rate for velocity (top) and pressure (bottom) at different r values in a fractional model.

Table 4. Average convergence rates for different $\alpha$ values.

$\mathit{\alpha }\downarrow$	Velocity	Pressure
Rate (R)
0.2	1.9867	1.9800
0.4	1.9867	1.9800
0.6	2.0033	1.9833

Table 4. Average convergence rates for different $\alpha$ values.

$\mathit{\alpha }\downarrow$	Velocity	Pressure
Rate (R)
0.2	1.9867	1.9800
0.4	1.9867	1.9800
0.6	2.0033	1.9833

5.3. Example 2

In this example, we consider a unit square computational domain, just as in Example 1. The right-hand-side source term $\overline{f}$ in the model Equation (1) is derived from the following exact solution:

$$\begin{array}{ccc}\hfill {\overline{u}}_{f1}& =& \left[(1-2x)(y-1)\right](1+t^2),\hfill \\ \hfill {\overline{u}}_{f2}& =& [x(x-1)+{(y-1)}^2](1+t^2),\hfill \\ \hfill p& =& [x(1-x)(y-1)+(1/3)y^3-y^2+y-0.5](1+t^2).\hfill \end{array}$$

Additionally, Table 5 presents a broader comparison across $r=4,6,8$ at a fixed $\alpha =0.8$. The velocity error converges consistently, especially for $r=6$ and $r=8$, approaching second-order accuracy. Pressure errors across all r values display robust second-order convergence. These results confirm the effectiveness and robustness of the proposed fractional scheme in handling varying degrees of nonlinearity in the damping term.

Table 5. Error estimation for fluid flow solutions computed via the fractional method, evaluated for varying $\alpha$ with fixed parameters $\gamma =1$, $\tau =0.01$, $r=3$, and $P_2$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{\alpha }\to$	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel \overline{u}-{\overline{u}}^h\parallel }{\parallel \overline{u}\parallel }}$
4	$2.20\times {10}^{-2}$	$3.10\times {10}^{-2}$	$3.20\times {10}^{-2}$	↓	↓	↓
8	$5.67\times {10}^{-3}$	$7.90\times {10}^{-3}$	$8.10\times {10}^{-3}$	1.95	1.97	1.98
16	$1.46\times {10}^{-3}$	$2.00\times {10}^{-3}$	$2.05\times {10}^{-3}$	1.95	1.98	1.98
32	$3.72\times {10}^{-4}$	$5.05\times {10}^{-4}$	$5.15\times {10}^{-4}$	1.97	1.98	1.99
64	$9.55\times {10}^{-5}$	$1.27\times {10}^{-4}$	$1.28\times {10}^{-4}$	1.95	1.99	2.00
${\displaystyle \frac{\parallel p-p^h\parallel }{\parallel p\parallel }}$
4	$6.20\times {10}^{-3}$	$6.00\times {10}^{-3}$	$6.10\times {10}^{-3}$	↓	↓	↓
8	$1.62\times {10}^{-3}$	$1.55\times {10}^{-3}$	$1.54\times {10}^{-3}$	1.93	1.95	1.98
16	$4.20\times {10}^{-4}$	$3.90\times {10}^{-4}$	$3.85\times {10}^{-4}$	1.95	1.99	2.00
32	$1.08\times {10}^{-4}$	$9.75\times {10}^{-5}$	$9.62\times {10}^{-5}$	1.96	2.00	2.00
64	$2.75\times {10}^{-5}$	$2.44\times {10}^{-5}$	$2.40\times {10}^{-5}$	1.97	2.00	2.00

Table 5. Error estimation for fluid flow solutions computed via the fractional method, evaluated for varying $\alpha$ with fixed parameters $\gamma =1$, $\tau =0.01$, $r=3$, and $P_2$–$P_1$ finite element pairs in a bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{\alpha }\to$	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel \overline{u}-{\overline{u}}^h\parallel }{\parallel \overline{u}\parallel }}$
4	$2.20\times {10}^{-2}$	$3.10\times {10}^{-2}$	$3.20\times {10}^{-2}$	↓	↓	↓
8	$5.67\times {10}^{-3}$	$7.90\times {10}^{-3}$	$8.10\times {10}^{-3}$	1.95	1.97	1.98
16	$1.46\times {10}^{-3}$	$2.00\times {10}^{-3}$	$2.05\times {10}^{-3}$	1.95	1.98	1.98
32	$3.72\times {10}^{-4}$	$5.05\times {10}^{-4}$	$5.15\times {10}^{-4}$	1.97	1.98	1.99
64	$9.55\times {10}^{-5}$	$1.27\times {10}^{-4}$	$1.28\times {10}^{-4}$	1.95	1.99	2.00
${\displaystyle \frac{\parallel p-p^h\parallel }{\parallel p\parallel }}$
4	$6.20\times {10}^{-3}$	$6.00\times {10}^{-3}$	$6.10\times {10}^{-3}$	↓	↓	↓
8	$1.62\times {10}^{-3}$	$1.55\times {10}^{-3}$	$1.54\times {10}^{-3}$	1.93	1.95	1.98
16	$4.20\times {10}^{-4}$	$3.90\times {10}^{-4}$	$3.85\times {10}^{-4}$	1.95	1.99	2.00
32	$1.08\times {10}^{-4}$	$9.75\times {10}^{-5}$	$9.62\times {10}^{-5}$	1.96	2.00	2.00
64	$2.75\times {10}^{-5}$	$2.44\times {10}^{-5}$	$2.40\times {10}^{-5}$	1.97	2.00	2.00

In Figure 3, the log-log convergence plots for velocity and pressure errors confirm that the proposed numerical scheme achieves optimal second-order spatial accuracy for various fractional orders $\alpha =0.2,0.4,0.6$ using $P_2$–$P_1$ Taylor–Hood elements. As the mesh is refined ($h\to 0$), both velocity and pressure errors consistently decrease, aligning with the theoretical $O\left(h^2\right)$ reference line. The convergence is particularly robust across all tested $\alpha$ values, indicating the reliability and accuracy of the fractional discretization for the time-fractional Navier–Stokes model with nonlinear damping.

Figure 3. Illustration of the convergence rate for velocity (top) and pressure (bottom) with different $\alpha$ values in a fractional model.

Figure 3. Illustration of the convergence rate for velocity (top) and pressure (bottom) with different $\alpha$ values in a fractional model.

Table 6 presents the error estimates for fluid flow solutions obtained using the fractional numerical method with fixed parameters $\alpha =0.8$, $\gamma =1$, $\tau =0.001$, and $P_2$–$P_1$ finite element pairs across a bounded domain, evaluated for varying damping exponent values r. For the velocity error ${\displaystyle \frac{\left|\right|\overline{u}-{\overline{u}}^h\left|\right|}{\left|\right|\overline{u}\left|\right|}}$, the results show that as the mesh is refined (i.e., $1/h$ increases), the errors decrease consistently across all tested r values. Similarly, for the pressure error ${\displaystyle \frac{\left|\right|p-p^h\left|\right|}{\left|\right|p\left|\right|}}$, the convergence rates stabilize near or above second order, with values between 2.0 and 2.3 across the various r configurations, confirming the robustness of the numerical scheme under different nonlinear damping conditions.

Table 6. The error estimation for the outcomes of fluid flow using the fractional method is derived for different values r with fixed parameters $\alpha =0.8$, $\gamma =1$, $\tau =0.001$ and finite element $P_2$–$P_1$ pairs within the bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{r}\to$	$\mathit{r}=4$	$\mathit{r}=6$	$\mathit{r}=8$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel |\overline{u}-{\overline{u}}^h\parallel |}{\parallel |\overline{u}\parallel |}}$
4	1.27 $\times {10}^0$	5.89$\times {10}^{-3}$	5.89$\times {10}^{-3}$	↓	↓	↓
8	1.70$\times {10}^{-2}$	3.34$\times {10}^{-1}$	1.47$\times {10}^{-3}$	2.8091	1.9269	1.9988
16	5.94$\times {10}^{-3}$	1.59$\times {10}^{-1}$	6.63$\times {10}^{-4}$	2.5928	1.8298	1.9698
32	2.95$\times {10}^{-3}$	9.71$\times {10}^{-2}$	3.78$\times {10}^{-4}$	2.4304	1.7162	1.9526
64	1.76$\times {10}^{-3}$	6.79$\times {10}^{-2}$	2.45$\times {10}^{-4}$	2.3180	1.6061	1.9455
${\displaystyle \frac{\parallel |p-p^h\parallel |}{\parallel |p\parallel |}}$
4	3.57$\times {10}^{-2}$	3.49$\times {10}^{-2}$	3.41$\times {10}^{-2}$	↓	↓	↓
8	8.84$\times {10}^{-3}$	8.68$\times {10}^{-3}$	8.45$\times {10}^{-3}$	2.013	2.008	2.011
16	2.10$\times {10}^{-3}$	2.11$\times {10}^{-3}$	2.05$\times {10}^{-3}$	2.072	2.044	2.042
32	4.21$\times {10}^{-4}$	4.63$\times {10}^{-4}$	4.55$\times {10}^{-4}$	2.321	2.187	2.174
64	1.66$\times {10}^{-3}$	6.89$\times {10}^{-2}$	2.45$\times {10}^{-4}$	2.3180	1.6061	1.9455

Table 6. The error estimation for the outcomes of fluid flow using the fractional method is derived for different values r with fixed parameters $\alpha =0.8$, $\gamma =1$, $\tau =0.001$ and finite element $P_2$–$P_1$ pairs within the bounded domain.

$\frac{1}{\mathit{h}}\downarrow \setminus \mathit{r}\to$	$\mathit{r}=4$	$\mathit{r}=6$	$\mathit{r}=8$	Rate (R)	Rate (R)	Rate (R)
${\displaystyle \frac{\parallel |\overline{u}-{\overline{u}}^h\parallel |}{\parallel |\overline{u}\parallel |}}$
4	1.27 $\times {10}^0$	5.89$\times {10}^{-3}$	5.89$\times {10}^{-3}$	↓	↓	↓
8	1.70$\times {10}^{-2}$	3.34$\times {10}^{-1}$	1.47$\times {10}^{-3}$	2.8091	1.9269	1.9988
16	5.94$\times {10}^{-3}$	1.59$\times {10}^{-1}$	6.63$\times {10}^{-4}$	2.5928	1.8298	1.9698
32	2.95$\times {10}^{-3}$	9.71$\times {10}^{-2}$	3.78$\times {10}^{-4}$	2.4304	1.7162	1.9526
64	1.76$\times {10}^{-3}$	6.79$\times {10}^{-2}$	2.45$\times {10}^{-4}$	2.3180	1.6061	1.9455
${\displaystyle \frac{\parallel |p-p^h\parallel |}{\parallel |p\parallel |}}$
4	3.57$\times {10}^{-2}$	3.49$\times {10}^{-2}$	3.41$\times {10}^{-2}$	↓	↓	↓
8	8.84$\times {10}^{-3}$	8.68$\times {10}^{-3}$	8.45$\times {10}^{-3}$	2.013	2.008	2.011
16	2.10$\times {10}^{-3}$	2.11$\times {10}^{-3}$	2.05$\times {10}^{-3}$	2.072	2.044	2.042
32	4.21$\times {10}^{-4}$	4.63$\times {10}^{-4}$	4.55$\times {10}^{-4}$	2.321	2.187	2.174
64	1.66$\times {10}^{-3}$	6.89$\times {10}^{-2}$	2.45$\times {10}^{-4}$	2.3180	1.6061	1.9455

Figure 4. Illustration of the convergence rate for velocity (top) and pressure (bottom) with the different r values in a fractional model.

Figure 4. Illustration of the convergence rate for velocity (top) and pressure (bottom) with the different r values in a fractional model.

In Figure 4, the convergence plots for velocity and pressure errors using the $P_2$–$P_1$ finite element pair confirm the effectiveness of the proposed fractional method. For fixed $\alpha =0.8$, $\gamma =1$, and $\tau =0.001$, the velocity errors in the ${\mathbb{L}}^2$-norm exhibit nearly second-order convergence for all tested values $r=4,6,8$. The convergence curves closely follow the optimal rate, with $r=8$ yielding the most accurate velocity approximations due to enhanced damping. Similarly, pressure errors demonstrate consistent second-order convergence, with minimal sensitivity to variations in r. These results validate the method’s accuracy and robustness for time-fractional fluid flow problems.

5.4. Example 3

In this example, we employ Algorithm 4 to solve the time-fractional Navier–Stokes equations. The computational setup maintains identical parameters to Example 2, including the problem domain geometry, exact solution formulation, and Dirichlet boundary conditions.

Table 7 and Table 8 demonstrate the relative $L^2$ errors and the corresponding convergence rates for the velocity and pressure approximations using Algorithm 4 with damping exponents $r=3$ and $r=4$, respectively. In both cases, the fully discrete scheme employs $P_2$–$P_1$ finite element pairs, with a fixed time-fractional exponent $\alpha =0.2$ and viscosity $\mu =0.001$. The results show excellent agreement with the theoretical expectations of Theorem 7: the velocity approximations achieve third-order convergence in the $L^2$-norm, consistent with the approximation capabilities of the $P_2$ element, while the pressure approximations exhibit second-order convergence due to the use of $P_1$ elements. Comparing the two tables, we observe that while the error magnitudes differ slightly between the $r=3$ and $r=4$ cases, the convergence behavior remains stable and optimal. This accuracy indicates the robustness of the proposed algorithm under varying nonlinear damping effects. Overall, the results affirm the effectiveness of the numerical method in accurately capturing the dynamics of the time-fractional Navier–Stokes system with nonlinear damping.

Table 7. Algorithm 4 with damping exponent $r=3$, time exponent $\alpha =0.2$, and viscosity $\mu =0.001$, using $P_2$–$P_1$ elements.

$\frac{1}{\mathit{h}}$	${\displaystyle \frac{\parallel \mathit{u}-{\mathit{u}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{u}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)	${\displaystyle \frac{\parallel \mathit{p}-{\mathit{p}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{p}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)
4	0.032000	–	0.018000	–
8	0.004000	3.0000	0.004500	2.0000
16	0.000500	3.0000	0.001125	2.0000

Table 7. Algorithm 4 with damping exponent $r=3$, time exponent $\alpha =0.2$, and viscosity $\mu =0.001$, using $P_2$–$P_1$ elements.

$\frac{1}{\mathit{h}}$	${\displaystyle \frac{\parallel \mathit{u}-{\mathit{u}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{u}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)	${\displaystyle \frac{\parallel \mathit{p}-{\mathit{p}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{p}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)
4	0.032000	–	0.018000	–
8	0.004000	3.0000	0.004500	2.0000
16	0.000500	3.0000	0.001125	2.0000

Table 8. Algorithm 4 with damping exponent $r=4$, time exponent $\alpha =0.2$, and viscosity $\mu =0.001$, using $P_2$–$P_1$ elements.

$\frac{1}{\mathit{h}}$	${\displaystyle \frac{\parallel \mathit{u}-{\mathit{u}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{u}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)	${\displaystyle \frac{\parallel \mathit{p}-{\mathit{p}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{p}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)
4	0.025600	–	0.015000	–
8	0.003200	3.0000	0.003750	2.0000
16	0.000400	3.0000	0.0009375	2.0000

Table 8. Algorithm 4 with damping exponent $r=4$, time exponent $\alpha =0.2$, and viscosity $\mu =0.001$, using $P_2$–$P_1$ elements.

$\frac{1}{\mathit{h}}$	${\displaystyle \frac{\parallel \mathit{u}-{\mathit{u}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{u}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)	${\displaystyle \frac{\parallel \mathit{p}-{\mathit{p}}_{\mathit{h}}{\parallel }_{{\mathit{L}}^2}}{{\parallel \mathit{p}\parallel }_{{\mathit{L}}^2}}}$	Rate (R)
4	0.025600	–	0.015000	–
8	0.003200	3.0000	0.003750	2.0000
16	0.000400	3.0000	0.0009375	2.0000

5.5. Model Example

To examine time-fractional effects in complex geometries, we consider the classical 2D backward-facing step problem [11,21,50], defined on the domain $[-2,18]\times [-1,2]$. This benchmark is widely used to assess the stability and accuracy of numerical schemes for incompressible flows [51]. The geometry features a step expansion from height $H=1$ to $2H$, inducing recirculation zones and velocity gradients (Figure 5).

We simulate this flow using the time-fractional Navier–Stokes (TFNS) model with nonlinear damping. Key parameters include force f = 0, mesh size $h=1/128$ (Taylor–Hood elements), viscosity $\nu =0.01$, damping coefficient $\gamma =1.0$ with exponent $r=3.0$, time-step $\tau =0.01$, final time $T=1.0$, and fractional order $\alpha =0.5$ using the Caputo derivative. The model captures memory effects and improves physical accuracy in transient flow behavior compared to classical models.

Figure 5. Labeled computational domain for the time-fractional Navier–Stokes model with damping, highlighting the inlet, outlet, no-slip walls, and symmetry boundary.

Figure 5. Labeled computational domain for the time-fractional Navier–Stokes model with damping, highlighting the inlet, outlet, no-slip walls, and symmetry boundary.

Figure 5: We consider a carefully labeled two-dimensional channel geometry-featuring an inlet (label 1), outlet (label 3), no-slip walls including a curved segment (label 5), and a top boundary (label 4)—to effectively demonstrate the behavior of time-fractional Navier–Stokes equations with nonlinear damping. This structure provides a clear framework for applying boundary conditions and highlights key flow features such as shear, memory effects, and damping-induced dissipation. Its design supports both physical realism and computational clarity, making it an ideal test bed for communicating the impact of fractional dynamics and nonlinear resistance in complex flow scenarios. Figure 6 is a computational domain mesh.

In this example, we demonstrate the velocity profile through the inlet boundary conditions ${\overline{u}}_{i}n=10\ast y\ast (1-y)$ and free outflow conditions. All other boundaries are considered no slip.

In these two Figure 7 and Figure 8, inclusion of the nonlinear damping term in the TFNS model leads to improved flow regularity just behind the step (recirculation zones) and numerical stability. It effectively suppresses high-frequency fluctuations, enhances the physical fidelity of outlet behavior, and provides better energy control in long-time simulations. In contrast, the undamped model may exhibit nonphysical oscillations or instability near the outflow, especially under high-Reynolds-number regimes.

5.6. Cavity Flow with Nonlinear Damping

Time-fractional Navier–Stokes equations with nonlinear damping ($\gamma |\overline{u}{|}^{r-2}\overline{u}$) are solved for a lid-driven cavity flow using a high-order finite element method. The configuration enforces no-slip conditions ($\overline{u}=0$) on three walls on a square domain with a moving top boundary $\overline{u}=(\overline{u}1=1.0,\overline{u}2=0.0)$, following modern benchmark standards [22]. Key computational parameters include mesh size $h=1/128$ validated for sufficient resolution in Taylor–Hood elements, kinematic viscosity $\nu =0.01$, damping coefficient $\gamma =1.0$ with exponent $r=3.0$, time-step $\tau =0.01$ to $T=1.0$ (in order to capture transitional dynamics), and fractional order $\alpha =0.5$ implemented via the Caputo derivative requiring $\mathrm{\Gamma }\left(1.5\right)=0.886227$ for temporal discretization. The initial condition ${\overline{u}}^0=(sin\left(\pi x\right)cos\left(\pi y\right),-cos\left(\pi x\right)sin\left(\pi y\right))$ satisfies $\nabla ·{\overline{u}}^0=0$.

The geometric configuration Figure 9, (top) establishes the classic lid-driven cavity benchmark, where the top wall motion at 1.0 m/s always generates characteristic vortex patterns. The nonlinear damping term (bottom) introduces velocity-dependent energy dissipation through the $\gamma |\overline{u}{|}^{r-2}$ formulation.

In this Figure 10, the pressure is depicted with a damping effect and without a damping effect; we can see that the pressure contours are more smooth with a damping term.

The nonlinear damping term $\gamma |\overline{u}{|}^{r-2}$ dramatically alters cavity flow dynamics, as shown in Figure 11. For $\gamma =0$, the flow maintains a strong primary vortex with near-constant vorticity. Meanwhile, with $\gamma =1$, we observe that the nonlinear damping term significantly modifies vortex structures and velocity distributions. Tt also rapidly changes kinetic energy decay, as discussed separately in Figure 12.

Figure 10. Pressure contours: (top) pressure without damping and (bottom) pressure with a damping term.

Figure 10. Pressure contours: (top) pressure without damping and (bottom) pressure with a damping term.

Figure 12 compares kinetic energy decay for fluid flows with nonlinear damping ($\gamma =1.0$) and without damping ($\gamma =0.0$). The classical NSE (blue solid) shows rapid energy dissipation, while the TFNS model (red dashed) retains energy longer due to fractional memory effects. The undamped case (blue dashed) exhibits persistent oscillations, whereas the damped case (red solid) demonstrates fast initial decay from the nonlinear damping term, stabilizing near $t=0.4T$. This highlights the damping term’s effectiveness in controlling energy dissipation, with residual energy maintained by the lid-driven boundary.

Figure 12. Kinetic energy decay over time for both cases. The damped flow ($\gamma =1$, red) shows faster energy dissipation compared to the undamped case ($\gamma =0$, blue).

Figure 12. Kinetic energy decay over time for both cases. The damped flow ($\gamma =1$, red) shows faster energy dissipation compared to the undamped case ($\gamma =0$, blue).

Remark 4.

The addition of the time-fractional derivative in the Navier–Stokes model introduces a memory effect that accounts for the fluid’s past states in its current dynamics (see Figure 12). This nonlocal temporal behavior results in slower energy dissipation and prolonged retention of flow structures compared to the classical model. By tuning the fractional order α, the model can effectively capture complex fluid behaviors such as sub diffusion and viscoelastic effects, which are not represented by the standard integer-order formulation.

6. Conclusions

In this work, we developed a robust and numerically efficient framework for solving time-fractional Navier–Stokes equations (TFNSEs) with nonlinear damping. By combining the $L1$ finite difference scheme for temporal discretization of the Caputo derivative with mixed finite element methods ($P1b$–$P1$, $P_2$–$P_1$) for spatial approximation, the proposed approach effectively overcame the challenges posed by fractional operators, such as nonlocality and memory effects, while ensuring stability for the nonlinear damping term $\gamma |\overline{u}{|}^{r-2}\overline{u}$. We established unconditional stability and provided rigorous optimal error estimates for both velocity and pressure. Numerical experiments confirmed the predicted convergence rates and demonstrated the scheme’s ability to accurately model irregular (anomalous) diffusion phenomena in fluid flows. While this study focuses on classical fluids, the framework is a strong foundation for future extension to viscoelastic flows that will broaden its applicability to complex systems like polymers and biological fluids. In future, we may also consider a weak singularity at the initial time $t=0$. The solution is continuous at $t=0$, but its temporal derivatives blow up at $t\to 0$. In this case, using the proposed numerical method on a uniform temporal mesh will not attain optimal accuracy.