Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model

Abstract

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

1. Introduction

The coupled Stokes–Darcy model has gained considerable attention due to its ability to describe interactions between free fluid flow and flows in a porous medium. Important examples include many engineering, biomedical and environmental problems such as surface–subsurface filtration transport of coastal aquifers [1], blood flow [2], reservoir engineering, flooding [3], and others. However, due to the complex behavior of these processes, some parameters in the governing equations may remain uncertain.

In the last decade, the stochastic Stokes–Darcy model has received increased interest to account for uncertainty and variability inherent in flows in porous media [4,5,6,7,8,9]. The challenge mainly arises from the difficulty in reliably determining the properties of a porous medium, such as hydraulic conductivity and porosity. By accounting for stochastic effects, the model better captures the spectrum of possible outcomes in porous media flow scenarios, thereby improving predictions and decision-making processes.

Recently, a new fractional-order generalization of the stochastic Stokes–Darcy model has been proposed [10] to capture hereditary properties of fluid flow process in both free flow domains and a porous medium. This phenomenon is referred to as memory, where previous states have an impact on the current flow. An important application of such generalization may pertain to intra- and extravascular blood flow where, specifically, intravascular flow adheres to the Stokes equation, while extravascular flow is governed by the Darcy equation [2]. The use of fractional derivatives is due to the complexity of blood flow phenomena as reported in recent studies [11,12].

There are numerous studies that confirm the effectiveness of applying fractional calculus to both Stokes [13,14,15] and Darcy models [16,17,18,19]. On top of that, recent studies report obtaining more realistic simulation results for various processes compared to classical models in such areas as epidemic propagation [20,21,22], blood flow [23,24,25], transport of drugs across biological materials [26], sound and shallow water wave propagation [27,28,29,30], complex multiphase flows in heterogeneous porous media [31,32,33], soil pollution [34], atmospheric processes [35,36], image and signal processing [37,38], economic processes [39], and many others [40].

Obtaining an analytical solution to the stochastic Stokes–Darcy equations is often infeasible due to their complexity. Consequently, numerical methods serve as the primary approach for solving these equations. In the scientific literature, various methods have been explored for implementing the stochastic Stokes–Darcy model such as the Monte Carlo method [5], the stochastic collocation method [4], the multigrid multilevel Monte Carlo method [8,41], and the mixed multiscale finite element method [42]. On top of that, the ensemble approach [5,6] allows solving Stokes–Darcy equations for all samples simultaneously, which considerably reduces computational costs. As for fractional-order stochastic Stokes–Darcy equations, the original paper [10] proposed a higher-order numerical scheme with the use of an ensemble-based approach and implemented a Monte Carlo method for uncertainty quantification. However, the authors of the work came to the conclusion that computing the discrete fractional derivative was the most resource-intensive procedure when implementing the algorithm.

Numerous studies have been devoted to the numerical solution of time-fractional stochastic equations, including the heat equation [43,44,45], the wave equation [46,47], the Korteweg–de Vries equation [48], the sine-Gordon equation [49], the gas dynamics equations [50], and the Navier–Stokes equations [51]. These contributions utilized various numerical methods, including the finite difference method [45], the Galerkin finite element method [43,46,51], a spectral collocation method [44], the implicit meshless method [48,49], and the conformable finite difference method [50].

This paper aims at efficient numerical implementation of the fractional-order stochastic Stokes–Darcy model proposed in [10]. The construction of the numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, fast numerical algorithm for the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the combined use of a reduced Karhunen–Loève expansion and a lognormal field. Further, a higher-order deterministic numerical method has been constructed, and its stability and convergence have been rigorously proved and confirmed by numerical tests. Finally, applying the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. This approach significantly reduced the computational cost and memory usage.

This paper has several key differences from previous studies. Firstly, we analyze the influence of fractional derivatives on the fluid flow process within the Stokes–Darcy model under uncertainty conditions. Despite existing research on the impact of fractional derivatives on fluid flow within the Darcy framework [16,17,18,19], this study specifically investigates their effects within the coupled Stokes–Darcy model. In our opinion, this is the first study devoted to such an analysis within the model that accounts for both memory and stochasticity effects. To this end, we compare the results of solving a few stochastic test problems with different orders of fractional derivatives as well as an integer-order derivative.

Secondly, we employ a fast implementation of the high-order discrete fractional derivative proposed in [52]. Compared to [10], this approach significantly reduced computation time and memory usage since it requires storing the solution only on the three latest time layers. This advantage is especially evident when solving stochastic problems due to the need to repeatedly solve the problem with different input data.

Thirdly, in contrast to the original paper [10], we opt for the stochastic collocation method over the Monte Carlo method due to its faster convergence.

The structure of the paper is as follows. The next section focuses on the description of the fractional-order stochastic Stokes–Darcy model. Section 2.2 briefly describes the sparse grid stochastic collocation method. Section 2.3 provides the deterministic numerical method and discusses its stability and convergence. Section 3 presents a few illustrative examples to confirm the results of theoretical analysis. In particular, Section 3.1 is dedicated to the verification of the convergence order on a model problem with a known exact solution, and Section 3.2 provides an analysis of the influence of fractional derivatives on the flow process in different scenarios. Section 3.3 presents some CPU time benchmarks and an efficiency comparison with previous studies. Finally, Section 4 discusses the results obtained and gives some concluding remarks.

2. Materials and Methods

2.1. Fractional-Order Stochastic Generalization of the Stokes–Darcy Model

To formulate the problem, let us denote by $D_f$ and $D_p$ two disjoint domains in ${\mathbb{R}}^d$, $d=2,3$ separated by an interface $I={\overline{D}}_f\cap {\overline{D}}_p$ (Figure 1). It is assumed that the fluid flow process in a porous medium $D_p$ is described by a fractional-order generalization of the continuity equation and linear Darcy law:

$$\begin{array}{cc}\hfill & S{\partial }_{0,t}^{\gamma }\varphi +\nabla ·\mathbf{w}=f_p,\gamma \in \left(0,1\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \mathbf{w}=-\mathbb{K}\nabla \varphi ,\hfill \end{array}$$

(2)

where ${\displaystyle \varphi =z+\frac{p_{\mathrm{dyn}}}{\rho g}}$ denotes the piezometric head, $p_{\mathrm{dyn}}$ is the dynamic pressure, z is height, $\rho$ is the density, g is the gravity constant, S is the water accumulation coefficient, $\mathbf{w}$ is flow velocity, $\mathbb{K}$ is the hydraulic conductivity tensor, $f_p$ is the source or sink term, and the temporal fractional derivative in the sense of Caputo is defined as

$${\partial }_{0,t}^{\gamma }f\left(·,t\right)={\displaystyle \frac{1}{\Gamma \left(1-\gamma \right)}}{\int }_0^t{\displaystyle \frac{{\partial }_{t}f\left(·,\theta \right)}{{\left(t-\theta \right)}^{\gamma }}}d\theta ,\gamma \in \left(0,1\right),t\in J,$$

(3)

$J=\left(0,T\right]$ is the time interval in which the fluid flow process is studied.

The inclusion of a fractional derivative in the continuity equation is due to the intention of taking into account the hereditary properties of the porous medium, known as memory. This is based on the assumption that the dynamics of fluid flow in a porous medium are influenced not only by the current state of the process but also by its previous states. We refer the reader to [17,18,19,53] and references therein for detailed information.

Further, we assume that a free fluid flow occurs in $D_f$, which is described by a fractional-order generalization of the non-stationary Stokes equations:

$$\begin{array}{cc}& {\partial }_{0,t}^{\gamma }\mathbf{u}-\nabla ·\mathbb{T}\left(\mathbf{u},p\right)={\mathbf{f}}_f,\hfill \end{array}$$

(4)

$$\begin{array}{cc}& \nabla ·\mathbf{u}=0,\hfill \end{array}$$

(5)

which are obtained by formally replacing the integer-order time derivative with the fractional-order derivative of order $\gamma$, where $\mathbf{u}$ and p are the velocity and kinematic pressure of the fluid, $\mathbb{T}\left(\mathbf{u},p\right)=-p\mathbb{I}+2\mu \mathbb{D}\left(\mathbf{u}\right)$ is the stress tensor, $\mathbb{D}\left(\mathbf{u}\right)={\displaystyle \frac{1}{2}}\left(\nabla \mathbf{u}+\nabla {\mathbf{u}}^{\top }\right)$ is the strain tensor, $\mu$ is the kinematic viscosity of the fluid, and ${\mathbf{f}}_f$ is the source or sink term. The inclusion of a fractional derivative in the Stokes equation has been the subject of previous studies [13,14,15].

The presented equations can be considered as a generalization of the classical Stokes–Darcy equations taking into account long-term changes in a porous medium and the process as a whole. For example, an important application of the combined fractional-order Equations (1)–(5) may be linked to understanding the complex process of blood circulation, where intravascular flow is governed by the fractional-order Stokes equation, whereas extravascular flow is subject to the fractional-order Darcy equation.

The following boundary conditions are imposed at the outer boundaries ${\Gamma }_f=\partial D_f\setminus I$ and ${\Gamma }_p=\partial D_p\setminus I$, where ${\Gamma }_p={\Gamma }_{pd}\cup {\Gamma }_{pn}$ and ${\Gamma }_{pd}\cap {\Gamma }_{pn}=\varnothing$:

$$\begin{array}{cccc}& \mathbf{u}=\mathbf{0}\hfill & \hfill & \mathrm{on}{\Gamma }_f\times J,\hfill \\ & \varphi ={\varphi }_d\hfill & & \mathrm{on}{\Gamma }_{pd}\times J,\hfill \\ & -\mathbb{K}\nabla \varphi ·{\mathbf{n}}_p=0\hfill & & \mathrm{on}{\Gamma }_{pn}\times J,\hfill \end{array}$$

where ${\mathbf{n}}_p$ is the outer unit normal to the boundary $\partial D_p$, and the following conditions are imposed on the interface I:

$$\begin{array}{cccc}& \mathbf{u}·{\mathbf{n}}_f-\mathbb{K}\nabla \varphi ·{\mathbf{n}}_p=0\hfill & \hfill & \mathrm{on}I\times J,\hfill \end{array}$$

(6)

$$\begin{array}{cccc}& -{\mathbf{n}}_f·\mathbb{T}\left(\mathbf{u},p\right)·{\mathbf{n}}_f=g\varphi \hfill & & \mathrm{on}I\times J,\hfill \end{array}$$

(7)

$$\begin{array}{cccc}& -{\mathbf{m}}_i·\mathbb{T}\left(\mathbf{u},p\right)·{\mathbf{n}}_f={\displaystyle \frac{{\alpha }_{\mathrm{BJS}}}{\sqrt{{\mathbf{m}}_i·\mathbb{K}{\mathbf{m}}_i}}}\mathbf{u}·{\mathbf{m}}_i\hfill & & \mathrm{on}I\times J,1\le i\le d-1,\hfill \end{array}$$

(8)

where ${\alpha }_{\mathrm{BJS}}$ is the dimensionless Beavers–Joseph–Saffman constant, ${\mathbf{n}}_f$ is the outer unit normal to the boundary $\partial D_f$, and $\left\{{\mathbf{m}}_1,{\mathbf{m}}_2,\cdots ,{\mathbf{m}}_{d-1}\right\}$ is the orthonormal system of unit tangent vectors to the interface I.

When studying fluid flow in complex heterogeneous media, a reliable determination of the hydraulic conductivity tensor is quite difficult. Firstly, the variability of hydraulic conductivity in different directions and locations within geological formations leads to difficulties in obtaining consistent and accurate measurements. Further, porous media often contain fractures, faults, and varying rock types, which complicate the flow paths and significantly impact the hydraulic conductivity. Therefore, it is reasonable to assume this tensor to be random. Let $\left(\Omega ,\mathcal{F},\mathcal{P}\right)$ be a complete probability space, where $\Omega$ is the space of elementary events, $\mathcal{F}$ is the sigma-algebra of events, $\mathcal{P}:\mathcal{F}\to \left[0,1\right]$ is a probability measure.

We model the hydraulic conductivity with a lognormal field such that

$$log\left(\mathbb{K}-{\mathbb{K}}_{min}\right)\left(x,\omega \right)=Z\left(x,\omega \right),$$

(9)

where $\omega \in \Omega$ is a Gaussian random field with zero mathematical expectation, $x\in {\overline{D}}_p$. Let us represent the lognormal hydraulic conductivity tensor using the reduced Karhunen–Loève expansion [4]:

$$Z\left(x,\omega \right)=\mathbb{E}\left[Z\right]+\sum _{n=1}^N\sqrt{{\lambda }_n}{v}_n\left(x\right)Y_n\left(\omega \right),N\in \mathbb{N},$$

(10)

where $\mathbb{E}\left[Z\right]$ is the expected value of the random variable Z, ${\left\{Y_n\left(\omega \right)\right\}}_{n=1}^{\infty }$ are pairwise uncorrelated real random variables with zero expectation and unit variance, ${\left\{{\lambda }_n\right\}}_{n=1}^N$ and ${\left\{v_n\right\}}_{n=1}^N$ are its eigenvalues and corresponding eigenvectors. Then the hydraulic conductivity tensor can be represented in the form

$${\mathbb{K}}_N\left(x,\omega \right)={\mathbb{K}}_{min}+exp\left(\mathbb{E}\left[Z\right]+\sum _{n=1}^N\sqrt{{\lambda }_n}{v}_n\left(x\right)Y_n\left(\omega \right)\right),N\in \mathbb{N},$$

(11)

where we explicitly indicate the dependence of $\mathbb{K}$ on N. Thus, the tensor ${\mathbb{K}}_N$ can be described by a random vector $\mathbf{Y}=\left[Y_1\left(\omega \right),\cdots ,Y_N\left(\omega \right)\right]$.

Assuming that the random variable $Y_n\left(\omega \right)$ is bounded, we denote its image $Y_n\left(\Omega \right)$ by $G_n$. Let us define ${\displaystyle G=\prod _{n=1}^N{G}_n}$ [54] and assume that the random vector $\mathbf{Y}$ has a bounded joint probability density function $\rho :G\to {\mathbb{R}}_{+}$, $\rho \in L^{\infty }\left(G\right)$.

The solution $\varphi$ corresponding to the tensor ${\mathbb{K}}_N$ with the chosen dimension of the stochastic space is denoted by

$${\varphi }_N={\varphi }_N\left(x,t,\omega \right)={\varphi }_N\left(x,t,Y_1\left(\omega \right),Y_2\left(\omega \right),\cdots ,Y_N\left(\omega \right)\right).$$

Note that the friction coefficient ${\varsigma }_{N,i}:={\displaystyle \frac{{\alpha }_{\mathrm{BJS}}}{\sqrt{{\mathbf{m}}_i·{\mathbb{K}}_N{\mathbf{m}}_i}}}$ in (8) depends on the hydraulic conductivity tensor along the interface I, so this quantity is also random. Due to the boundary conditions (6)–(8), the equations in the subdomains are strongly coupled; therefore, all unknown quantities $\mathbf{u}$, p, $\varphi$ are random:

$$\begin{array}{cc}& {\mathbf{u}}_N={\mathbf{u}}_N\left(x,t,\omega \right)={\mathbf{u}}_N\left(x,t,Y_1\left(\omega \right),Y_2\left(\omega \right),\cdots ,Y_N\left(\omega \right)\right),\hfill \\ & p_N=p_N\left(x,t,\omega \right)=p_N\left(x,t,Y_1\left(\omega \right),Y_2\left(\omega \right),\cdots ,Y_N\left(\omega \right)\right).\hfill \end{array}$$

As a result, we arrive at the following initial boundary value problem for time-fractional stochastic Stokes–Darcy equations.

Problem 1. 

Find random variables ${\varphi }_N:{\overline{D}}_p\times \overline{J}\times \Omega \to \mathbb{R}$, ${\mathbf{u}}_N:{\overline{D}}_f\times \overline{J}\times \Omega \to {\mathbb{R}}^d$ and $p_N:{\overline{D}}_f\times \overline{J}\times \Omega \to \mathbb{R}$ satisfying the following conditions almost certainly in Ω:

$$\begin{array}{llll}& S{\partial }_{0,t}^{\gamma }{\varphi }_N-\nabla ·\left({\mathbb{K}}_N\nabla {\varphi }_N\right)=f_p\hfill & & \mathrm{in}{D}_p\times J\times \Omega ,\gamma \in \left(0,1\right),\hfill \end{array}$$

(12)

$$\begin{array}{cccc}& {\varphi }_N={\varphi }_d\hfill & & \mathrm{on}{\Gamma }_{pd}\times J\times \Omega ,\hfill \end{array}$$

(13)

$$\begin{array}{cccc}& -{\mathbb{K}}_N\nabla {\varphi }_N·{\mathbf{n}}_p=0\hfill & & \mathrm{on}{\Gamma }_{pn}\times J\times \Omega ,\hfill \\ \hfill \end{array}$$

(14)

$$\begin{array}{cccc}& {\partial }_{0,t}^{\gamma }{\mathbf{u}}_N-\nabla ·\mathbb{T}\left({\mathbf{u}}_N,p_N\right)={\mathbf{f}}_f\hfill & & \mathrm{in}{D}_f\times J\times \Omega ,\hfill \end{array}$$

(15)

$$\begin{array}{cccc}& \nabla ·{\mathbf{u}}_N=0\hfill & & \mathrm{in}{D}_f\times J\times \Omega ,\hfill \end{array}$$

(16)

$$\begin{array}{cccc}& {\mathbf{u}}_N=\mathbf{0}\hfill & & \mathrm{on}{\Gamma }_f\times J\times \Omega ,\hfill \\ \hfill \end{array}$$

(17)

$$\begin{array}{cccc}& {\mathbf{u}}_N·{\mathbf{n}}_f-{\mathbb{K}}_N\nabla {\varphi }_N·{\mathbf{n}}_p=0\hfill & & \mathrm{on}I\times J\times \Omega ,\hfill \end{array}$$

(18)

$$\begin{array}{cccc}& -{\mathbf{n}}_f·\mathbb{T}\left({\mathbf{u}}_N,p_N\right)·{\mathbf{n}}_f=g{\varphi }_N\hfill & & \mathrm{on}I\times J\times \Omega ,\hfill \end{array}$$

(19)

$$\begin{array}{cccc}& -{\mathbf{m}}_i·\mathbb{T}\left({\mathbf{u}}_N,p_N\right)·{\mathbf{n}}_f={\displaystyle \frac{{\alpha }_{\mathrm{BJS}}}{\sqrt{{\mathbf{m}}_i·{\mathbb{K}}_N{\mathbf{m}}_i}}}{\mathbf{u}}_N·{\mathbf{m}}_i\hfill & & \mathrm{on}I\times J\times \Omega ,1\le i\le d-1,\hfill \\ \hfill \end{array}$$

(20)

$$\begin{array}{cccc}& {\mathbf{u}}_N={\mathbf{u}}_0\hfill & & \mathrm{in}{\overline{D}}_f\times \left\{0\right\}\times \Omega ,\hfill \end{array}$$

(21)

$$\begin{array}{cccc}& {\varphi }_N={\varphi }_0\hfill & & \mathrm{in}{\overline{D}}_p\times \left\{0\right\}\times \Omega .\hfill \end{array}$$

(22)

2.2. Sparse Grid Stochastic Collocation Method

To solve stochastic Problem 1, we apply the sparse grid stochastic collocation method. The construction of the method is carried out in several stages [4,54,55].

(1) First, Problem 1 is discretized using a deterministic numerical method. The numerical approach in this study combines a finite element method for spatial analysis with a finite difference approximation with respect to the temporal variable. As a result, we obtain a semi-discrete approximation of the stochastic problem.

(2) Consider a set of points ${\left\{{\mathbf{Y}}^{\left(k\right)}\right\}}_{k=1}^M$, ${\mathbf{Y}}^{\left(k\right)}\in G$, called collocation points, and calculate approximate solutions ${\mathbf{u}}_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)$, $p_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)$, ${\varphi }_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)$, where h represents the discretization parameters. Numerical approximation of the solution $\mathbf{u}$, p, $\varphi$ is sought in a finite-dimensional space $H_{h,\mathcal{L}}=H_h\otimes H_{\mathcal{L}}$, where

(a) $H_h\equiv X_{fh}\times Q_{fh}\times Q_{ph}$ is the finite-dimensional subspace of the space $X_h\times Q_h\times Q_p$ of a semi-discrete solution in time and space variables for a constant $\mathbf{Y}\in G$,

$$\begin{array}{cc}& X_f=\left\{\mathbf{v}\in {\left(H^1\left(D_f\right)\right)}^d|\mathbf{v}=\mathbf{0}\mathrm{on}{\Gamma }_f\right\},\hfill \end{array}$$

(23)

$$\begin{array}{cc}& Q_f=\left\{q\in L^2\left(D_f\right)|{\int }_{D_f}qdx=0\right\},\hfill \end{array}$$

(24)

$$\begin{array}{cc}& Q_p=\left\{\psi \in H^1\left(D_p\right)|\psi =0\mathrm{on}{\Gamma }_{pd}\right\}.\hfill \end{array}$$

(25)

(b) $H_{\mathcal{L}}\left(G\right)\subset L_{\rho }^2\left(G\right)$ is the linear span of orthogonal polynomials of degree at most $\mathbf{P}=\left[P_1\left(\mathcal{L}\right),P_2\left(\mathcal{L}\right),\cdots ,P_N\left(\mathcal{L}\right)\right]$, where $P_n$ is the largest degree of the polynomial in the nth direction, $n=1,2,\cdots ,N$, and $\mathcal{L}$ is a parameter called the level of approximation. For every $Y_n$ characterized by the density function ${\rho }_n$, $n=1,2,\cdots ,N$ we refer to $H_{P_n}\left(G_n\right)$ as the linear space generated by ${\rho }_n$-orthogonal polynomials. Then ${\displaystyle H_{\mathcal{L}}\left(G\right)=\underset{n=1}{\overset{N}{⨂}}{H}_{P_n}\left(G_n\right)}$, and its dimension is ${\displaystyle dim\left(H_{\mathcal{L}}\right)=\prod _{n=1}^N\left(P_n+1\right)}$. As can be seen, an increase in N leads to an exponential growth in dimensionality.

To reduce computational costs, the sparse grid stochastic collocation method involves using a subset of the full tensor product satisfying S. A. Smolyak’s rule [56]. This significantly reduces the number of collocation points without a notable loss in accuracy.

(3) A global polynomial approximation based on the obtained approximate solutions is constructed as follows [4]:

$$\begin{array}{cc}& {\mathbf{u}}_{N,h,P}\left(x,t,\mathbf{Y}\right)=\sum _{k=1}^M{\mathbf{u}}_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)L_k\left(\mathbf{Y}\right),\hfill \\ & p_{N,h,P}\left(x,t,\mathbf{Y}\right)=\sum _{k=1}^M{p}_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)L_k\left(\mathbf{Y}\right),\hfill \\ & {\varphi }_{N,h,P}\left(x,t,\mathbf{Y}\right)=\sum _{k=1}^M{\varphi }_{N,h}\left(x,t,{\mathbf{Y}}^{\left(k\right)}\right)L_k\left(\mathbf{Y}\right),\hfill \end{array}$$

where ${\left\{L_k\right\}}_{k=1}^M$ are orthogonal polynomials, and P is the degree of the polynomial.

(4) To predict the statistical moments of the solution, such as mathematical expectation and variance, we use the Gaussian quadrature formula. For example [55],

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\mathbf{u}}_N\left(·,·,\mathbf{Y}\right)\right]\approx \mathbb{E}\left[{\mathbf{u}}_{N,h,P}\left(·,·,\mathbf{Y}\right)\right]={\int }_G{\mathbf{u}}_{N,h,P}\left(·,·,\mathbf{Y}\right)\rho \left(\mathbf{Y}\right)d\mathbf{Y}\approx \sum _{k=1}^M{\theta }_k{\mathbf{u}}_{N,h}\left(·,·,{\mathbf{Y}}^{\left(k\right)}\right),\hfill \\ \hfill & \mathbb{E}\left[p_N\left(·,·,\mathbf{Y}\right)\right]\approx \mathbb{E}\left[p_{N,h,P}\left(·,·,\mathbf{Y}\right)\right]={\int }_G{p}_{N,h,P}\left(·,·,\mathbf{Y}\right)\rho \left(\mathbf{Y}\right)d\mathbf{Y}\approx \sum _{k=1}^M{\theta }_k{p}_{N,h}\left(·,·,{\mathbf{Y}}^{\left(k\right)}\right),\hfill \\ \hfill & \mathbb{E}\left[{\varphi }_N\left(·,·,\mathbf{Y}\right)\right]\approx \mathbb{E}\left[{\varphi }_{N,h,P}\left(·,·,\mathbf{Y}\right)\right]={\int }_G{\varphi }_{N,h,P}\left(·,·,\mathbf{Y}\right)\rho \left(\mathbf{Y}\right)d\mathbf{Y}\approx \sum _{k=1}^M{\theta }_k{\varphi }_{N,h}\left(·,·,{\mathbf{Y}}^{\left(k\right)}\right),\hfill \end{array}$$

where the weights are defined as

$$\begin{array}{cc}& {\theta }_k=\prod _{n=1}^N{\int }_{G_n}{L}_{k_n}\left(Y_n\right){\rho }_n\left(Y_n\right)d{Y}_n,L_{k_n}\left(Y_n\right)=\prod _{\begin{array}{c}i=0\\ i\ne k_n\end{array}}^M{\displaystyle \frac{Y_n-Y_n^{\left(i\right)}}{Y_n^{\left(k_n\right)}-Y_n^{\left(i\right)}}}.\hfill \end{array}$$

2.3. Construction and Analysis of the Semi-Discrete Scheme

2.3.1. Construction of the Semi-Discrete Scheme

Let us first dwell on the construction of a deterministic numerical method. Denote the inner product in $L^2\left(D_f\right)$ by ${\left(·,·\right)}_f$, let ${∥·∥}_{k,f}={∥·∥}_{H^k\left(D_f\right)}$, and introduce similar notations for $D_p$ and I. Further, let $\mathbf{X}=X_f\times Q_p$. Then the weak formulation of Problem 1 for a specific sample of the hydraulic conductivity tensor is as follows.

Problem 2. 

For a given $\mathbf{Y}\in G$, find $\left({\widehat{\mathbf{u}}}_N\left(·,\mathbf{Y}\right),p_N\left(·,\mathbf{Y}\right)\right):J\to \mathbf{X}\times Q_f$ such that

$${\mathcal{D}}^{\gamma }\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)+\mathcal{A}\left({\mathbb{K}}_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)+{\mathcal{A}}_I\left({\varsigma }_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)+\mathcal{B}\left({\widehat{\mathbf{u}}}_N,q\right)-\mathcal{B}\left(\widehat{\mathbf{v}},p_N\right)+\mathcal{I}\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)=\left(\widehat{\mathbf{f}},\widehat{\mathbf{v}}\right)$$

(26)

for all $\left(\widehat{\mathbf{v}},q\right)\in \mathbf{X}\times Q_f$, where ${\mathbb{K}}_N={\mathbb{K}}_N\left(·,\mathbf{Y}\right)$,

$$\begin{array}{cccc}\hfill & {\widehat{\mathbf{u}}}_N={\left[{\mathbf{u}}_N,{\varphi }_N\right]}^{\top },\widehat{\mathbf{v}}={\left[\mathbf{v},\psi \right]}^{\top },\widehat{\mathbf{f}}={\left[{\mathbf{f}}_f,g{f}_p\right]}^{\top },\hfill & \hfill & \hfill \\ \hfill & {\mathcal{D}}^{\gamma }\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)={\left(gS{\partial }_{0,t}^{\gamma }{\varphi }_N,\psi \right)}_p+{\left({\partial }_{0,t}^{\gamma }{\mathbf{u}}_N,\mathbf{v}\right)}_f,\hfill & \hfill & {\mathcal{A}}_I\left({\varsigma }_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)=\sum _{i=1}^{d-1}{\int }_I{\varsigma }_{N,i}\left({\mathbf{u}}_N·{\mathbf{m}}_i\right)\left(\mathbf{v}·{\mathbf{m}}_i\right)ds,\hfill \\ \hfill & \mathcal{A}\left({\mathbb{K}}_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)={\mathcal{A}}_p\left({\mathbb{K}}_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)+{\mathcal{A}}_f\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right),\hfill & \hfill & \mathcal{I}\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)=g{\int }_I\left({\varphi }_N\mathbf{v}-\psi {\mathbf{u}}_N\right)·{\mathbf{n}}_{f}ds,\hfill \\ \hfill & {\mathcal{A}}_p\left({\mathbb{K}}_N;{\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)=g{\left({\mathbb{K}}_N\nabla {\varphi }_N,\nabla \psi \right)}_p,\hfill & \hfill & \mathcal{B}\left(\widehat{\mathbf{v}},q\right)={\left(q,\nabla ·\mathbf{v}\right)}_f,\hfill \\ \hfill & {\mathcal{A}}_f\left({\widehat{\mathbf{u}}}_N,\widehat{\mathbf{v}}\right)=2\mu {\left(\mathbb{D}\left({\mathbf{u}}_N\right),\mathbb{D}\left(\mathbf{v}\right)\right)}_f,\hfill & \hfill & \left(\widehat{\mathbf{f}},\widehat{\mathbf{v}}\right)={\left({\mathbf{f}}_f,\mathbf{v}\right)}_f+{\left(g{f}_p,\psi \right)}_p,\hfill \end{array}$$

and ${\varsigma }_N={\left\{{\varsigma }_{N,i}\right\}}_{i=1,2,\cdots ,d-1}$.

We start by discretizing the problem with respect to the time variable. For a given time step $\tau >0$, let $J_{\tau }=\left\{t_n\in \mathbb{R}:t_n=n\tau ,n=0,1,\cdots ,N_T,N_T\tau =T\right\}$ be a set of equidistant points in $\overline{J}$, and denote $f^n=f\left(·,t_n,\omega \right)$ for brevity. Further, we utilize the following approximation formulas from [52] for fast evaluation of the Caputo fractional derivatives in (26):

$$\begin{array}{cc}\hfill {\left.{\Delta }_{\tau }^{\gamma }f\left(t\right)\right|}_{t=t_1}& ={\displaystyle \frac{1}{{\tau }^{\gamma }\Gamma \left(2-\gamma \right)}}\left(f\left(t_1\right)-f\left(t_0\right)\right),\hfill \\ \hfill {\left.{\Delta }_{\tau }^{\gamma }f\left(t\right)\right|}_{t=t_n}& ={\displaystyle \frac{1}{{\tau }^{\gamma }\Gamma \left(3-\gamma \right)}}\left({\displaystyle \frac{\gamma }{2}}f\left(t_{n-2}\right)-2f\left(t_{n-1}\right)+{\displaystyle \frac{4-\gamma }{2}}f\left(t_n\right)\right)\hfill \end{array}$$

(27)

$$\begin{array}{cc}& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{\Gamma \left(1-\gamma \right)}}\left({\displaystyle \frac{f\left(t_{n-1}\right)}{{\tau }^{\gamma }}}-{\displaystyle \frac{f\left(t_0\right)}{t_n^{\gamma }}}-\gamma \sum _{i=1}^{N_{\epsilon }}{\omega }_i^{\left(\gamma \right)}{U}_{\tau ,i}^{n,\gamma }\left[f\right]\right),n\ge 2,\hfill \end{array}$$

(28)

which are based on applying different approximations of the integral in (3) on the intervals $\left(0,t_1\right)$ and $\left(t_1,t_n\right)$, where

$$\begin{array}{cc}& U_{\tau ,i}^{1,\gamma }\left[f\right]=0,\hfill \\ & U_{\tau ,i}^{n,\gamma }\left[f\right]=e^{-s_i^{\left(\gamma \right)}\tau }{U}_{\tau ,i}^{n-1,\gamma }+{\widehat{a}}_i^{\left(\gamma \right)}f\left(t_{n-2}\right)-{\widehat{b}}_i^{\left(\gamma \right)}f\left(t_{n-1}\right)+{\widehat{c}}_i^{\left(\gamma \right)}f\left(t_n\right),\hfill \\ & {\widehat{a}}_i^{\left(\gamma \right)}={\displaystyle \frac{e^{-2s_i^{\left(\gamma \right)}\tau }}{2{\left(s_i^{\left(\gamma \right)}\right)}^3{\tau }^2}}\left[2e^{s_i^{\left(\gamma \right)}\tau }-2{\left(s_i^{\left(\gamma \right)}\tau \right)}^2-s_i^{\left(\gamma \right)}\tau \left(3-e^{s_i^{\left(\gamma \right)}\tau }\right)-2\right],\hfill \\ & {\widehat{b}}_i^{\left(\gamma \right)}={\displaystyle \frac{e^{-2s_i^{\left(\gamma \right)}\tau }}{{\left(s_i^{\left(\gamma \right)}\right)}^3{\tau }^2}}\left[2e^{s_i^{\left(\gamma \right)}\tau }-2s_i^{\left(\gamma \right)}\tau -{\left(s_i^{\left(\gamma \right)}\tau \right)}^2{e}^{s_i^{\left(\gamma \right)}\tau }-2\right],\hfill \\ & {\widehat{c}}_i^{\left(\gamma \right)}={\displaystyle \frac{e^{-2s_i^{\left(\gamma \right)}\tau }}{2{\left(s_i^{\left(\gamma \right)}\right)}^3{\tau }^2}}\left[2e^{s_i^{\left(\gamma \right)}\tau }-s_i^{\left(\gamma \right)}\tau \left(1+e^{s_i^{\left(\gamma \right)}\tau }\right)-2\right].\hfill \end{array}$$

The derivation of Equation (28) is based on the observation that for a given $\epsilon >0$, the power function $t^{-1-\gamma }$ can be approximated by a sum-of-exponentials (SOE) approximation such that

$$\left|{\displaystyle \frac{1}{t^{1+\gamma }}}-\sum _{i=1}^{N_{\epsilon }}{\omega }_i^{\left(\gamma \right)}{e}^{-s_i^{\left(\gamma \right)}t}\right|<\epsilon$$

for some positive real numbers $s_i^{\left(\gamma \right)}$, ${\omega }_i^{\left(\gamma \right)}$, $i=1,2,\cdots ,N_{\epsilon }$. We refer the reader to the original work [52] for details.

Despite the significant reduction in computational costs associated with utilizing Smolyak’s formulas in the stochastic collocation method, Problem 1 still needs to be solved multiple times for each sample of the hydraulic conductivity tensor. To further reduce computational costs, we utilize an ensemble strategy [57], which allows solving the deterministic problem once for all samples, rather than solving it separately for each sample. This is achieved by applying a special approximation of the second and third terms of (26):

$$\begin{array}{cc}\hfill & \mathcal{A}\left({\mathbb{K}}_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)=\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)-{\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\mathcal{A}}_I\left({\varsigma }_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)={\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)-{\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right),\hfill \end{array}$$

(30)

where ${\displaystyle {\overline{\mathbb{K}}}_N\left(x\right)=\frac{1}{M}\sum _{k=1}^M{\mathbb{K}}_N\left(x,{\mathbf{Y}}^{\left(k\right)}\right)}$ and ${\displaystyle {\overline{\varsigma }}_N\left(x\right)=\frac{1}{M}\sum _{k=1}^M{\varsigma }_N\left(x,{\mathbf{Y}}^{\left(k\right)}\right)}$ denote the mean of all samples of ${\mathbb{K}}_N$ and ${\varsigma }_N$, respectively, and $\sigma {\widehat{\mathbf{u}}}_N^n$ is the third-order approximation of ${\widehat{\mathbf{u}}}_N^n$ on the nth time layer, $n\ge 3$, which will be defined later. Due to this approximation, the coefficients of ${\widehat{\mathbf{u}}}_N^n$ remain independent of the current hydraulic conductivity sample. As a result, the finite element procedure is reduced to solving a large number of systems of linear algebraic equations sharing the same stiffness matrix but having different right-hand sides. This approach enables the application of efficient factorization methods, which significantly accelerate finding a solution to the problem.

Utilizing the approximations (27)–(30), we rewrite Problem 2 in the following way.

Problem 3. 

For a given $\mathbf{Y}\in G$, let $\left({\widehat{\mathbf{u}}}_N^k,p_N^k\right)\in \mathbf{X}\times Q_f$, $k=0,1,\cdots ,n-1$ be known; in particular, ${\widehat{\mathbf{u}}}_N^0={\left[{\mathbf{u}}_0,{\varphi }_0\right]}^{\top }$. Find $\left({\widehat{\mathbf{u}}}_N^n,p_N^n\right)\in \mathbf{X}\times Q_f$, $n\ge 1$, satisfying the identity

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)+\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)\hfill \\ \hfill & +\mathcal{B}\left({\widehat{\mathbf{u}}}_N^n,q\right)-\mathcal{B}\left(\widehat{\mathbf{v}},p_N^n\right)+\mathcal{I}\left(\sigma {\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)\hfill \\ \hfill & =\left({\widehat{\mathbf{f}}}^n,\widehat{\mathbf{v}}\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)+{\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)\hfill \\ \hfill & +{\left({\mathbf{r}}_{11}^n,\mathbf{v}\right)}_f+{\left(g{r}_{12}^n,\psi \right)}_p+\sum _{i=1}^{d-1}{\int }_I\left({\overline{\varsigma }}_{N,i}-{\varsigma }_{N,i}\right)\left({\mathbf{r}}_{21}^n·{\mathbf{m}}_i\right)\left(\mathbf{v}·{\mathbf{m}}_i\right)ds\hfill \\ \hfill & +g{\left(\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N\right)\nabla r_{22}^n,\nabla \psi \right)}_p+{\int }_I{\mathbf{r}}_{21}^n·\psi {\mathbf{n}}_{f}ds-g{\int }_I{r}_{22}^n\mathbf{v}·{\mathbf{n}}_{f}ds\hfill \end{array}$$

(31)

for all $\left(\widehat{\mathbf{v}},q\right)\in \mathbf{X}\times Q_f$, where

$${\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_N^n,\widehat{\mathbf{v}}\right)={\left(gS{\Delta }_{\tau }^{\gamma }{\varphi }_N^n,\psi \right)}_p+{\left({\Delta }_{\tau }^{\gamma }{\mathbf{u}}_N^n,\mathbf{v}\right)}_f,$$

(32)

$$\sigma {\mathbf{u}}_N^n=\left\{\begin{array}{cc}{\mathbf{u}}_N^0,\hfill & n=1,\hfill \\ 2{\mathbf{u}}_N^1-{\mathbf{u}}_N^0,\hfill & n=2,\hfill \\ 3{\mathbf{u}}_N^{n-1}-3{\mathbf{u}}_N^{n-2}+{\mathbf{u}}_N^{n-3},\hfill & n\ge 3,\hfill \end{array}\right.$$

(33)

$$\sigma {\varphi }_N^n=\left\{\begin{array}{cc}{\varphi }_N^0,\hfill & n=1,\hfill \\ 2{\varphi }_N^1-{\varphi }_N^0,\hfill & n=2,\hfill \\ 3{\varphi }_N^{n-1}-3{\varphi }_N^{n-2}+{\varphi }_N^{n-3},\hfill & n\ge 3,\hfill \end{array}\right.$$

(34)

$${\mathbf{r}}_{11}^n={\partial }_{0,t}^{\gamma }{\mathbf{u}}_N^n-{\Delta }_{\tau }^{\gamma }{\mathbf{u}}_N^n,r_{12}^n={\partial }_{0,t}^{\gamma }{\varphi }_N^n-{\Delta }_{\tau }^{\gamma }{\varphi }_N^n,$$

$${\mathbf{r}}_{21}^n=\sigma {\mathbf{u}}_N^n-{\mathbf{u}}_N^n,r_{22}^n=\sigma {\varphi }_N^n-{\varphi }_N^n,\left(\widehat{\mathbf{f}},\widehat{\mathbf{v}}\right)={\left(g{f}_p,\psi \right)}_p+{\left({\mathbf{f}}_f,\mathbf{v}\right)}_f.$$

To discretize (31) by spatial variables, we introduce quasi-uniform triangulations $D_{fh}$ and $D_{ph}$ with the diameter $h>0$ in $D_f$ and $D_p$, respectively. Let $X_{fh}\subset X_f$, $Q_{fh}\subset Q_f$, $Q_{ph}\subset Q_p$ be finite element spaces with the following approximation properties:

$$\begin{array}{cccc}& \underset{{\mathbf{v}}_h\in X_{fh}}{inf}{∥\mathbf{v}-{\mathbf{v}}_h∥}_{0,f}\le C{h}^{l+1}{∥\mathbf{v}∥}_{l+1,f}\hfill & \hfill & \forall \mathbf{v}\in {\left(H^{l+1}\left(D_f\right)\right)}^d,\hfill \\ & \underset{{\mathbf{v}}_h\in X_{fh}}{inf}{∥\nabla \left(\mathbf{v}-{\mathbf{v}}_h\right)∥}_{0,f}\le C{h}^l{∥\mathbf{v}∥}_{l+1,f}\hfill & & \forall \mathbf{v}\in {\left(H^{l+1}\left(D_f\right)\right)}^d,\hfill \\ & \underset{q_h\in Q_{fh}}{inf}{∥q-q_h∥}_{0,f}\le C{h}^l{∥q∥}_{l,f}\hfill & & \forall q\in H^l\left(D_f\right),\hfill \\ & \underset{{\psi }_h\in Q_{ph}}{inf}{∥\psi -{\psi }_h∥}_{0,p}\le C{h}^{m+1}{∥\psi ∥}_{m+1,p}\hfill & & \forall \psi \in H^{m+1}\left(D_p\right),\hfill \\ & \underset{{\psi }_h\in Q_{ph}}{inf}{∥\nabla \left(\psi -{\psi }_h\right)∥}_{0,p}\le C{h}^m{∥\psi ∥}_{m+1,p}\hfill & & \forall \psi \in H^{m+1}\left(D_p\right),\hfill \end{array}$$

(35)

and let ${\mathbf{X}}_h=X_{fh}\times Q_{ph}$.

Discarding the error terms in (31), we obtain the following semi-discrete problem.

Problem 4. 

For a given $\mathbf{Y}\in G$, let $\left({\widehat{\mathbf{u}}}_{N,h}^k,p_{N,h}^k\right)\in {\mathbf{X}}_h\times Q_{fh}$, $k=0,1,\cdots ,n-1$ be known; in particular, let ${\widehat{\mathbf{u}}}_{N,h}^0$ contain $L^2$-projections of the initial values (21), (22). Find $\left({\widehat{\mathbf{u}}}_{N,h}^n,p_{N,h}^n\right)\in {\mathbf{X}}_h\times Q_{fh}$, $n\ge 1$, satisfying the identity

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)+\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)\hfill \\ \hfill & +\mathcal{B}\left({\widehat{\mathbf{u}}}_{N,h}^n,q_h\right)-\mathcal{B}\left({\widehat{\mathbf{v}}}_h,p_{N,h}^n\right)+\mathcal{I}\left(\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)\hfill \\ \hfill & ={\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)+{\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{v}}}_h\right)+\left({\widehat{\mathbf{f}}}^n,{\widehat{\mathbf{v}}}_h\right)\hfill \end{array}$$

(36)

for all $\left({\widehat{\mathbf{v}}}_h,q_h\right)\in {\mathbf{X}}_h\times Q_{fh}$.

2.3.2. Preliminary Results

Before discussing the stability and convergence of the constructed numerical scheme (36), let us first recall some known results that will be used in the subsequent analysis.

Lemma 1 

([58]). Let $\left\{y_n\right\}$ be a positive real number sequence such that

$$y_n\le \sum _{i=0}^{n-1}{z}_i{y}_i+f_n\forall n\ge 0,$$

where $\left\{z_n\right\}$ and $\left\{f_n\right\}$ are two given real numbers sequences and $\left\{z_n\right\}$ is furthermore positive. Then

$$y_n\le f_n+\sum _{i=0}^{n-1}{f}_i{z}_{i}exp\left(\sum _{j=i}^{n-1}{z}_j\right).$$

We often use the following inequalities:

$${∥\mathbf{v}∥}_I\le C_{\mathrm{tr}}\left(D_f\right){∥\mathbf{v}∥}_{0,f}^{1/2}{∥\nabla \mathbf{v}∥}_{0,f}^{1/2},{∥\psi ∥}_I\le C_{\mathrm{tr}}\left(D_p\right){∥\psi ∥}_{0,p}^{1/2}{∥\nabla \psi ∥}_{0,p}^{1/2},$$

(37)

$${∥\mathbf{v}∥}_{0,f}\le C_{P,f}{∥\nabla \mathbf{v}∥}_{0,f},{∥\psi ∥}_{0,p}\le C_{P,p}{∥\nabla \psi ∥}_{0,p},$$

(38)

$$ab\le \epsilon a^2+{\displaystyle \frac{1}{4\epsilon }}{b}^2,a>0,b>0,\epsilon >0.$$

(39)

2.3.3. Stability of the Semi-Discrete Scheme

Lemma 2. 

For a given $\mathbf{Y}\in G$, the following inequalities hold under the condition $\tau <{\tau }_0$, where ${\tau }_0={\left({\displaystyle \frac{12{\varpi }_2^{\left(\gamma \right)}}{ϵ{\varpi }_1^{\left(\gamma \right)}}}\right)}^{{\displaystyle \frac{1}{\gamma +1}}}$, $0<ϵ<<1$:

$$\begin{array}{cc}\hfill {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_{N,h}^1,{\widehat{\mathbf{u}}}_{N,h}^1\right)& \ge {\displaystyle \frac{gS}{2{\tau }^{\gamma }{\varpi }_0^{\left(\gamma \right)}}}\left({∥{\varphi }_{N,h}^1∥}_{0,p}^2-{∥{\varphi }_{N,h}^0∥}_{0,p}^2\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\displaystyle \frac{1}{2{\tau }^{\gamma }{\varpi }_0^{\left(\gamma \right)}}}\left({∥{\mathbf{u}}_{N,h}^1∥}_{0,f}^2-{∥{\mathbf{u}}_{N,h}^0∥}_{0,f}^2\right),\hfill \\ \hfill {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)& \ge {\displaystyle \frac{gS}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}{∥{\varphi }_{N,h}^n∥}_{0,p}^2-{\displaystyle \frac{gS}{4ϵ{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\varphi }_{N,h}^{n-s}∥}_{0,p}^2\hfill \\ \hfill & -{\displaystyle \frac{3}{2}}gS{\left({\displaystyle \frac{7ϵ}{6{\varpi }_2^{\left(\gamma \right)}}}\right)}^2{\tau }^{\gamma +1}{\varpi }_1^{\left(\gamma \right)}\sum _{s=1}^n{∥\nabla {\varphi }_{N,h}^{n-s}∥}_{0,p}^2\hfill \\ \hfill & +{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}{∥{\mathbf{u}}_{N,h}^n∥}_{0,f}^2-{\displaystyle \frac{1}{4ϵ{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\mathbf{u}}_{N,h}^{n-s}∥}_{0,f}^2\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{3}{2}}{\left({\displaystyle \frac{7ϵ}{6{\varpi }_2^{\left(\gamma \right)}}}\right)}^2{\tau }^{\gamma +1}{\varpi }_1^{\left(\gamma \right)}\sum _{s=1}^n{∥\nabla {\mathbf{u}}_{N,h}^{n-s}∥}_{0,f}^2,n\ge 2,\hfill \end{array}$$

(41)

where

$$\begin{array}{cc}\hfill & {\varpi }_0^{\left(\gamma \right)}=\Gamma \left(2-\gamma \right),\hfill \\ \hfill & {\varpi }_1^{\left(\gamma \right)}=\Gamma \left(3-\gamma \right){\left(c_1^{\left(\gamma \right)}+2-{\displaystyle \frac{\gamma }{3}}\right)}^{-1},\hfill \\ \hfill & {\varpi }_2^{\left(\gamma \right)}=\Gamma \left(3-\gamma \right){\left(\gamma \left(1-\gamma \right)\left(2-\gamma \right)\right)}^{-1},\hfill \\ \hfill & c_1^{\left(\gamma \right)}=-{\displaystyle \frac{1}{2}}\left(2-\gamma \right)\left(2^{1-\gamma }+1\right)+2^{2-\gamma }-1.\hfill \end{array}$$

Proof. 

The inequality (40) immediately follows from (27) and the elementary inequality $\left(a-b,a\right)\ge {\displaystyle \frac{1}{2}}\left(a^2-b^2\right)$. The inequality (41) relies on the equivalent representation of (28) considered in [52]:

$$\begin{array}{cc}\hfill & {\Delta }_{\tau }^{\gamma }{\varphi }_{N,h}^n={\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\left({\varphi }_{N,h}^n-\sum _{s=1}^n{d}_{n-s}^{n,\gamma }{\varphi }_{N,h}^{n-s}\right)+{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_2^{\left(\gamma \right)}}}\sum _{s=0}^n{\chi }_{n-s}^{n,\gamma }{\varphi }_{N,h}^{n-s},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\Delta }_{\tau }^{\gamma }{\mathbf{u}}_{N,h}^n={\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\left({\mathbf{u}}_{N,h}^n-\sum _{s=1}^n{d}_{n-s}^{n,\gamma }{\mathbf{u}}_{N,h}^{n-s}\right)+{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_2^{\left(\gamma \right)}}}\sum _{s=0}^n{\chi }_{n-s}^{n,\gamma }{\mathbf{u}}_{N,h}^{n-s},\hfill \end{array}$$

(43)

where the coefficients $d_{n-s}^{n,\gamma }$ and ${\chi }_{n-s}^{n,\gamma }$ are defined in [52] and have the following properties:

$$\begin{array}{cc}\hfill & d_{n-s}^{n,\gamma }>0\mathrm{for}s\ne 2,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & d_{n-1}^{n,\gamma }\in \left(0,{\displaystyle \frac{4}{3}}\right),d_{n-2}^{n,\gamma }\in \left(-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}}\right),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & \sum _{s=1}^n{d}_{n-s}^{n,\gamma }=1,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & \left|{\chi }_{n-s}^{n,\gamma }\right|<{\displaystyle \frac{7}{6}}ϵ{\tau }^{\gamma +1}.\hfill \end{array}$$

(47)

Taking the inner product of (42) with ${\varphi }_{N,h}^n$ and using Cauchy inequality results in the following inequality:

$${\left({\Delta }_{\tau }^{\gamma }{\varphi }_{N,h}^n,{\varphi }_{N,h}^n\right)}_p\ge {\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\left({∥{\varphi }_{N,h}^n∥}_{0,p}^2-\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\varphi }_{N,h}^{n-s}∥}_{0,p}{∥{\varphi }_{N,h}^n∥}_{0,p}\right)$$

$$-{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_2^{\left(\gamma \right)}}}\left|{\chi }_n^{n,\gamma }\right|{∥{\varphi }_{N,h}^n∥}_{0,p}^2-{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_2^{\left(\gamma \right)}}}\sum _{s=1}^n\left|{\chi }_{n-s}^{n,\gamma }\right|{∥{\varphi }_{N,h}^{n-s}∥}_{0,p}{∥{\varphi }_{N,h}^n∥}_{0,p}.$$

(48)

Considering (47) and using the inequalities (38), (39), we have

$${\left({\Delta }_{\tau }^{\gamma }{\varphi }_{N,h}^n,{\varphi }_{N,h}^n\right)}_p\ge {\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\left(1-3\epsilon -{\displaystyle \frac{ϵ{\varpi }_1^{\left(\gamma \right)}}{12{\varpi }_2^{\left(\gamma \right)}}}{\tau }^{\gamma +1}\right){∥{\varphi }_{N,h}^n∥}_{0,p}^2$$

$$-{\displaystyle \frac{1}{4\epsilon }}{\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_1^{\left(\gamma \right)}}}\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\varphi }_{N,h}^{n-s}∥}_{0,p}^2-{\displaystyle \frac{1}{4\epsilon }}{\left({\displaystyle \frac{7ϵ}{6{\varpi }_2^{\left(\gamma \right)}}}\right)}^2{\tau }^{\gamma +1}{\varpi }_1^{\left(\gamma \right)}\sum _{s=1}^n{∥\nabla {\varphi }_{N,h}^{n-s}∥}_{0,p}^2,\epsilon >0.$$

(49)

Obtaining a similar inequality for ${\mathbf{u}}_{N,h}^n$ by utilizing Equation (43) and combining it with (49), we arrive at the statement of the lemma. □

Theorem 1 

(Stability of the semi-discrete scheme). Let $\left({\widehat{\mathbf{u}}}_{N,h}^n,p_{N,h}^n\right)\in {\mathbf{X}}_h\times Q_{fh}$, $n\ge 1$ be the solution of Problem 4. Then the following inequality holds for sufficiently small τ:

$${\mathcal{E}}^n\le C\left({\mathcal{E}}^0+{\tau }^{\gamma }\sum _{s=1}^n\left({∥{\mathbf{f}}_f^n∥}_{0,f}^2+{∥f_p^n∥}_{0,p}^2\right)\right)$$

(50)

with

$${\mathcal{E}}^n={∥{\varphi }_{N,h}^n∥}_{0,p}^2+{∥{\mathbf{u}}_{N,h}^n∥}_{0,f}^2+{\tau }^{\gamma }{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2+{\tau }^{\gamma }{∥\nabla {\mathbf{u}}_{N,h}^n∥}_{0,f}^2,$$

which yields the stability of the numerical scheme (36) with respect to the initial data and the right-hand sides of the equations, where C is a positive constant depending on the order of the fractional derivatives, γ, but independent of the mesh parameters.

Proof. 

Take $\left({\widehat{\mathbf{v}}}_h,q_h\right)=\left({\widehat{\mathbf{u}}}_{N,h}^n,p_{N,h}^n\right)$ in (36) to obtain:

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)+\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)+\mathcal{I}\left(\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\hfill \\ \hfill & ={\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)+{\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)+\left({\widehat{\mathbf{f}}}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right).\hfill \end{array}$$

(51)

The first term on the left-hand side of (51) is estimated with Lemma 2. Further, according to Korn’s inequality [59], we have

$$\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\ge {\mu }_0{∥{\mathbf{u}}_{N,h}^n∥}_{1,f}^2+g{k}_{min}{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2.$$

By applying a Cauchy inequality to the last term in the left-hand side of (51), we arrive at

$$\mathcal{I}\left(\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\le g{∥\sigma {\varphi }_{N,h}^n∥}_{0,I}{∥{\mathbf{u}}_{N,h}^n·{\mathbf{n}}_f∥}_{0,I}+g{∥\sigma {\mathbf{u}}_{N,h}^n·{\mathbf{n}}_f∥}_{0,I}{∥{\varphi }_{N,h}^n∥}_{0,I}.$$

Further, by utilizing the expansion

$${\displaystyle \mathbf{v}=\mathbf{v}·{\mathbf{n}}_f+\sum _{i=1}^{d-1}\mathbf{v}·{\mathbf{m}}_i}$$

(52)

and the inequalities (37)–(39), we obtain

$$\mathcal{I}\left(\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\le {\mu }_0ϵ{∥\nabla {\mathbf{u}}_{N,h}^n∥}_{0,f}^2+g{k}_{min}ϵ{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2+C{∥\nabla \sigma {\varphi }_{N,h}^n∥}_{0,p}^2+C{∥\nabla \sigma {\mathbf{u}}_{N,h}^n∥}_{0,f}^2.$$

The first term on the right-hand side is estimated in a similar way:

$${\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\le C{∥\nabla \sigma {\mathbf{u}}_{N,h}^n∥}_{0,f}^2+{\mu }_0ϵ{∥\nabla {\mathbf{u}}_{N,h}^n∥}_{0,f}^2.$$

The remaining terms are estimated as follows:

$${\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathbf{u}}}_{N,h}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\le C{∥\nabla \sigma {\varphi }_{N,h}^n∥}_{0,p}^2+g{k}_{min}ϵ{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2,$$

$$\left({\widehat{\mathbf{f}}}^n,{\widehat{\mathbf{u}}}_{N,h}^n\right)\le {\mu }_0ϵ{∥\nabla {\mathbf{u}}_{N,h}^n∥}_{0,f}^2+g{k}_{min}ϵ{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2+C{∥{\mathbf{f}}_f^n∥}_{0,f}^2+C{∥f_p^n∥}_{0,p}^2.$$

Considering the above inequalities, it follows from (51) that

$${∥{\varphi }_{N,h}^1∥}_{0,p}^2+{∥{\mathbf{u}}_{N,h}^1∥}_{0,f}^2+{\tau }^{\gamma }{∥\nabla {\varphi }_{N,h}^1∥}_{0,p}^2+{\tau }^{\gamma }{∥{\mathbf{u}}_{N,h}^1∥}_{1,f}^2$$

$$\le {∥{\varphi }_{N,h}^0∥}_{0,p}^2+{∥{\mathbf{u}}_{N,h}^0∥}_{0,f}^2+C{\tau }^{\gamma }\left({∥\nabla {\varphi }_{N,h}^0∥}_{0,p}^2+{∥\nabla {\mathbf{u}}_{N,h}^0∥}_{0,f}^2+{∥{\mathbf{f}}_f^1∥}_{0,f}^2+{∥f_p^1∥}_{0,p}^2\right)$$

(53)

and

$${∥{\varphi }_{N,h}^n∥}_{0,p}^2+{∥{\mathbf{u}}_{N,h}^n∥}_{0,f}^2+{\tau }^{\gamma }{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2+{\tau }^{\gamma }{∥{\mathbf{u}}_{N,h}^n∥}_{1,f}^2$$

$$\le \sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\varphi }_{N,h}^{n-s}∥}_{0,p}^2+\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\mathbf{u}}_{N,h}^{n-s}∥}_{0,f}^2$$

$$+C_1{\tau }^{\gamma }\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥\nabla {\varphi }_{N,h}^{n-s}∥}_{0,p}^2+{\tau }^{\gamma }\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥\nabla {\mathbf{u}}_{N,h}^{n-s}∥}_{0,f}^2+C{\tau }^{\gamma }{∥{\mathbf{f}}_f^n∥}_{0,f}^2+C{\tau }^{\gamma }{∥f_p^n∥}_{0,p}^2,$$

(54)

where ${\displaystyle C_1=\underset{0\le s\le n-1}{max}\left\{max\left\{{\left|d_s^{n,\gamma }\right|}^{-1},1\right\}\right\}}$. Combining the inequalities (53) and (54), considering (46), and applying Lemma 1, we arrive at the assertion of the theorem. □

2.3.4. Convergence of the Semi-Discrete Scheme

Consider the projectors ${\mathsf{\Phi }}_h:Q_p\to Q_{ph}$, ${\mathsf{\Pi }}_h:X_f\to X_{fh}$, ${\mathsf{\Xi }}_h:Q_f\to Q_{fh}$ satisfying the identities

$$\begin{array}{cccc}& \left(\nabla \left(\psi -{\mathsf{\Phi }}_h\psi \right),\nabla {\psi }_h\right)=0,\hfill & \hfill & \forall {\psi }_h\in Q_{ph},\hfill \end{array}$$

(55)

$$\begin{array}{cccc}& \left(\nabla ·\left(\mathbf{v}-{\mathsf{\Pi }}_h\mathbf{v}\right),q_h\right)=0,\hfill & & \forall q_h\in Q_{fh},\hfill \end{array}$$

(56)

$$\begin{array}{cccc}& \left(\nabla ·{\mathbf{v}}_h,q_h-{\mathsf{\Xi }}_h{q}_h\right)=0,\hfill & & \forall {\mathbf{v}}_h\in X_{fh}.\hfill \end{array}$$

(57)

To study the convergence of the numerical scheme (36) introduce the notation

$$\begin{array}{cc}& {\mathbf{u}}_N^n-{\mathbf{u}}_{N,h}^n=\left({\mathbf{u}}_N^n-{\mathsf{\Pi }}_h{\mathbf{u}}_N^n\right)+\left({\mathsf{\Pi }}_h{\mathbf{u}}_N^n-{\mathbf{u}}_{N,h}^n\right)={\xi }_{\mathbf{u}}^n+{\theta }_{\mathbf{u}}^n,\hfill \\ & p_N^n-p_{N,h}^n=\left(p_N^n-{\mathsf{\Xi }}_h{p}_N^n\right)+\left({\mathsf{\Xi }}_h{p}_N^n-p_{N,h}^n\right)={\xi }_p^n+{\theta }_p^n,\hfill \\ & {\varphi }_N^n-{\varphi }_{N,h}^n=\left({\varphi }_N^n-{\mathsf{\Phi }}_h{\varphi }_N^n\right)+\left({\mathsf{\Phi }}_h{\varphi }_N^n-{\varphi }_{N,h}^n\right)={\xi }_{\varphi }^n+{\theta }_{\varphi }^n.\hfill \end{array}$$

(58)

Theorem 2 

(Convergence of the semi-discrete scheme). Let $\left({\widehat{\mathbf{u}}}_N^n,p_N^n\right)\in \mathbf{X}\times Q_f$ be the solution of Problem 2, and $\left({\widehat{\mathbf{u}}}_{N,h}^n,p_{N,h}^n\right)\in {\mathbf{X}}_h\times Q_{fh}$ be the solution of Problem 4. Moreover, let the time steps ${\tau }_1$ and ${\tau }_2$ on the intervals $\left[t_0,t_1\right]$ and $\left[t_1,t_2\right]$ be determined by the relations

$${\tau }_1={\tau }^{{\displaystyle \frac{6-2\gamma }{2+\gamma }}}\mathrm{and}{\tau }_2={\tau }^{{\displaystyle \frac{6-2\gamma }{4+\gamma }}}.$$

(59)

Then the following inequality holds for sufficiently small τ:

$${∥{\mathbf{u}}_N\left(t_n\right)-{\mathbf{u}}_{N,h}^n∥}_{0,f}^2+{∥{\varphi }_N\left(t_n\right)-{\varphi }_{N,h}^n∥}_{0,p}^2+{\tau }^{\gamma }{∥{\mathbf{u}}_N\left(t_n\right)-{\mathbf{u}}_{N,h}^n∥}_{1,f}^2$$

$$+{\tau }^{\gamma }{∥\nabla \left({\varphi }_N\left(t_n\right)-{\varphi }_{N,h}^n\right)∥}_{0,p}^2\le C_{\gamma }\left(h^{2min\left\{m,l\right\}+1}+{\tau }^{6-2\gamma }\right),$$

(60)

which yields the convergence of the approximate solution to the solution of the differential problem, where l and m are defined in (35) and $C_{\gamma }$ is a generic positive constant depending on the order of the fractional derivatives γ but independent of mesh parameters.

Proof. 

Denote ${\widehat{\mathit{\theta }}}^n={\left[{\theta }_{\mathbf{u}}^n,{\theta }_{\varphi }^n\right]}^{\top }$, ${\widehat{\mathit{\xi }}}^n={\left[{\xi }_{\mathbf{u}}^n,{\xi }_{\varphi }^n\right]}^{\top }$ and consider the difference of identities (36) and (31), then choose $\left({\widehat{\mathbf{v}}}_h,q_h\right)=\left({\widehat{\mathit{\theta }}}^n,{\theta }_p^n\right)$ to obtain

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)+\mathcal{A}\left({\overline{\mathbb{K}}}_N;{\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)+\mathcal{I}\left(\sigma {\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)\hfill \\ \hfill & =-{\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)-{\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)-\mathcal{I}\left(\sigma {\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)\hfill \\ \hfill & +{\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)+{\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)+{\mathcal{A}}_p\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N;\sigma {\widehat{\mathit{\theta }}}^n,{\widehat{\mathit{\theta }}}^n\right)\hfill \\ \hfill & +{\left({\mathbf{r}}_{11}^n,{\theta }_{\mathbf{u}}^n\right)}_f+{\left(g{r}_{12}^n,{\theta }_{\varphi }^n\right)}_p+\sum _{i=1}^{d-1}{\int }_I\left({\overline{\varsigma }}_{N,i}-{\varsigma }_{N,i}\right)\left({\mathbf{r}}_{21}^n·{\mathbf{m}}_i\right)\left({\theta }_{\mathbf{u}}^n·{\mathbf{m}}_i\right)ds\hfill \\ \hfill & +g{\left(\left({\overline{\mathbb{K}}}_N-{\mathbb{K}}_N\right)\nabla r_{22}^n,\nabla {\theta }_{\varphi }^n\right)}_p+{\int }_I{\mathbf{r}}_{21}^n·{\theta }_{\varphi }^n{\mathbf{n}}_{f}ds-g{\int }_I{r}_{22}^n{\theta }_{\mathbf{u}}^n·{\mathbf{n}}_{f}ds.\hfill \end{array}$$

(61)

The estimation of the terms mainly repeats the reasoning carried out in Theorem 1 up to notation. Therefore, we focus on the terms that are not in present in (51). Note that

$$\begin{array}{cc}\hfill {\left({\Delta }_{\tau }^{\gamma }{\xi }_{\mathbf{u}}^1,{\theta }_{\mathbf{u}}^1\right)}_f& \le C{∥{\Delta }_{\tau }^{\gamma }{\xi }_{\mathbf{u}}^1∥}_{0,f}^2+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^1∥}_{0,f}^2\hfill \\ \hfill & \le C{\int }_{D_f}{\left({\displaystyle \frac{1}{{\tau }^{\gamma }{\varpi }_{0,}^{\left(\gamma \right)}}}{\int }_0^{\tau }{\partial }_t{\xi }_{\mathbf{u}}\left(s\right)ds\right)}^{2}dx+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^1∥}_{0,f}^2\hfill \\ \hfill & \le C{\tau }^{2-2\gamma }\underset{0\le t\le \tau }{max}{∥{\partial }_t{\xi }_{\mathbf{u}}∥}_{0,f}^2+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^1∥}_{0,f}^2\hfill \\ \hfill & \le C{T}^{2-2\gamma }{h}^{2l+2}+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^1∥}_{0,f}^2,\hfill \\ \hfill -{\tau }^{\gamma }{\left({\Delta }_{\tau }^{\gamma }{\xi }_{\mathbf{u}}^n,{\theta }_{\mathbf{u}}^n\right)}_f& =-{\left({\displaystyle \frac{1}{{\varpi }_1^{\left(\gamma \right)}}}\left({\xi }_{\mathbf{u}}^n-\sum _{s=1}^n{d}_{n-s}^{n,\gamma }{\xi }_{\mathbf{u}}^{n-s}\right)+{\displaystyle \frac{1}{{\varpi }_2^{\left(\gamma \right)}}}\sum _{s=0}^n{\chi }_{n-s}^{n,\gamma }{\xi }_{\mathbf{u}}^{n-s},{\theta }_{\mathbf{u}}^n\right)}_f\hfill \\ \hfill & \le {\displaystyle \frac{1}{{\varpi }_1^{\left(\gamma \right)}}}\left({∥{\xi }_{\mathbf{u}}^n∥}_{0,f}+\sum _{s=1}^n\left|d_{n-s}^{n,\gamma }\right|{∥{\xi }_{\mathbf{u}}^{n-s}∥}_{0,f}\right){∥{\theta }_{\mathbf{u}}^n∥}_{0,f}\hfill \\ \hfill & +{\displaystyle \frac{1}{{\varpi }_2^{\left(\gamma \right)}}}\sum _{s=0}^n\left|{\chi }_{n-s}^{n,\gamma }\right|{∥{\xi }_{\mathbf{u}}^{n-s}∥}_{0,f}{∥{\theta }_{\mathbf{u}}^n∥}_{0,f}\hfill \\ \hfill & \le C{h}^{l+1}{∥{\theta }_{\mathbf{u}}^n∥}_{0,f}+C{h}^{l+1}{\tau }^{\gamma }{∥{\theta }_{\mathbf{u}}^n∥}_{0,f},n\ge 2,\hfill \end{array}$$

where we used the properties (46), (47) of the coefficients $d_{n-s}^{n,\gamma }$ and ${\chi }_{n-s}^{n,\gamma }$. Obtaining similar estimates for ${\left({\Delta }_{\tau }^{\gamma }{\xi }_{\varphi }^n,{\theta }_{\varphi }^n\right)}_p$, we obtain

$$-{\tau }^{\gamma }{\mathcal{D}}_{\tau }^{\gamma }\left({\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)\le C{h}^{2l+2}+C{h}^{2m+2}+{\mu }_0ϵ{\tau }^{\gamma }{∥\nabla {\theta }_{\mathbf{u}}^1∥}_{0,f}^2+g{k}_{min}ϵ{\tau }^{\gamma }{∥\nabla {\varphi }_{N,h}^n∥}_{0,p}^2.$$

Further, using (52), (37), (38) and (39), we have

$$\begin{array}{cc}\hfill & {\mathcal{A}}_I\left({\overline{\varsigma }}_N;{\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)\le C{\tau }^{\gamma }\sum _{i=1}^{d-1}{∥{\xi }_{\mathbf{u}}^n·{\mathbf{m}}_i∥}_{0,I}{∥{\theta }_{\mathbf{u}}^n·{\mathbf{m}}_i∥}_{0,I}\le {\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+C{∥{\xi }_{\mathbf{u}}^n∥}_{0,f}{∥\nabla {\xi }_{\mathbf{u}}^n∥}_{0,f},\hfill \\ \hfill & \mathcal{I}\left(\sigma {\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)\le {\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+g{k}_{min}{∥\nabla {\theta }_{\varphi }^n∥}_{0,p}^2+C{∥\sigma {\xi }_{\varphi }^n∥}_{0,p}{∥\nabla \sigma {\xi }_{\varphi }^n∥}_{0,p}+C{∥\sigma {\xi }_{\mathbf{u}}^n∥}_{0,f}{∥\nabla \sigma {\xi }_{\mathbf{u}}^n∥}_{0,f},\hfill \\ \hfill & {\mathcal{A}}_I\left({\overline{\varsigma }}_N-{\varsigma }_N;\sigma {\widehat{\mathit{\xi }}}^n,{\widehat{\mathit{\theta }}}^n\right)\le {\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+C{∥\sigma {\xi }_{\mathbf{u}}^n∥}_{0,f}{∥\nabla \sigma {\xi }_{\mathbf{u}}^n∥}_{0,f},\hfill \\ \hfill & {\left({\mathbf{r}}_{11}^n,{\theta }_{\mathbf{u}}^n\right)}_f+{\left(g{r}_{12}^n,{\theta }_{\varphi }^n\right)}_p\le C{∥{\mathbf{r}}_{11}^n∥}_{0,f}^2+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+C{∥r_{12}^n∥}_{0,p}^2+g{k}_{min}ϵ{∥\nabla {\theta }_{\varphi }^n∥}_{0,p}^2,\hfill \\ \hfill & \sum _{i=1}^{d-1}{\int }_I\left({\overline{\varsigma }}_{N,i}-{\varsigma }_{N,i}\right)\left({\mathbf{r}}_{21}^n·{\mathbf{m}}_i\right)\left({\theta }_{\mathbf{u}}^n·{\mathbf{m}}_i\right)ds\le C{∥\nabla {\mathbf{r}}_{21}^n∥}_{0,f}^2+{\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2,\hfill \\ \hfill & {\int }_I{\mathbf{r}}_{21}^n·{\theta }_{\varphi }^n{\mathbf{n}}_{f}ds-g{\int }_I{r}_{22}^n{\theta }_{\mathbf{u}}^n·{\mathbf{n}}_{f}ds\le {\mu }_0ϵ{∥\nabla {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+g{k}_{min}ϵ{∥\nabla {\theta }_{\varphi }^n∥}_{0,p}^2+C{∥\nabla {\mathbf{r}}_{21}^n∥}_{0,f}^2+C{∥\nabla r_{22}^n∥}_{0,p}^2.\hfill \end{array}$$

Considering the above inequalities, (61) yields

$${∥{\theta }_{\varphi }^1∥}_{0,p}^2+{∥{\theta }_{\mathbf{u}}^1∥}_{0,f}^2+{\tau }^{\gamma }{∥{\theta }_{\mathbf{u}}^1∥}_{1,f}^2+{\tau }^{\gamma }{∥\nabla {\theta }_{\varphi }^1∥}_{0,p}^2\le C{\tau }^{2+\gamma }+C{h}^{2l+1}+C{h}^{2m+2},$$

(62)

$$\begin{array}{cc}\hfill & {∥{\theta }_{\varphi }^n∥}_{0,p}^2+{∥{\theta }_{\mathbf{u}}^n∥}_{0,f}^2+{\tau }^{\gamma }{∥{\theta }_{\mathbf{u}}^n∥}_{1,f}^2+{\tau }^{\gamma }{∥\nabla {\theta }_{\varphi }^n∥}_{0,p}^2\hfill \\ \hfill & \le C\left(\sum _{s=1}^{n-1}\left|d_{n-s}^{n,\gamma }\right|{∥{\theta }_{\varphi }^{n-s}∥}_{0,p}^2+\sum _{s=1}^{n-1}\left|d_{n-s}^{n,\gamma }\right|{∥{\theta }_{\mathbf{u}}^{n-s}∥}_{0,f}^2+\tau \sum _{s=1}^{n-1}{∥\nabla {\theta }_{\varphi }^{n-s}∥}_{0,p}^2+\tau \sum _{s=1}^{n-1}{∥\nabla {\theta }_{\mathbf{u}}^{n-s}∥}_{0,f}^2\right)\hfill \\ \hfill & +C{\tau }^{\gamma }\left({∥\nabla \sigma {\theta }_{\varphi }^n∥}_{0,p}^2+{∥\nabla \sigma {\theta }_{\mathbf{u}}^n∥}_{0,f}^2+{∥\nabla {\mathbf{r}}_{21}^n∥}_{0,f}^2+{∥\nabla r_{22}^n∥}_{0,p}^2+{\tau }^{6-2\gamma }\right)+C\left(h^{2l+1}+h^{2m+1}\right),n\ge 2.\hfill \end{array}$$

(63)

The inequalities (62) and (63) and relations (33) and (34) imply that the error at the first and second time layers is of the order lower than $O\left({\tau }^{3-\gamma }\right)$. To increase the convergence order, let us introduce subpartitions on the intervals $\left(0,t_1\right)$ and $\left(t_1,t_2\right)$ with the time steps (59). Then, it follows from (63) that

$${∥{\theta }_{\varphi }^n∥}_{0,p}^2+{∥{\theta }_{\mathbf{u}}^n∥}_{0,f}^2+{\tau }^{\gamma }{∥{\theta }_{\mathbf{u}}^n∥}_{1,f}^2+{\tau }^{\gamma }{∥\nabla {\theta }_{\varphi }^n∥}_{0,p}^2\le C\sum _{s=1}^{n-1}\left|d_s^{n,\gamma }\right|\left({∥{\theta }_{\varphi }^s∥}_{0,p}^2+{∥{\theta }_{\mathbf{u}}^s∥}_{0,f}^2\right)$$

$$+C_1{\tau }^{\gamma }\sum _{s=1}^{n-1}\left|d_s^{n,\gamma }\right|\left({∥\nabla {\theta }_{\varphi }^s∥}_{0,p}^2+{∥\nabla {\theta }_{\mathbf{u}}^s∥}_{0,f}^2\right)+C{\tau }^{6-2\gamma }+C{h}^{2l+1}+C{h}^{2m+1},$$

where ${\displaystyle C_1=\underset{0\le s\le n-1}{max}\left\{max\left\{{\left|d_s^{n,\gamma }\right|}^{-1},1\right\}\right\}}$. Finally, taking into account (46), and applying Lemma 1, we arrive at the assertion of the theorem. □

3. Results

3.1. Verification of the Convergence Order

To verify the theoretical convergence estimate obtained in Theorem 2 for the semi-discrete scheme, numerical tests have been conducted on the example of a model problem with a sufficiently smooth known exact solution.

Example 1. 

Consider Problem 1 in domains $D_f=\left(0,1\right)\times \left(0,1\right)$, $D_p=\left(0,1\right)\times \left(1,2\right)$ with the following set of parameters: $\mu =1$, $g=1$, $S=1$, ${\alpha }_{\mathrm{BJS}}=0.01$, $T=1$, and ${\mathbb{K}}_N=k\mathbb{I}$, where k is a constant. The right-hand sides of Equations (12) and (15) are defined as follows:

$$\begin{array}{cc}\hfill f_p& =-{\displaystyle \frac{6S{t}^{3-\gamma }}{k\Gamma \left(4-\gamma \right)}}\left({\displaystyle \frac{x_2^3}{3}}-x_2^2+x_1{x}_2\left(x_1-1\right)+x_2\right)+2t^3\left(2x_2-1\right),\hfill \\ \hfill {\mathbf{f}}_f& =\left[{\displaystyle \frac{6t^{3-\gamma }\left(1-2x_1\right)\left(x_2-1\right)}{\Gamma \left(4-\gamma \right)}}-{\displaystyle \frac{g{t}^3{x}_2\left(2x_1-1\right)}{k}},\right.\hfill \\ \hfill & {\left.{\displaystyle \frac{6t^{3-\gamma }\left(x_1\left(x_1-1\right)+{\left(x_2-1\right)}^2\right)}{\Gamma \left(4-\gamma \right)}}-{\displaystyle \frac{g{t}^3\left(x_2^2-2x_2+x_1\left(x_1-1\right)+1\right)}{k}}\right]}^{\top },\hfill \end{array}$$

and first-kind boundary conditions are imposed both on ${\mathbf{u}}_N$ and ${\varphi }_N$ on outer boundaries of the corresponding domains which can be derived directly from the known exact solution:

$$\begin{array}{cc}& {\varphi }_N=-{\displaystyle \frac{t^3}{k}}\left({\displaystyle \frac{x_2^3}{3}}-x_2^2+x_1{x}_2\left(x_1-1\right)+x_2\right),\hfill \\ & {\mathbf{u}}_N={\left[t^3\left(1-2x_1\right)\left(x_2-1\right),t^3\left(x_1\left(x_1-1\right)+{\left(x_2-1\right)}^2\right)\right]}^{\top },\hfill \\ & p_N=4\mu t^3\left(x_2-1\right)-{\displaystyle \frac{g{t}^3}{k}}\left({\displaystyle \frac{x_2^3}{3}}-x_2^2+x_1{x}_2\left(x_1-1\right)+x_2\right).\hfill \end{array}$$

The numerical test was conducted to examine the relationship between the empirical convergence order and fractional derivative orders, while also assessing its alignment with the theoretical convergence order obtained in Theorem 2. To this end, we considered a sequence of time steps $\left\{{\tau }_i\right\}$ decreasing by half, a fragment of which is shown in the first column of

Table 1. The errors ${\displaystyle \underset{n}{max}{∥{\mathbf{u}}_N^n-{\mathbf{u}}_{N,h}^n∥}_{0,f}}$ and ${\displaystyle \underset{n}{max}{∥{\mathbf{u}}_N^n-{\mathbf{u}}_{N,h}^n∥}_{1,f}}$, and corresponding convergence orders computed for different values of $\tau$ and orders of fractional derivatives.

$\mathit{\tau }$	$\mathit{\gamma }=0.1$		$\mathit{\gamma }=0.5$		$\mathit{\gamma }=0.9$
	${\mathit{L}}^{\mathbf{2}}$-Error	Order	${\mathit{L}}^{\mathbf{2}}$-Error	Order	${\mathit{L}}^{\mathbf{2}}$-Error	Order
$\tau =1/10$	$7.9412\times {10}^{-6}$	–	$1.1255\times {10}^{-5}$	–	$2.2792\times {10}^{-5}$	–
$\tau =1/20$	$1.0386\times {10}^{-6}$	2.93	$1.7146\times {10}^{-6}$	2.71	$5.2935\times {10}^{-6}$	2.11
$\tau =1/40$	$1.3684\times {10}^{-7}$	2.92	$2.7390\times {10}^{-7}$	2.65	$1.2354\times {10}^{-6}$	2.10
$\tau =1/80$	$1.8160\times {10}^{-8}$	2.91	$4.6990\times {10}^{-8}$	2.54	$2.8804\times {10}^{-7}$	2.10
$\tau =1/160$	$2.4187\times {10}^{-9}$	2.91	$8.2064\times {10}^{-9}$	2.52	$6.7237\times {10}^{-8}$	2.10
Expected		2.90		2.50		2.10
	${\mathit{H}}^{\mathbf{1}}$-Error	Order	${\mathit{H}}^{\mathbf{1}}$-Error	Order	${\mathit{H}}^{\mathbf{1}}$-Error	Order
$\tau =1/10$	$1.3885\times {10}^{-4}$	–	$1.1440\times {10}^{-4}$	–	$2.2641\times {10}^{-4}$	–
$\tau =1/20$	$1.8676\times {10}^{-5}$	2.89	$2.0072\times {10}^{-5}$	2.51	$6.4810\times {10}^{-5}$	1.80
$\tau =1/40$	$2.5366\times {10}^{-6}$	2.88	$3.5815\times {10}^{-6}$	2.49	$1.9139\times {10}^{-5}$	1.76
$\tau =1/80$	$3.4522\times {10}^{-7}$	2.88	$6.7946\times {10}^{-7}$	2.40	$5.8900\times {10}^{-6}$	1.70
$\tau =1/160$	$4.7485\times {10}^{-8}$	2.86	$1.3905\times {10}^{-7}$	2.29	$1.8343\times {10}^{-6}$	1.68
Expected		2.85		2.25		1.65

and

Table 2. The errors ${\displaystyle \underset{n}{max}{∥{\varphi }_N^n-{\varphi }_{N,h}^n∥}_{0,f}}$ and ${\displaystyle \underset{n}{max}{∥{\varphi }_N^n-{\varphi }_{N,h}^n∥}_{1,f}}$, and corresponding convergence orders computed for different values of $\tau$ and orders of fractional derivatives.

$\mathit{\tau }$	$\mathit{\gamma }=0.1$		$\mathit{\gamma }=0.5$		$\mathit{\gamma }=0.9$
	${\mathit{L}}^{\mathbf{2}}$-Error	Order	${\mathit{L}}^{\mathbf{2}}$-Error	Order	${\mathit{L}}^{\mathbf{2}}$-Error	Order
$\tau =1/10$	$2.5172\times {10}^{-4}$	–	$1.0767\times {10}^{-4}$	–	$1.8017\times {10}^{-4}$	–
$\tau =1/20$	$3.2791\times {10}^{-5}$	2.94	$1.4532\times {10}^{-5}$	2.89	$4.1377\times {10}^{-5}$	2.12
$\tau =1/40$	$4.2806\times {10}^{-6}$	2.94	$1.9790\times {10}^{-6}$	2.88	$9.5844\times {10}^{-6}$	2.11
$\tau =1/80$	$5.6339\times {10}^{-7}$	2.93	$2.8631\times {10}^{-7}$	2.79	$2.2380\times {10}^{-6}$	2.10
$\tau =1/160$	$7.4955\times {10}^{-8}$	2.91	$4.6679\times {10}^{-8}$	2.62	$5.2317\times {10}^{-7}$	2.10
Expected		2.90		2.50		2.10
	${\mathit{H}}^{\mathbf{1}}$-Error	Order	${\mathit{H}}^{\mathbf{1}}$-Error	Order	${\mathit{H}}^{\mathbf{1}}$-Error	Order
$\tau =1/10$	$4.6635\times {10}^{-4}$	–	$3.0754\times {10}^{-4}$	–	$6.1386\times {10}^{-4}$	–
$\tau =1/20$	$6.2985\times {10}^{-5}$	2.89	$5.3492\times {10}^{-5}$	2.52	$1.7309\times {10}^{-4}$	1.83
$\tau =1/40$	$8.6070\times {10}^{-6}$	2.87	$9.5507\times {10}^{-6}$	2.49	$5.0645\times {10}^{-5}$	1.77
$\tau =1/80$	$1.1793\times {10}^{-6}$	2.87	$1.9000\times {10}^{-6}$	2.33	$1.5543\times {10}^{-5}$	1.70
$\tau =1/160$	$1.6319\times {10}^{-7}$	2.85	$3.9935\times {10}^{-7}$	2.25	$4.9018\times {10}^{-6}$	1.66
Expected		2.85		2.25		1.65

. Each of the subdomains $D_f$ and $D_p$ contained 1024 finite elements, and the diameter of the triangulation was $h\approx 0.044194$.

We considered 25 samples of the parameter k generated from the uniform distribution in the segment $\left[0.9,1.1\right]$. Having found the mean of the solutions corresponding to each sample k, we calculated the errors $E_{{\tau }_i}$ in the $L^2$ and $H^1$-norms for each ${\tau }_i$, $i\ge 1$, and computed the empirical convergence orders by the formula

$$R_i={\displaystyle \frac{log{E}_{{\tau }_i}-log{E}_{{\tau }_{i-1}}}{log{\tau }_i-log{\tau }_{i-1}}},i\ge 2.$$

In the experiments, we considered a few fractional derivative orders, $\gamma =0.1,0.5,0.9$, and presented the convergence analysis in Table 1 for ${\mathbf{u}}_{N,h}$ and Table 2 for ${\varphi }_{N,h}$, respectively. Both empirical and theoretical convergence orders predicted in Theorem 2 are indicated in the “Order” column.

The empirical convergence order in the $L^2$-norm, as inferred from Table 1 and Table 2, was significantly affected by the order of the fractional derivatives. Notably, the convergence order was around 2.90 at $\gamma =0.1$, but it decreased significantly as the fractional derivative orders increased. The observed behavior was consistent with the theoretical predictions derived in Theorem 2. Similar observations can be made for $H^1$-norm analysis.

3.2. Impact of the Orders of Fractional Derivatives on Flow Pattern

Now let us assess the impact of the orders of fractional derivatives on the fluid flow pattern on a few test problems without a known exact solution. In the following example, we perform such an analysis by solving a deterministic problem with the only constant hydraulic conductivity tensor.

Example 2. 

Consider Problem 1 in domains $D_f=\left(0,1\right)\times \left(0,0.5\right)$, $D_p=\left(0,1\right)\times \left(0.5,1\right)$ with initial conditions ${\mathbf{u}}_N=\mathbf{0}$, $x\in {\overline{D}}_f$, ${\varphi }_N=0$, $x\in {\overline{D}}_p$, $t=0$, and the following boundary conditions:

$${\mathbf{u}}_N=\left\{\begin{array}{cc}{\left[\left(x_2-0.5\right)\left(1-x_2\right),0\right]}^{\top },\hfill & x\in \left\{0\right\}\times \left(0.5,1\right),\hfill \\ {\left[0,0\right]}^{\top },\hfill & x\in \left[0,1\right]\times \left\{1\right\}\cup \left\{1\right\}\times \left(0.5,1\right),\hfill \end{array}\right.$$

$$\begin{array}{cccc}& {\varphi }_N=0,\hfill & \hfill & x\in \left[0,1\right]\times \left\{0\right\},t>0,\hfill \\ & \nabla {\varphi }_N·{\mathbf{n}}_p=0,\hfill & & x\in \left\{0,1\right\}\times \left(0,0.5\right).\hfill \end{array}$$

The rest of the parameters are chosen as follows: ${\mathbf{f}}_f\equiv \mathbf{0}$, $f_p\equiv 0$, ${\mathbb{K}}_N=0.01\mathbb{I}$, $\mu =S=1$, $g=0.05$, ${\alpha }_{\mathrm{BJS}}=0.01$, $T=20$.

We solved Problem 1 with $\gamma \in \left\{0.05,0.3,0.5,0.8,0.95\right\}$. Each of the subdomains $D_f$ and $D_p$ contained 2500 finite elements, the diameter of the triangulation was $h\approx 0.022361$, and the time step was approximately equal to 0.02. After solving the problem, the velocity field in the porous medium $D_p$ was recovered using Equation (2). The fluid velocity magnitude in both subdomains at final time $T=20$ are depicted in Figure 2b–f.

Further, we solved Problem 1 with integer-order temporal derivatives, i.e., in the case of $\gamma =1$. To this end, the following approximations of the first-order derivative were utilized in (32) instead of (27) and (28):

$${\Delta }_{\tau }^1{\mathbf{u}}_N^n=\left\{\begin{array}{cc}{\displaystyle \frac{{\mathbf{u}}_N^1-{\mathbf{u}}_N^0}{\tau },}\hfill & n=1,\hfill \\ {\displaystyle \frac{3{\mathbf{u}}_N^n-4{\mathbf{u}}_N^{n-1}+{\mathbf{u}}_N^{n-2}}{2\tau },}\hfill & n\ge 2,\hfill \end{array}\right.{\Delta }_{\tau }^1{\varphi }_N^n=\left\{\begin{array}{cc}{\displaystyle \frac{{\varphi }_N^1-{\varphi }_N^0}{\tau },}\hfill & n=1,\hfill \\ {\displaystyle \frac{3{\varphi }_N^n-4{\varphi }_N^{n-1}+{\varphi }_N^{n-2}}{2\tau },}\hfill & n\ge 2.\hfill \end{array}\right.$$

The solution result at $T=20$ is presented in Figure 2a.

Overall, a simple analysis shows that the velocity of fluid flow through a porous medium slows down significantly as the order of the fractional derivative decreases. This behavior can also be observed in a slice of the subdomains at $x_1=0.15$, as shown in Figure 3.

By comparing results corresponding to cases $\gamma =0.95$ and $\gamma =1$ and observing the consistency of the curves, it is important to note that a small change in the order of the derivative leads to a small change in the solution, which may indicate that the model under study is indeed a generalization of the classical Stokes–Darcy model.

Let us now study the impact of the fractional derivative orders on the flow pattern in the case of a non-homogeneous hydraulic conductivity tensor.

Example 3. 

Consider Problem 2 with the diagonal hydraulic conductivity tensor, $\mathbb{K}\left(x,\omega \right)=\mathrm{diag}\left(k_1\left(x,\omega \right),k_2\left(x,\omega \right)\right)$ [54], where

$$k_s\left(x,\omega \right)=0.5+exp\left(1+Y_1\left(\omega \right){\left({\displaystyle \frac{\sqrt{\pi }L}{2}}\right)}^{1/2}+\sum _{n=2}^N{\zeta }_n{\gamma }_n\left(x_s\right)Y_n\left(\omega \right)\right),$$

(64)

where

$${\zeta }_n={\left(\sqrt{\pi }L\right)}^{1/2}exp\left({\displaystyle \frac{-{\left(⌊n/2⌋\pi L\right)}^2}{8}}\right),n\ge 2,$$

and

$${\gamma }_n\left(x\right)=\left\{\begin{array}{cc}sin\left({\displaystyle \frac{⌊n/2⌋\pi x}{L_p}}\right),\hfill & \mathrm{if}n\mathrm{is}\mathrm{even},\hfill \\ cos\left({\displaystyle \frac{⌊n/2⌋\pi x}{L_p}}\right),\hfill & \mathrm{if}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

The problem was solved for the following set of parameters: $T=1$, $\mu =1$, $g=1$, ${\alpha }_{\mathrm{BJS}}=1$, $S=1$. Each of the subdomains $D_f$ and $D_p$ contained 400 finite elements, the diameter of the triangulation was $h\approx 0.055902$, and the time step was approximately equal to 0.05. The simulation results are presented in Figure 4 for $\gamma =0.50$ and $\gamma =0.95$. One can observe a slight slowdown in the velocity and dynamics of the piezometric head as the order of the fractional derivative decreases.

Then, we solved two variations of Example 2 with a randomly generated conductivity tensor without employing the expansion (11).

Example 4. 

Consider Problem 2 with the conductivity tensor $\mathbb{K}\left(x,\omega \right)=\mathrm{diag}\left(k\left(x,\omega \right),k\left(x,\omega \right)\right)$, where

$$k\left(x,\omega \right)=a+exp\left(\lambda \sum _{i=1}^N\prod _{s=1}^2\left(Y_{4(i-1)+2s-1}\left(\omega \right)cos\left(i\pi x_s\right)+Y_{4(i-1)+2s}\left(\omega \right)sin\left(i\pi x_s\right)\right)\right)$$

(65)

with $x=\left(x_1,x_2\right)$ and the following two cases of parameters:

$\mathrm{Case}\mathrm{I}.$ $a=0.01$, $N=7$, $\lambda =exp\left(-{\displaystyle \frac{1}{8}}\right)$.

$\mathrm{Case}\mathrm{II}.$ $a=0.001$, $N=9$, $\lambda =exp\left(-{\displaystyle \frac{1}{8}}\right)$.

The rest of the parameters remained the same as in Problem 2.

Example 5. 

Consider Problem 4 in the case when ${\Gamma }_{\mathrm{in}}:=\left\{0<x_1<1,x_2=1\right\}$ is the inlet boundary on which ${\mathbf{u}}_N={\left[0,x_1\left(1-x_1\right)\right]}^{\top }$; homogeneous boundary conditions of the second kind are imposed on the left and right boundaries of $D_f$, and the boundary conditions for ${\varphi }_N$, as well as the remaining parameters, are the same as in Problem 4.

We first generated hydraulic conductivity samples according to a set of parameters defined in Example 4. The resulting hydraulic conductivity plots are shown in Figure 5. Numerical tests were conducted for $\gamma \in \left\{0.05,0.5,0.95\right\}$. The expectation of the velocity magnitude at final time $T=20$ corresponding to Cases I and II is presented in Figure 6 and Figure 7, respectively.

As in Example 2, a comparison of the presented plots also indicates that a decrease in the order of fractional derivatives leads to a deceleration in the flow velocity.

3.3. Efficiency Tests

Now we conduct some CPU time benchmarks of the algorithm depending on the dimension of the stochastic space in Example 3. The numerical algorithm described in the study was implemented in the Julia programming language [60] and was run on a Macbook Pro powered by the Apple M2 Max processor with 12 cores and equipped with 32 GB of unified memory. The finite element analysis was implemented with the use of the Ferrite.jl package [61]. The assembly of the stiffness matrix in parallel was achieved through a graph coloring strategy that computes non-adjacent local elements simultaneously. The solution of the resulting systems of linear equations was implemented using the Panua Pardiso library [62,63,64]. Furthermore, we used the SmolyakApprox.jl package [65] to determine sparse grid collocation points and the quadrature weights.

Table 3. CPU time benchmark conducted for $N=5$ and various levels of approximation $\mathcal{L}$.

Level of Approximation, $\mathcal{L}$	3	4	5	6	7
Number of samples	241	801	2433	6993	19,313
CPU Time (s)	5.59	16.05	49.05	158.59	645.85

outlines CPU time benchmarks conducted for a fixed number of terms $N=5$ in the expansion (64) and various levels of approximation $\mathcal{L}$ used in the Smolyak method.

In the second test, we fixed the level of approximation $\mathcal{L}=5$ and considered various numbers of terms N in the expansion (64). The results of the numerical test are presented in

Table 4. CPU time benchmark conducted for a fixed level of approximation $\mathcal{L}=5$ and various numbers of terms N in the expansion (64).

N	3	4	5	6	7
Number of samples	441	1105	2433	4865	9017
CPU Time (sec.)	12.04	22.26	49.05	109.41	285.84

.

Finally, let us present a CPU time benchmark analysis of two algorithms for solving the stochastic problem described in Example 2. The first algorithm is taken from [10], which uses an approximation formula of order $O\left({\tau }^{3-\gamma }\right)$ for the Caputo fractional derivative, and it is compared with the algorithm proposed in this study. In this numerical test, we considered 100 samples of the hydraulic conductivity tensor generated from a uniform distribution in the segment $\left[0.8,1.2\right]$. Each of the subdomains $D_f$ and $D_p$ contained 100 finite elements, and time steps were equal to $1/100$, $1/1000$ and 1/10,000.

It follows from the analysis presented in

Table 5. Comparison of CPU time benchmarks (in seconds).

Time Step, $\mathit{\tau }$	$1/100$	$1/1000$	1/10,000
Parallel algorithm from [10]	5.30	77.84	4115.23
Our algorithm	14.27	154.01	1792.39

that at relatively large time steps, the use of the approximation formula applied in [10] is approximately 2–2.5 times faster than that utilized in our study. However, when choosing a finer time step, utilizing approximation Formulas (27) and (28) turned out to be 2.3 times faster than the parallel algorithm used in [10]. Moreover, the memory usage remained nearly constant across considered time step values when applying our algorithm. In contrast, the first algorithm required significantly larger arrays to store the solution on all time layers.

4. Conclusions

Having performed a theoretical study on the proposed numerical method and carefully studying the computational experiment findings, we can outline the subsequent conclusions.

(1) Based on the results of Example 2, one can conclude that the replacement of integer-order derivatives, ${\partial }_t\mathbf{u}$ and ${\partial }_t\varphi$, with fractional-order derivatives, ${\partial }_{0,t}^{0.95}\mathbf{u}$ and ${\partial }_{0,t}^{0.95}\varphi$, in the governing equations led to a slight deceleration in the flow velocity. Due to the ability of fractional derivatives to capture long-term changes in the medium properties [17,18,19], this observation may confirm the fact that the porous medium exhibits a retarding effect on the flow process over time. Further decrease in the order of the fractional derivatives led to a more significant deceleration effect. This may explain the fact that the orders of the fractional derivatives determine the degree of memory impact on the flow pattern. This behavior is in good agreement with previous studies conducted in [17,66].

(2) The results obtained in Example 1 show that the orders of fractional derivatives also significantly affect the convergence order of the numerical scheme. Specifically, an increase in the orders of the fractional derivatives decreases the convergence order.

(3) The use of the fast algorithm for the Caputo fractional derivative allowed us to significantly reduce computational resources. In fact, a comparison with the numerical algorithm considered in [10] showed a significant reduction in both computational time and memory requirements while maintaining the order of convergence.

In subsequent papers, the authors intend to consider more realistic examples within the model under study. Specifically, it is interesting to analyze the impact of memory on various natural and industrial phenomena such as flows in karst aquifers, blood circulation and transport in surface–subsurface systems.