Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model

Abstract

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (${\mathbf{P}}_1$-$P_1$-$P_1$) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.

1. Introduction

The Stokes–Darcy system describes a model that is composed of the Stokes equations for the fluid flow and Darcy’s law for the porous media flow with coupled interface conditions. The finite element method is widely used for solving this model since it is more applicable to the complex region and its numerical analysis is perfect. Until now, there have been lots of works on this coupled model, for instance, coupled finite element methods [1,2,3], two-grid methods [4,5,6], multi-grid methods [7], domain decomposition methods [8,9,10,11], and so on. Compared with coupled methods, decoupled methods could save a lot of computing time by dividing the coupled problem into two sub-problems. Among these decoupled methods, the local and parallel finite element methods were first proposed by Xu and Zhou to solve the elliptic boundary value problems by combining the two-grid method and domain decomposition technique in [12,13]. Subsequently, many researchers generalized them to the Stokes problem [14], the Navier–Stokes problem [15,16,17], the Stokes–Darcy model [18], the Navier–Stokes–Darcy model [19], and the MHD problem [20].

In this paper, the local and parallel finite element methods are considered to solve the coupled problem. The steps of the local and parallel finite element method can be presented as follows. Firstly, the Stokes–Darcy model is approximated to obtain the numerical solution by using standard finite element method on a coarse grid. Secondly, the coupled problem is decoupled into two individual sub-problems and the complete domain is divided into a series of sub-domains. Then, the residual problems are solved on a fine grid in these sub-domains. To avoid the effect of the artificial Dirichlet boundary condition, each sub-domain is properly enlarged to a larger domain. Finally, the numerical solution is assembled on the coarse grid and the residual together. However, the solution for the Stokes–Darcy model using local and parallel finite element methods can be improved since the solutions are, in general, globally discontinuous. To overcome this drawback, the most popular idea is to introduce the partition of unity technique, which is adaptable and controllable, to decompose the computational domain [21,22].

It is well known that finite element spaces utilized for the coupled Stokes–Darcy model should satisfy the inf-sup (or LBB) condition. Although the lowest equal-order finite element pairs do not satisfy the inf-sup condition, they are computationally convenient in practical applications because of the identical degree distribution for the velocity and pressure. Therefore, the lowest equal-order finite element pairs have attracted much more attention in recent years. To circumvent the inf-sup condition, many stabilized techniques have been researched, such as local pressure projection stabilized methods [23,24,25] and the stabilized methods based on two local Gauss integrations [26,27,28,29,30]. Among these stabilized methods, the stabilized method based on two local Gauss integrations does not need to calculate high-order derivatives and construct the projection operator; however, it can be computed at the element level.

In this paper, two parallelized stabilized finite element algorithms are proposed and analyzed for the mixed Stokes–Darcy model by combining the classical local and parallel finite element methods and the stabilized method based on two local Gauss integration techniques. Compared to our previous work, for instance [18], the lowest equal-order finite element pairs are considered. The algorithms in this study and those in [18] were devised with the understanding that, for a solution to the mixed problem, the low-frequency components have the global property while the high-frequency components have the local property. Hence, the low-frequency components are computed on a coarse mesh and the high-frequency components are obtained on a fine mesh by some local and parallel procedures. The theoretical results indicate that our methods could derive the same error convergence orders as the parallel methods provided in [18]. On the other hand, the numerical results show that algorithms in this study could achieve a better error accuracy and take less time compared with the parallel methods provided in [18].

The rest of this article is organized as follows. In Section 2, the Stokes–Darcy model is introduced. The finite element spaces and some useful notations are described in Section 3. In Section 4, two local and parallel finite element methods are proposed. In Section 5, the theoretical analysis is presented. Some numerical results are reported to verify the validity and efficiency of the presented algorithms in Section 6. Finally, a conclusion is derived in Section 7.

2. The Stokes–Darcy Model

Let ${\Omega }_f\subset R^d(d=2,3)$ be a fluid region and ${\Omega }_p\subset R^d$ be a porous media region with ${\Omega }_f\cap {\Omega }_p=\varnothing ,\overline{{\Omega }_f}\cap \overline{{\Omega }_p}=\Gamma ,\overline{{\Omega }_f}\cup \overline{{\Omega }_p}=\overline{\Omega }.$ Denote ${\Gamma }_f=\partial {\Omega }_f\setminus \Gamma ,{\Gamma }_p=\partial {\Omega }_p\setminus \Gamma$.

In the fluid region ${\Omega }_f$, the fluid is governed by the Stokes equations as follows:

$$\left\{\begin{array}{c}-\nabla ·\mathbb{T}(\mathit{u},p)={\mathit{f}}_1\mathrm{in}{\Omega }_f,\hfill \\ \nabla ·\mathit{u}=0\mathrm{in}{\Omega }_f,\hfill \end{array}\right.$$

(1)

where $\mathbb{T}=-p\mathbb{I}+2\nu \mathbb{D}\left(\mathit{u}\right)$ is the stress tensor, $\mathbb{I}$ is the identity matrix, $\nu$ is the kinematic viscosity, $\mathbb{D}\left(\mathit{u}\right)=\frac{1}{2}(\nabla \mathit{u}+{\nabla }^T\mathit{u})$ is the velocity deformation tensor, $\mathit{u}$ represents the velocity, p represents the kinematic pressure, and ${\mathit{f}}_1$ represents the external force.

In the porous media region ${\Omega }_p$, the fluid is governed by the equations as follows:

$$\left\{\begin{array}{c}\nabla ·{\mathit{u}}_p=f_2\mathrm{in}{\Omega }_p,\hfill \\ {\mathit{u}}_p=-\mathbb{K}\nabla \varphi \mathrm{in}{\Omega }_p,\hfill \end{array}\right.$$

(2)

where ${\mathit{u}}_p$ denotes the fluid velocity, $f_2$ denotes a source term, $\mathbb{K}$ denotes the hydraulic conductivity, and $\varphi =z+\frac{p_p}{\rho g}$ denotes the piezometric head, with z being the height from a reference level, $p_p$ being the dynamic pressure, $\rho$ being the density of the fluid, and g being the gravitational acceleration.

Then, Equation (2) is rewritten in the following formalization:

$$\begin{array}{c}\hfill -\nabla ·(\mathbb{K}\nabla \varphi )=f_2\mathrm{in}{\Omega }_p,\end{array}$$

(3)

which we shall consider in the following paper.

On the interface $\Gamma$, the following conditions are considered:

$$\left\{\begin{array}{c}\mathit{u}·{\mathit{n}}_f+{\mathit{u}}_p·{\mathit{n}}_p=0,\hfill \\ -[\mathbb{T}(\mathit{u},p)·{\mathit{n}}_f]·{\mathit{n}}_f=\rho g\varphi ,\hfill \\ -[\mathbb{T}(\mathit{u},p)·{\mathit{n}}_f]·{\mathit{\tau }}_i=\alpha \sqrt{\frac{\nu g}{tr\left(\mathbb{K}\right)}}\mathit{u}·{\mathit{\tau }}_i,\hfill \end{array}\right.$$

(4)

where ${\mathit{n}}_f$ is the unit normal vector on $\Gamma$ from ${\Omega }_f$ to ${\Omega }_p$, ${\mathit{n}}_p$ is the unit normal vector on $\Gamma$ from ${\Omega }_p$ to ${\Omega }_f$, ${\left\{{\mathit{\tau }}_i\right\}}_{i=1}^{d-1}$ are the tangential unit vectors on $\Gamma$, and $\alpha$ is a positive constant, which is dependent on the property of the porous media region. The first equation is the mass conservation, the second one denotes the balance of normal forces, and the last one is the well-known Beavers–Joseph–Saffman interface condition, which is the simplification of the Beavers–Joseph interface condition.

For the sake of simplicity, the following Dirichlet boundary conditions are considered:

$$\mathit{u}=0\mathrm{on}{\Gamma }_f,\varphi =0\mathrm{on}{\Gamma }_p.$$

In the following, the standard Sobolev spaces and related norms are utilized. Furthermore, for a domain D, let ${(·,·)}_D$ stand for the usual $L^2$ inner product on D. To derive the weak formulation of the mixed Stokes–Darcy problem, the following spaces are introduced:

$$\begin{array}{cc}\hfill & H_f=\{\mathit{v}\in H^1{\left({\Omega }_f\right)}^d:\mathit{v}=0\mathrm{on}{\Gamma }_f\},\hfill \\ \hfill & H_p=\{\psi \in H^1\left({\Omega }_p\right):\psi =0\mathrm{on}{\Gamma }_p\},\hfill \\ \hfill & Q=L^2\left({\Omega }_f\right),\hfill \\ \hfill & W=H_f\times H_p,\hfill \\ \hfill & X=W\times Q.\hfill \end{array}$$

Then, the weak formulation of the coupled Stokes–Darcy model with the Beavers–Joseph–Saffman interface condition reads as follows: find $\overrightarrow{u}=(\mathit{u},\varphi )\in W$, $p\in Q,$ such that

$$\left\{\begin{array}{c}a(\overrightarrow{u},\overrightarrow{v})+b(\overrightarrow{v},p)=(\overrightarrow{f},\overrightarrow{v})\forall \overrightarrow{v}=(\mathit{v},\psi )\in W,\hfill \\ b(\overrightarrow{u},q)=0\forall q\in Q,\hfill \end{array}\right.$$

(5)

where

$$\begin{array}{cc}\hfill & a(\overrightarrow{u},\overrightarrow{v})=a_{\Omega }(\overrightarrow{u},\overrightarrow{v})+a_{\Gamma }(\overrightarrow{u},\overrightarrow{v})=a_f(\mathit{u},\mathit{v})+a_p(\varphi ,\psi )+a_{\Gamma }(\overrightarrow{u},\overrightarrow{v}),\hfill \\ \hfill & a_f(\mathit{u},\mathit{v})=2\nu {(\mathbb{D}\left(\mathit{u}\right),\mathbb{D}\left(\mathit{v}\right))}_{{\Omega }_f}+\alpha \sqrt{\frac{\nu g}{tr\left(\mathbb{K}\right)}}{\int }_{\Gamma }{P}_{\tau }\mathit{u}·P_{\tau }\mathit{v},P_{\tau }\mathit{v}=\sum _{j=1}^{d-1}(\mathit{v}·{\mathit{\tau }}_j){\mathit{\tau }}_j,\hfill \\ \hfill & a_p(\varphi ,\psi )=\rho g{(\mathbb{K}\nabla \varphi ,\nabla \psi )}_{{\Omega }_p},a_{\Gamma }(\overrightarrow{u},\overrightarrow{v})=\rho g{\int }_{\Gamma }(\varphi \mathit{v}-\psi \mathit{u})·{\mathit{n}}_{\mathit{f}},\hfill \\ \hfill & b(\overrightarrow{v},p)\equiv b(\mathit{v},p)=-{(p,\nabla ·\mathit{v})}_{{\Omega }_f},(\overrightarrow{f},\overrightarrow{v})={({\mathit{f}}_1,\mathit{v})}_{{\Omega }_f}+\rho g{(f_2,\psi )}_{{\Omega }_p}.\hfill \end{array}$$

3. Stabilized Finite Element Approximation

Let ${\tau }_h$ be a regular triangulation of $\Omega$. The triangles $K_i$ ($i=1,\cdots ,M$) satisfy $\overline{\Omega }=\overline{K_1}\cup \overline{K_2}\cup \cdots \cup \overline{K_M}$, $h=\underset{K\in {\tau }_h}{max}diam\left(K\right)$. Assume the triangulation ${\tau }_h\left({\Omega }_f\right)$ is compatible with ${\tau }_h\left({\Omega }_p\right)$ on the interface $\Gamma$. Define the following finite element spaces as

$$\begin{array}{cc}\hfill & H_{f,h}=\{\mathit{v}\in H_f:\mathit{v}{|}_K\in {\mathbf{P}}_1\triangleq P_1^d,\forall K\in {\tau }_h\left({\Omega }_f\right)\},\hfill \\ \hfill & H_{p,h}=\{\psi \in H_p:\psi {|}_K\in P_1,\forall K\in {\tau }_h\left({\Omega }_p\right)\},\hfill \\ \hfill & Q_h=\{q\in Q:q{|}_K\in P_1,\forall K\in {\tau }_h\left({\Omega }_f\right)\},\hfill \\ \hfill & W_h=H_{f,h}\times H_{p,h},\hfill \\ \hfill & X_h=W_h\times Q_h.\hfill \end{array}$$

It is well known that the above finite element spaces $H_{f,h}\times Q_h$ do not satisfy the discrete inf-sup condition. To derive a stable numerical solution, the following stabilization term is introduced:

$$G(p,q)=\lambda \left(\right(I-\Pi )p,(I-\Pi \left)q\right),$$

(6)

where the stabilization parameter $\lambda$ satisfies $0<\lambda <1$. For the local pressure projection $\Pi :L^2(\Omega )\to R$, $R\subset \Omega$, there holds

$$\begin{array}{cc}\hfill & (p,q)=(\Pi p,q)\forall p\in L^2(\Omega ),q\in R,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\parallel \Pi p\parallel }_0\le c{\parallel p\parallel }_0\forall p\in L^2(\Omega ),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \left|(\mathit{I}-\Pi )p\right|\le c{h}^m{\parallel p\parallel }_m\forall p\in H^m(\Omega ),m=0,1.\hfill \end{array}$$

(9)

Define the discrete form of Equation (6) with two Gauss integrals

$$G(p_h,q_h)=\lambda \sum _{K\in K_h}\left\{{\int }_{K,2}{p}_h{q}_{h}dx-{\int }_{K,1}{p}_h{q}_{h}dx\right\}\forall p_h,q_h\in Q_h,$$

(10)

where ${\int }_{K,m}·d\mathbf{x}$ denotes a Gauss integral over K, K is exact for polynomials of degree m, and $m=1,2$.

Under the above notations, the stabilized finite element method of Equation (5) reads as follows: find $(\overrightarrow{u_h},p_h)=({\mathit{u}}_h,{\varphi }_h,p_h)\in X_h$, such that

$$\left\{\begin{array}{c}a(\overrightarrow{u_h},\overrightarrow{v_h})+b(\overrightarrow{v_h},p_h)=(\overrightarrow{f},\overrightarrow{v_h})\forall \overrightarrow{v_h}=({\mathit{v}}_h,{\psi }_h)\in W_h,\hfill \\ b(\overrightarrow{u_h},q_h)+G(p_h,q_h)=0\forall q_h\in Q_h.\hfill \end{array}\right.$$

(11)

Then, recall some error estimates of the stabilized finite element method deduced by Li et al., in [29].

$$\begin{array}{cc}\hfill & \parallel \mathit{u}-{\mathit{u}}_h{\parallel }_1+\parallel \varphi -{\varphi }_h{\parallel }_1+\parallel p-p_h{\parallel }_0\le {ch(\parallel \mathit{u}\parallel }_2+{\parallel \varphi \parallel }_2+{\parallel p\parallel }_1),\hfill \\ \hfill & \parallel \mathit{u}-{\mathit{u}}_h{\parallel }_0+\parallel \varphi -{\varphi }_h{\parallel }_0\le c{h}^2{(\parallel \mathit{u}\parallel }_2+{\parallel \varphi \parallel }_2+{\parallel p\parallel }_1).\hfill \end{array}$$

(12)

Since $0<\lambda <1$, the above estimates still hold for Equation (11).

Then, the Stokes equation can be rewritten as

$$\begin{array}{c}\hfill {\mathcal{B}}_s(({\mathit{u}}_h,p_h);({\mathit{v}}_h,q_h))=({\mathit{f}}_1,{\mathit{v}}_h),\end{array}$$

(13)

where

$$\begin{array}{c}\hfill {\mathcal{B}}_s(({\mathit{u}}_h,p_h);({\mathit{v}}_h,q_h))=a_f({\mathit{u}}_h,{\mathit{v}}_h)+b({\mathit{v}}_h,p_h)-b({\mathit{u}}_h,q_h)-G(p_h,q_h).\end{array}$$

(14)

It is easy to verify that

$$\begin{array}{cc}\hfill & \left|{\mathcal{B}}_s((\mathit{u},p);(\mathit{v},q))\right|\le c{\left|\left|\left|(\mathit{u},p)\right|\right|\right|}_{{\Omega }_f}{\left|\left|\left|(\mathit{v},q)\right|\right|\right|}_{{\Omega }_f},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \beta {\left|\left|\left|({\mathit{u}}_h,p_h)\right|\right|\right|}_{{\Omega }_f}\le \underset{({\mathit{v}}_h,q_h)\in H_{f,h}\times Q_h}{sup}\frac{\left|{\mathcal{B}}_s(({\mathit{u}}_h,p_h);({\mathit{v}}_h,q_h))\right|}{{\left|\left|\left|({\mathit{v}}_h,q_h)\right|\right|\right|}_{{\Omega }_f}}\forall ({\mathit{u}}_h,p_h)\in H_{f,h}\times Q_h,\hfill \end{array}$$

(16)

where ${\left|\left|\left|({\mathit{u}}_h,p_h)\right|\right|\right|}_{{\Omega }_f}=\parallel {\mathit{u}}_h{\parallel }_{1,{\Omega }_f}+{\parallel p_h\parallel }_{0,{\Omega }_f}$.

4. Numerical Algorithm

Divide ${\Omega }_f$ into a series of disjoint sub-domains $D_j$, and then enlarge $D_j$ to ${\Omega }_j$ such that $D_j\subset \subset {\Omega }_j\subset \subset {\Omega }_f$ ($D_j\subset \subset {\Omega }_j$ means that dist($\partial D_j\setminus \partial {\Omega }_f,\partial {\Omega }_j\setminus \partial {\Omega }_f)>0$). Define ${\Gamma }_{{\Omega }_j}=\Gamma \cap \partial {\Omega }_j$. Furthermore, divide ${\Omega }_p$ into a series of disjoint sub-domains $D^i$, then enlarge $D^i$ to ${\Omega }^i$. Then, obtain $D^i\subset \subset {\Omega }^i\subset \subset {\Omega }_p$ and define ${\Gamma }_{{\Omega }^i}=\Gamma \cap \partial {\Omega }^i$ similarly.

Algorithm 1 Local and parallel stabilized finite element method

Step 1. On a coarse grid, solve the following coupled model to find $(\overrightarrow{u_H},p_H)\in X_H$ satisfying $$\left\{\begin{array}{c}a(\overrightarrow{u_H},\overrightarrow{v_H})+b(\overrightarrow{v_H},p_H)=(\overrightarrow{f},\overrightarrow{v_H})\forall \overrightarrow{v_H}=({\mathit{v}}_H,{\psi }_h)\in W_H,\hfill \\ b(\overrightarrow{u_H},q_H)+G(p_H,q_H)=0\forall q_H\in Q_H.\hfill \end{array}\right.$$ (17) Step 2. On a fine mesh, solve a series of local Darcy sub-problems in parallel as follows: Find the local residuals ${ϵ}_h^i\in H_{p,h}\left({\Omega }^i\right)$ ($i=1,2,\cdots ,M_p$, $h<H$) satisfying $$a_p({ϵ}_h^i,{\psi }_h)=\rho g{(f_2,{\psi }_h)}_{{\Omega }^i}-a_p({\varphi }_H,{\psi }_h)+\rho g{\int }_{{\Gamma }_{{\Omega }^i}}{\psi }_h{\mathit{u}}_H·{\mathit{n}}_f\forall {\psi }_h\in H_{p,h}\left({\Omega }^i\right),$$ (18) and we set ${\varphi }^h={\varphi }_H+{ϵ}_h^i$ in $D^i$. On a fine mesh, solve the following local Stokes sub-problems in parallel. Find local residuals $({\mathit{e}}_h^j,{\eta }_h^j)\in H_{f,h}\left({\Omega }_j\right)\times Q_h\left({\Omega }_j\right)$ ($j=1,2,\cdots ,M_f$), such that, for all $({\mathit{v}}_h,q_h)\in H_{f,h}\left({\Omega }_j\right)\times Q_h\left({\Omega }_j\right)$, $$\begin{array}{cc}\hfill a_f({\mathit{e}}_h^j,{\mathit{v}}_h)+b({\mathit{v}}_h,{\eta }_h^j)& ={({\mathit{f}}_1,{\mathit{v}}_h)}_{{\Omega }_j}-(a_f({\mathit{u}}_H,{\mathit{v}}_h)+b({\mathit{v}}_h,p_H))-\rho g{\int }_{{\Gamma }_{{\Omega }_j}}{\varphi }_H{\mathit{v}}_h·{\mathit{n}}_f,\hfill \\ \hfill b({\mathit{e}}_h^j,q_h)+G({\eta }_h^j,q_h)& =-b({\mathit{u}}_H,q_h)-G(p_H,q_h),\hfill \end{array}$$ (19) and then set $({\mathit{u}}^h,p^h)=({\mathit{u}}_H+{\mathit{e}}_h^j,p_H+{\eta }_h^j)$ in $D_j$.

However, the solution of the coupled Stokes–Darcy model Equation (11) derived by Algorithm 1 is globally discontinuous. By combining the local and parallel finite element method and the partition of unity method, a new local and parallel finite element method is obtained. Let ${\left\{{\Omega }_f^j\right\}}_{j=1}^{M_f}$ be an open cover of ${\Omega }_f$ and ${\left\{{\varphi }_j^f\right\}}_{j=1}^{M_f}$ be the partition of unity subordinate to ${\left\{{\Omega }_f^j\right\}}_{j=1}^{M_f}$. Let ${\left\{{\Omega }_p^i\right\}}_{i=1}^{M_p}$ be an open cover of ${\Omega }_p$ and ${\left\{{\delta }_i^p\right\}}_{i=1}^{M_p}$ be the partition of unity subordinate to ${\left\{{\Omega }_p^i\right\}}_{i=1}^{M_p}$. ${\varphi }_j^f$ and ${\delta }_i^p$ could be chosen as piecewise linear Lagrange basis functions. Then, there holds the following results [31]:

$$\begin{array}{cc}\hfill \mathrm{supp}{\varphi }_j^f& \subset {\overline{\Omega }}_f^j\forall j=1,\cdots ,M_f,\hfill \\ \hfill \sum _{j=1}^{M_f}{\varphi }_j^f& =1\mathrm{on}{\Omega }_f,\hfill \\ \hfill \parallel {\varphi }_j^f{\parallel }_{L^{\infty }\left(R^d\right)}& \le c,\hfill \\ \hfill \mathrm{supp}{\delta }_i^p& \subset {\overline{\Omega }}_p^i\forall i=1,\cdots ,M_p,\hfill \\ \hfill \sum _{i=1}^{M_p}{\delta }_i^p& =1\mathrm{on}{\Omega }_p,\hfill \\ \hfill \parallel {\delta }_i^p{\parallel }_{L^{\infty }\left(R^d\right)}& \le c.\hfill \end{array}$$

Furthermore, construct the partition of unity as follows: Define a regular triangulation ${\tau }_{H_p}$ in $\Omega$ such that $h<H\le H_p$, where $H_p$ is fixed and independent of $h,H$. For the triangulation ${\tau }_{H_p}$, let $D_f^j$ be the union of triangles in ${\Omega }_f$, and let $D_p^i$ be the union of triangles in ${\Omega }_p$. Define ${\Gamma }_{{\Omega }_f^j}=\Gamma \cap \partial {\Omega }_f^j$, ${\Gamma }_{{\Omega }_p^i}=\Gamma \cap \partial {\Omega }_p^i$.

Algorithm 2 Local and parallel partition of unity stabilized finite element method

Step 1. On a coarse grid, solve the following coupled model to obtain $(\overrightarrow{u_H},p_H)\in X_H$, such that $$\left\{\begin{array}{c}a(\overrightarrow{u_H},\overrightarrow{v_H})+b(\overrightarrow{v_H},p_H)=(\overrightarrow{f},\overrightarrow{v_H})\forall \overrightarrow{v_H}=({\mathit{v}}_H,{\psi }_h)\in W_H,\hfill \\ b(\overrightarrow{u_H},q_H)+G(p_H,q_H)=0\forall q_H\in Q_H.\hfill \end{array}\right.$$ (20) Step 2. On a fine mesh, find local fine grid correction ${ϵ}_h^i\in H_{p,h}\left({\Omega }_p^i\right)(i=1,2,\cdots ,M_p$, $h<H$), such that for all ${\psi }_h\in H_{p,h}\left({\Omega }_p^i\right)$, $$\begin{array}{c}\hfill a_p({ϵ}_h^i,{\psi }_h)=\rho g{(f_2,{\psi }_h)}_{{\Omega }_p^i}-a_p({\varphi }_H,{\psi }_h)+\rho g{\int }_{{\Gamma }_{{\Omega }_p^i}}{\psi }_h{\mathit{u}}_H·{\mathit{n}}_f,\end{array}$$ (21) and then assemble them together to derive a continuous solution as $${\varphi }_H^h={\varphi }_H+\sum _{i=1}^{M_p}{\delta }_i^p{ϵ}_h^i.$$ (22) On a fine mesh, find the local corrections $({\mathit{e}}_h^j,{\eta }_h^j)\in H_{f,h}\left({\Omega }_f^j\right)\times Q_h\left({\Omega }_f^j\right)(j=1,2,\cdots ,M_f)$, $\forall ({\mathit{v}}_h,q_h)\in H_{f,h}\left({\Omega }_f^j\right)\times Q_h\left({\Omega }_f^j\right)$ such that $$\begin{array}{cc}\hfill & a_f({\mathit{e}}_h^j,{\mathit{v}}_h)+b({\mathit{v}}_h,{\eta }_h^j)={({\mathit{f}}_1,{\mathit{v}}_h)}_{{\Omega }_f^j}-(a_f({\mathit{u}}_H,{\mathit{v}}_h)+b({\mathit{v}}_h,p_H))-\rho g{\int }_{{\Gamma }_{{\Omega }_f^j}}{\varphi }_H{\mathit{v}}_h·{\mathit{n}}_f,\hfill \\ \hfill & b({\mathit{e}}_h^j,q_h)+G({\eta }_h^j,q_h)=-b({\mathit{u}}_H,q_h)-G(p_H,q_h),\hfill \end{array}$$ (23) and then obtain the final approximation as $$({\mathit{u}}_H^h,p_H^h)=({\mathit{u}}_H+\sum _{j=1}^{M_f}{\varphi }_j^f{\mathit{e}}_h^j,p_H+\sum _{j=1}^{M_f}{\varphi }_j^f{\eta }_h^j).$$ (24)

5. Theoretical Analysis

In this section, the error estimates of the proposed algorithms are derived. Firstly, a lemma, which is crucial for the later analysis, is introduced. Then, the main results based upon the provided lemma are derived. The proof of the following lemma is similar to lemma 3.2 in [26] and so it will be omitted.

Lemma 1.

Let $D\subset \subset {\Omega }_0\subset {\Omega }_f$, for $\mathit{f}\in L^2\left({\Gamma }_{{\Omega }_0}\right)$, and $\lambda =\mathcal{O}\left(h\right)$, if there exists $({\mathit{w}}_h,r_h)\in H_{f,h}\times Q_h$ such that

$$\begin{array}{c}\hfill a_f({\mathit{w}}_h,{\mathit{v}}_h)+b({\mathit{v}}_h,r_h)-b({\mathit{w}}_h,q_h)-G(r_h,q_h)=(\mathit{f},{\mathit{v}}_h),\forall ({\mathit{v}}_h,q_h)\in H_{f,h}\left({\Omega }_0\right)\times Q_h\left({\Omega }_0\right),\end{array}$$

then there holds

$$\begin{array}{c}\hfill {\parallel {\mathit{w}}_h\parallel }_{1,D}+\parallel r_h{\parallel }_{0,D}\le c(\parallel {\mathit{w}}_h{\parallel }_{0,{\Omega }_0}+\parallel r_h{\parallel }_{-1,{\Omega }_0}+{\parallel \mathit{f}\parallel }_{L^2\left({\Gamma }_{{\Omega }_0}\right)}).\end{array}$$

(25)

Theorem 1.

Assume that Lemma 1 holds, $(\mathit{u},\varphi ,p)$ is the exact solution of (5), $({\mathit{u}}_h,{\varphi }_h,p_h)$ is the solution of the standard finite element method, and $({\mathit{u}}^h,{\varphi }^h,p^h)$ is the solution of Algorithm 1. The following estimates hold:

$$\begin{array}{cc}\hfill & {\parallel {\varphi }_h-{\varphi }^h\parallel }_{1,D^i}\le c{H}^2,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & {\parallel {\mathit{u}}_h-{\mathit{u}}^h\parallel }_{1,D_j}+{\parallel p_h-p^h\parallel }_{0,D_j}\le c{H}^2.\hfill \end{array}$$

(27)

Consequently, there holds

$$\begin{array}{cc}\hfill & \parallel \varphi -{\varphi }^h{\parallel }_{1,D^i}\le c(h+H^2),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \parallel \mathit{u}-{\mathit{u}}^h{\parallel }_{1,D_j}+{\parallel p-p^h\parallel }_{0,D_j}\le c(h+H^2).\hfill \end{array}$$

(29)

Proof. 

Taking $({\mathit{v}}_h,{\psi }_h)=(0,{\psi }_h)$ into Equation (11) yields

$$\begin{array}{c}\hfill a_p({\varphi }_h,{\psi }_h)=\rho g(f_2,{\psi }_h)+\rho g{\int }_{\Gamma }{\psi }_h{\mathit{u}}_h·{\mathit{n}}_f.\end{array}$$

(30)

Since

$$\begin{array}{c}\hfill a_p({\varphi }^h,{\psi }_h)=\rho g(f_2,{\psi }_h)+\rho g{\int }_{{\Gamma }_{{\Omega }^i}}{\psi }_h{\mathit{u}}_H·{\mathit{n}}_f,\end{array}$$

(31)

and setting ${\psi }_h={\varphi }_h-{\varphi }^h$, there holds

$$\begin{array}{c}\hfill a_p({\varphi }_h-{\varphi }^h,{\varphi }_h-{\varphi }^h)=\rho g{\int }_{{\Gamma }_{{\Omega }^i}}({\varphi }_h-{\varphi }^h)({\mathit{u}}_h-{\mathit{u}}_H)·{\mathit{n}}_f.\end{array}$$

(32)

Then, the auxiliary problem similar to [6] is introduced as follows: find $\delta \in H^1\left({\Omega }_f\right)$, such that

$$\left\{\begin{array}{c}-\Delta \delta =0\mathrm{in}{\Omega }_f,\hfill \\ \delta ={\varphi }_h-{\varphi }^h\mathrm{on}{\Gamma }_{{\Omega }^i},\hfill \\ \delta =0\mathrm{on}\partial {\Omega }_f/{\Gamma }_{{\Omega }^i}.\hfill \end{array}\right.$$

Recalling the interpolation space $H_{00}^{1/2}\left({\Gamma }_{{\Omega }^i}\right)={[L^2\left({\Gamma }_{{\Omega }^i}\right),H_0^1\left({\Gamma }_{{\Omega }^i}\right)]}_{1/2}$ presented in [32], it follows that

$$\begin{array}{c}\hfill {\parallel \delta \parallel }_{1,{\Omega }_f}\le c\parallel {\varphi }_h-{\varphi }^h{\parallel }_{H_{00}^{1/2}\left({\Gamma }_{{\Omega }^i}\right)}\le c{\parallel {\varphi }_h-{\varphi }^h\parallel }_{1,{\Omega }^i}.\end{array}$$

(33)

Combining Equation (11) with Equation (17) yields

$$\begin{array}{c}\hfill b({\mathit{u}}_h-{\mathit{u}}_H,q_H)+G(p_h-p_H,q_H)=0.\end{array}$$

(34)

Hence,

$$\begin{array}{cc}\hfill & \rho g{\int }_{{\Gamma }_{{\Omega }^i}}({\varphi }_h-{\varphi }^h)({\mathit{u}}_h-{\mathit{u}}_H)·{\mathit{n}}_f\hfill \\ \hfill & =\rho g{\int }_{\partial {\Omega }_f}\delta ({\mathit{u}}_h-{\mathit{u}}_H)·{\mathit{n}}_f\hfill \\ \hfill & =\rho g{\int }_{{\Omega }_f}\nabla \delta ·({\mathit{u}}_h-{\mathit{u}}_H)+\rho g{\int }_{{\Omega }_f}\delta \nabla ·({\mathit{u}}_h-{\mathit{u}}_H)\hfill \\ \hfill & =\rho g{\int }_{{\Omega }_f}\nabla \delta ·({\mathit{u}}_h-{\mathit{u}}_H)+\rho g{\int }_{{\Omega }_f}(\delta -q_H)\nabla ·({\mathit{u}}_h-{\mathit{u}}_H)+\rho gG(p_h-p_H,q_H).\hfill \end{array}$$

(35)

Noting that Equation (12) holds, therefore, for $\delta \in H^1\left({\Omega }_f\right)$, it can be derived that

$$\begin{array}{cc}\hfill \parallel {\varphi }_h-{\varphi }^h{\parallel }_{1,{\Omega }^i}^2& =a_p({\varphi }_h-{\varphi }^h,{\varphi }_h-{\varphi }^h)\hfill \\ \hfill & \le {c\parallel \delta \parallel }_{1,{\Omega }_f}{\parallel {\mathit{u}}_h-{\mathit{u}}_H\parallel }_{0,{\Omega }_f}+c\underset{\forall q_H\in Q_H}{inf}\left|{\int }_{{\Omega }_f}(\delta -q_H)\nabla ·({\mathit{u}}_h-{\mathit{u}}_H)\right|\hfill \\ \hfill & +cG(p_h-p_H,q_H-\delta )+cG(p_h-p_H,\delta )\hfill \\ \hfill & \le c{H}^2{\parallel \delta \parallel }_{1,{\Omega }_f}+c\underset{\forall q_H\in Q_H}{inf}\parallel \delta -q_H{\parallel }_{0,{\Omega }_f}{\parallel {\mathit{u}}_h-{\mathit{u}}_H\parallel }_{1,{\Omega }_f}\hfill \\ \hfill & +c\parallel p_h-p_H{\parallel }_{0,{\Omega }_f}\underset{\forall q_H\in Q_H}{inf}\parallel q_H{-\delta \parallel }_{0,{\Omega }_f}+c\parallel p_h-p_H{\parallel }_{0,{\Omega }_f}{\parallel (I-\Pi )\delta \parallel }_{0,{\Omega }_f}\hfill \\ \hfill & \le c{H}^2{\parallel \delta \parallel }_{1,{\Omega }_f}\hfill \\ \hfill & \le c{H}^2{\parallel {\varphi }_h-{\varphi }^h\parallel }_{1,{\Omega }^i}.\hfill \end{array}$$

(36)

Then, Equation (26) is established.

Analogously, an auxiliary problem in the porous media domain is introduced: find $\Phi \in H^1\left({\Omega }_p\right)$ such that

$$\left\{\begin{array}{c}-\nabla ·(\mathbb{K}\nabla \Phi )=0\mathrm{in}{\Omega }_p,\hfill \\ \mathbb{K}\nabla \Phi ·{\mathit{n}}_p=\rho g{\mathit{v}}_h·{\mathit{n}}_p\mathrm{on}{\Gamma }_{{\Omega }_j},\hfill \\ \mathbb{K}\nabla \Phi ·{\mathit{n}}_p=0\mathrm{on}\partial {\Omega }_p/{\Gamma }_{{\Omega }_j}.\hfill \end{array}\right.$$

It is classical that

$$\begin{array}{c}\hfill {\parallel {\mathbb{K}}^{1/2}\nabla \Phi \parallel }_{0,{\Omega }_p}\le c{\parallel {\mathit{v}}_h\parallel }_{1,{\Omega }_j},\end{array}$$

(37)

and

$$\begin{array}{c}\hfill {\parallel \Phi \parallel }_{2,{\Omega }_p}\le c{\parallel {\mathit{v}}_h\parallel }_{1,{\Omega }_j}.\end{array}$$

(38)

Taking $({\mathit{v}}_h,{\psi }_h,q_h)=({\mathit{v}}_h,0,q_h)$ into Equation (11) yields

$$\begin{array}{c}\hfill a_f({\mathit{u}}_h,{\mathit{v}}_h)+b({\mathit{v}}_h,p_h)-b({\mathit{u}}_h,q_h)-G(p_h,q_h)=({\mathit{f}}_1,{\mathit{v}}_h)-\rho g{\int }_{\Gamma }{\varphi }_h{\mathit{v}}_h·{\mathit{n}}_f.\end{array}$$

(39)

Subtracting Equation (19) from Equation (39) yields

$$\begin{array}{cc}\hfill & a_f({\mathit{u}}_h-{\mathit{u}}^h,{\mathit{v}}_h)+b({\mathit{v}}_h,p_h-p^h)-b({\mathit{u}}_h-{\mathit{u}}^h,q_h)-G(p_h-p^h,q_h)\hfill \\ \hfill & +\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f=0\forall ({\mathit{v}}_h,q_h)\in H_{f,h}\left({\Omega }_j\right)\times Q_h\left({\Omega }_j\right).\hfill \end{array}$$

(40)

Using Lemma 1, it follows that

$$\begin{array}{cc}& {\parallel {\mathit{u}}_h-{\mathit{u}}^h\parallel }_{1,D_j}+{\parallel p_h-p^h\parallel }_{0,D_j}\hfill \\ & \le c\left(\parallel {\mathit{u}}_h-{\mathit{u}}^h{\parallel }_{0,{\Omega }_j}+{\parallel p_h-p^h\parallel }_{-1,{\Omega }_j}+\frac{|\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f|}{\parallel {\mathit{v}}_h{\parallel }_{1,{\Omega }_j}}\right)\hfill \\ & \le c\left(\parallel {\mathit{u}}_h-{\mathit{u}}_H{\parallel }_{0,{\Omega }_j}+\parallel {\mathit{u}}_H-{\mathit{u}}_h{\parallel }_{0,{\Omega }_j}+\parallel p_h-p_H{\parallel }_{-1,{\Omega }_j}+{\parallel p_H-p^h\parallel }_{-1,{\Omega }_j}+\frac{|\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f|}{\parallel {\mathit{v}}_h{\parallel }_{1,{\Omega }_j}}\right)\hfill \\ & =c\left(\parallel {\mathit{u}}_h-{\mathit{u}}_H{\parallel }_{0,{\Omega }_j}+\parallel p_h-p_H{\parallel }_{-1,{\Omega }_j}+\parallel {\mathit{e}}_h^j{\parallel }_{0,{\Omega }_j}+{\parallel {\eta }_h^j\parallel }_{-1,{\Omega }_j}+\frac{|\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f|}{\parallel {\mathit{v}}_h{\parallel }_{1,{\Omega }_j}}\right).\hfill \end{array}$$

(41)

Now, estimate the last term on the right side of the above inequality. Following the idea in [33], there holds

$$\begin{array}{cc}\hfill \rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f& =-{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H)\mathbb{K}\nabla \Phi ·{\mathit{n}}_p\hfill \\ \hfill & =-{\int }_{\partial {\Omega }_p}({\varphi }_h-{\varphi }_H)\mathbb{K}\nabla \Phi ·{\mathit{n}}_p\hfill \\ \hfill & =-{\int }_{{\Omega }_p}({\varphi }_h-{\varphi }_H)\nabla ·(\mathbb{K}\nabla \Phi )-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }_h-{\varphi }_H)·\nabla \Phi \hfill \\ \hfill & =-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }_h-{\varphi }_H)·\nabla \Phi \hfill \\ \hfill & =-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }_h-{\varphi }^h)·\nabla \Phi -{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }^h-{\varphi }_H)·\nabla \Phi .\hfill \end{array}$$

(42)

For the first term on the right side of the above inequality, it can be derived that

$$\begin{array}{c}\hfill \left|-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }_h-{\varphi }^h)·\nabla \Phi \right|\le c\parallel {\varphi }_h-{\varphi }^h{\parallel }_{1,{\Omega }^i}\parallel {\mathbb{K}}^{1/2}{\nabla \Phi \parallel }_{0,{\Omega }_p}\le c{H}^2{\parallel {\mathit{v}}_h\parallel }_{1,{\Omega }_j}.\end{array}$$

(43)

From Equation (11), it is easy to obtain

$$\begin{array}{c}\hfill a_p({\varphi }^h,{\psi }_H)=\rho g(f_2,{\psi }_H)+\rho g{\int }_{{\Gamma }_{{\Omega }_j}}{\psi }_H{\mathit{u}}_H·{\mathit{n}}_f.\end{array}$$

(44)

Taking $({\mathit{v}}_H,{\psi }_H)=(0,{\psi }_H)$ into Equation (11) and using Equation (44) yields

$$\begin{array}{c}\hfill -{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }^h-{\varphi }_H)·\nabla {\psi }_H=0.\end{array}$$

(45)

Hence,

$$\begin{array}{cc}\hfill \left|-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }^h-{\varphi }_H)·\nabla \Phi \right|& =\left|-{\int }_{{\Omega }_p}\mathbb{K}\nabla ·({\varphi }^h-{\varphi }_H)·\nabla (\Phi -{\psi }_H)\right|\hfill \\ \hfill & \le c\parallel {\varphi }^h-{\varphi }_H{\parallel }_{1,{\Omega }_p}\underset{\forall {\psi }_H\in H_{p,H}}{inf}{\parallel \Phi -{\psi }_H\parallel }_{1,{\Omega }_p}\hfill \\ \hfill & \le cH(\parallel {\varphi }^h-{\varphi }_h{\parallel }_{1,{\Omega }_p}+\parallel {\varphi }_h-{\varphi }_H{\parallel }_{1,{\Omega }_p}{)\parallel \Phi \parallel }_{2,{\Omega }_p}\hfill \\ \hfill & \le c{H}^2{\parallel {\mathit{v}}_h\parallel }_{1,{\Omega }_j}.\hfill \end{array}$$

(46)

By using the above inequalities, it can be derived that

$$\begin{array}{c}\hfill \left|\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{v}}_h·{\mathit{n}}_f\right|\le c{H}^2{\parallel {\mathit{v}}_h\parallel }_{1,{\Omega }_j}.\end{array}$$

(47)

Next, a dual problem is introduced to estimate $\parallel {\mathit{e}}_h^j{\parallel }_{0,{\Omega }_j}$ and $\parallel {\eta }_h^j{\parallel }_{-1,{\Omega }_j}$. For $\mathit{\theta }\in L^2{\left({\Omega }_j\right)}^d,\phi \in H_0^1\left({\Omega }_j\right)$, find $(\mathit{w},r)\in (H_f\left({\Omega }_j\right)\cap H^2{\left({\Omega }_j\right)}^d)\times L^2\left({\Omega }_j\right)$ such that for all $(\mathit{v},q)\in H_f\left({\Omega }_j\right)\times L^2\left({\Omega }_j\right)$,

$$\begin{array}{c}\hfill a_f(\mathit{v},\mathit{w})+b(\mathit{w},q)-b(\mathit{v},r)-G(q,r)={(\mathit{\theta },\mathit{v})}_{{\Omega }_j}+{(\phi ,q)}_{{\Omega }_j}.\end{array}$$

(48)

And there holds

$$\begin{array}{c}\hfill {\parallel \mathit{w}\parallel }_{2,{\Omega }_j}+{\parallel r\parallel }_{1,{\Omega }_j}\le {c(\parallel \mathit{\theta }\parallel }_{0,{\Omega }_j}+{\parallel \phi \parallel }_{1,{\Omega }_j}).\end{array}$$

(49)

Assume $({\mathit{w}}_{\mu },r_{\mu })\in H_{f,\mu }\left({\Omega }_j\right)\times Q_{\mu }\left({\Omega }_j\right)$ is derived by using the stabilized finite element method; then, there holds

$$\begin{array}{c}\hfill a_f(\mathit{v},\mathit{w}-{\mathit{w}}_{\mu })+b(\mathit{w}-{\mathit{w}}_{\mu },q)-b(\mathit{v},r-r_{\mu })-G(q,r-r_{\mu })=0\forall (\mathit{v},q)\in H_{f,\mu }\left({\Omega }_j\right)\times Q_{\mu }\left({\Omega }_j\right),\end{array}$$

(50)

where $\mu =h$ or H.

Apparently,

$$\begin{array}{cc}\hfill \parallel \mathit{w}-{\mathit{w}}_{\mu }{\parallel }_{1,{\Omega }_j}+{\parallel r-r_{\mu }\parallel }_{0,{\Omega }_j}& \le c\mu (\parallel \mathit{w}{\parallel }_{2,{\Omega }_j}+\parallel r{\parallel }_{1,{\Omega }_j})\hfill \\ \hfill & \le c\mu (\parallel \mathit{\theta }{\parallel }_{0,{\Omega }_j}+\parallel \phi {\parallel }_{1,{\Omega }_j}).\hfill \end{array}$$

(51)

Then, there holds

$$\begin{array}{c}\hfill {\parallel {\mathit{w}}_h-{\mathit{w}}_H\parallel }_{1,{\Omega }_j}+\parallel r_h-r_H{\parallel }_{0,{\Omega }_j}\le {cH(\parallel \mathit{\theta }\parallel }_{0,{\Omega }_j}+{\parallel \phi \parallel }_{1,{\Omega }_j}).\end{array}$$

(52)

From Equation (17), it is easy to obtain

$$\begin{array}{cc}\hfill & a_f({\mathit{u}}_h-{\mathit{u}}_H,{\mathit{w}}_H)+b({\mathit{w}}_H,p_h-p_H)-b({\mathit{u}}_h-{\mathit{u}}_H,r_H)-G(p_h-p_H,r_H)\hfill \\ \hfill & +\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{w}}_H·{\mathit{n}}_f=0\forall ({\mathit{w}}_H,r_H)\in H_{f,H}\left({\Omega }_j\right)\times Q_H\left({\Omega }_j\right).\hfill \end{array}$$

(53)

Taking $\mathit{v}={\mathit{e}}_h^j,q={\eta }_h^j$ into Equation (48), and together with Equations (40), (50), and (53) yields

$$\begin{array}{cc}& {(\mathit{\theta },{\mathit{e}}_h^j)}_{{\Omega }_j}+{(\phi ,{\eta }_h^j)}_{{\Omega }_j}\hfill \\ & =a_f({\mathit{e}}_h^j,\mathit{w})+b(\mathit{w},{\eta }_h^j)-b({\mathit{e}}_h^j,r)-G({\eta }_h^j,r)\hfill \\ & =a_f({\mathit{e}}_h^j,{\mathit{w}}_h)+b({\mathit{w}}_h,{\eta }_h^j)-b({\mathit{e}}_h^j,r_h)-G({\eta }_h^j,r_h)\hfill \\ & =a_f({\mathit{u}}_h-{\mathit{u}}_H,{\mathit{w}}_h)+b({\mathit{w}}_h,p_h-p_H)-b({\mathit{u}}_h-{\mathit{u}}_H,r_h)-G(p_h-p_H,r_h)+\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{w}}_h·{\mathit{n}}_f\hfill \\ & =a_f({\mathit{u}}_h-{\mathit{u}}_H,{\mathit{w}}_h-{\mathit{w}}_H)+b({\mathit{w}}_h-{\mathit{w}}_H,p_h-p_H)-b({\mathit{u}}_h-{\mathit{u}}_H,r_h-r_H)-G(p_h-p_H,r_h-r_H)\hfill \\ & +\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H)·({\mathit{w}}_h-{\mathit{w}}_H)·{\mathit{n}}_f.\hfill \end{array}$$

(54)

Based on Equations (12) and (51), it is valid that

$$\begin{array}{cc}\hfill & \left|{(\mathit{\theta },{\mathit{e}}_h^j)}_{{\Omega }_j}+{(\phi ,{\eta }_h^j)}_{{\Omega }_j}\right|\hfill \\ \hfill & \le (\parallel {\mathit{u}}_h-{\mathit{u}}_H{\parallel }_{1,{\Omega }_f}+\parallel p_h-p_H{\parallel }_{0,{\Omega }_f}+\parallel {\varphi }_h-{\varphi }_H{\parallel }_{1,{\Omega }_p})(\parallel {\mathit{w}}_h-{\mathit{w}}_H{\parallel }_{1,{\Omega }_j}+\parallel r_h-r_H{\parallel }_{0,{\Omega }_j})\hfill \\ \hfill & \le cH(\parallel {\mathit{u}}_h-{\mathit{u}}_H{\parallel }_{1,{\Omega }_f}+\parallel p_h-p_H{\parallel }_{0,{\Omega }_f}+\parallel {\varphi }_h-{\varphi }_H{\parallel }_{1,{\Omega }_p}{\left)\right(\parallel \mathit{\theta }\parallel }_{0,{\Omega }_j}+{\parallel \phi \parallel }_{1,{\Omega }_j})\hfill \\ \hfill & \le c{H}^2{(\parallel \mathit{\theta }\parallel }_{0,{\Omega }_j}+{\parallel \phi \parallel }_{1,{\Omega }_j}).\hfill \end{array}$$

(55)

Consequently,

$$\begin{array}{c}\hfill {\parallel {\mathit{e}}_h^j\parallel }_{0,{\Omega }_j}+{\parallel {\eta }_h^j\parallel }_{-1,{\Omega }_j}\le c{H}^2.\end{array}$$

(56)

In the following, the estimate of $\parallel p_h-p_H{\parallel }_{-1,{\Omega }_j}$ is deduced. Setting $(\mathit{v},q)=({\mathit{u}}_h-{\mathit{u}}_H,p_h-p_H)$ in Equation (48), together with Equations (47), (51), and (53), and using the fact that ${\parallel {\mathit{w}}_H\parallel }_{1,{\Omega }_j}\le {\parallel \mathit{w}-{\mathit{w}}_H\parallel }_{1,{\Omega }_j}+{\parallel \mathit{w}\parallel }_{1,{\Omega }_j}\le c{\parallel \mathit{w}\parallel }_{2,{\Omega }_j}$, it is easy to obtain

$$\begin{array}{cc}\hfill & \left|{(\mathit{\theta },{\mathit{u}}_h-{\mathit{u}}_H)}_{{\Omega }_j}+{(\phi ,p_h-p_H)}_{{\Omega }_j}\right|\hfill \\ \hfill & =\left|a_f({\mathit{u}}_h-{\mathit{u}}_H,\mathit{w})+b(\mathit{w},p_h-p_H)-b({\mathit{u}}_h-{\mathit{u}}_H,r)-G(p_h-p_H,r)\right|\hfill \\ \hfill & =|a_f({\mathit{u}}_h-{\mathit{u}}_H,\mathit{w}-{\mathit{w}}_H)+b(\mathit{w}-{\mathit{w}}_H,p_h-p_H)-b({\mathit{u}}_h-{\mathit{u}}_H,r-r_H)\hfill \\ \hfill & -G(p_h-p_H,r-r_H)-\rho g{\int }_{{\Gamma }_{{\Omega }_j}}({\varphi }_h-{\varphi }_H){\mathit{w}}_H·{\mathit{n}}_f|\hfill \\ \hfill & \le c(\parallel {\mathit{u}}_h-{\mathit{u}}_H{\parallel }_{1,{\Omega }_j}+\parallel p_h-p_H{\parallel }_{0,{\Omega }_j}\left)\right(\parallel \mathit{w}-{\mathit{w}}_H{\parallel }_{1,{\Omega }_j}+\parallel r-r_H{\parallel }_{0,\Omega })+c{H}^2{\parallel w_H\parallel }_{1,{\Omega }_j}\hfill \\ \hfill & \le c{H}^2{(\parallel \mathit{w}\parallel }_{2,{\Omega }_j}+{\parallel r\parallel }_{1,{\Omega }_j})\hfill \\ \hfill & \le c{H}^2{(\parallel \mathit{\theta }\parallel }_{0,{\Omega }_j}+{\parallel \phi \parallel }_{1,{\Omega }_j}).\hfill \end{array}$$

(57)

Hence,

$$\begin{array}{c}\hfill {\parallel {\mathit{u}}_h-{\mathit{u}}_H\parallel }_{0,{\Omega }_j}+{\parallel p_h-p_H\parallel }_{-1,{\Omega }_j}\le c{H}^2.\end{array}$$

(58)

Therefore, together with Equations (41), (47), (56), and (58), (27) is derived. By combing the triangle inequality with Equations (26) and (27), Equations (28) and (29) could be directly derived. □

Define the following norm as follows:

$$\begin{array}{cc}\hfill & {\left|\left|\left|{\mathit{u}}_h-{\mathit{u}}^h\right|\right|\right|}_{1,{\Omega }_f}=(\sum _{j=1}^{M_f}\parallel {\mathit{u}}_h-{\mathit{u}}^h{{\parallel }_{1,D_j})}^{1/2},\hfill \\ \hfill & {\left|\left|\left|p_h-p^h\right|\right|\right|}_{0,{\Omega }_f}=(\sum _{j=1}^{M_f}\parallel p_h-p^h{{\parallel }_{0,D_j})}^{1/2},\hfill \\ \hfill & {\left|\left|\left|{\varphi }_h-{\varphi }^h\right|\right|\right|}_{1,{\Omega }_p}=(\sum _{i=1}^{M_p}\parallel {\varphi }_h-{\varphi }^h{{\parallel }_{1,D^i})}^{1/2}.\hfill \end{array}$$

Then, the following theoretical results can be derived directly:

Theorem 2.

Based on Theorem 1, there holds

$$\begin{array}{c}\hfill {\left|\left|\left|{\mathit{u}}_h-{\mathit{u}}^h\right|\right|\right|}_{1,{\Omega }_f}+{\parallel p_h-p^h\parallel }_{0,{\Omega }_f}+{\left|\left|\left|{\varphi }_h-{\varphi }^h\right|\right|\right|}_{1,{\Omega }_p}\le c{H}^2.\end{array}$$

(59)

Proof. 

By collecting the sub-domains $D_j$ ($D^i$), $j=1,\cdots ,M_f$, $i=1,\cdots ,M_p$, the proof is finished. □

Theorem 3.

Assume that the condition of Theorem 2 holds, then for the solutions using Algorithm 2, there holds the following estimate:

$$\begin{array}{c}\hfill {\parallel {\mathit{u}}_h-{\mathit{u}}_H^h\parallel }_{1,{\Omega }_f}+\parallel p_h-p_H^h{\parallel }_{0,{\Omega }_f}+{\parallel {\varphi }_h-{\varphi }_H^h\parallel }_{1,{\Omega }_p}\le c{H}^2.\end{array}$$

(60)

Proof. 

Since ${\mathit{u}}_{\mu }={\displaystyle \sum _{j=1}^{M_f}}{\varphi }_j^f{\mathit{u}}_{\mu }$, $p_{\mu }={\displaystyle \sum _{j=1}^{M_f}}{\varphi }_j^f{p}_{\mu }$, ${\varphi }_{\mu }={\displaystyle \sum _{i=1}^{M_p}}{\delta }_i^p{\varphi }_{\mu }$, $\mu =h,H$, it is easy to obtain that

$$\begin{array}{cc}\hfill & {\parallel {\mathit{u}}_h-{\mathit{u}}_H^h\parallel }_{1,{\Omega }_f}+\parallel p_h-p_H^h{\parallel }_{0,{\Omega }_f}+{\parallel {\varphi }_h-{\varphi }_H^h\parallel }_{1,{\Omega }_p}\hfill \\ \hfill & =\parallel {\mathit{u}}_h-({\mathit{u}}_H+\sum _{j=1}^{M_f}{\varphi }_j^f{\mathit{e}}_h^j){\parallel }_{1,{\Omega }_f}+\parallel p_h-(p_H+\sum _{j=1}^{M_f}{\varphi }_j^f{\eta }_h^j){\parallel }_{0,{\Omega }_f}+{\parallel {\varphi }_h-({\varphi }_H+\sum _{i=1}^{M_p}{\delta }_i^p{ϵ}_h^i)\parallel }_{1,{\Omega }_p}\hfill \\ \hfill & =\parallel \sum _{j=1}^{M_f}{\varphi }_j^f({\mathit{u}}_h-{\mathit{u}}_H-{\mathit{e}}_h^j){\parallel }_{1,{\Omega }_f}+\parallel \sum _{j=1}^{M_f}{\varphi }_j^f(p_h-p_H-{\eta }_h^j){\parallel }_{0,{\Omega }_f}+{\parallel \sum _{i=1}^{M_p}{\delta }_i^p({\varphi }_h-{\varphi }_H-{ϵ}_h^i)\parallel }_{1,{\Omega }_p}\hfill \\ \hfill & \le \sum _{j=1}^{M_f}\parallel {\varphi }_j^f({\mathit{u}}_h-{\mathit{u}}^h){\parallel }_{1,D_f^j}+\sum _{j=1}^{M_f}\parallel {\varphi }_j^f(p_h-p^h){\parallel }_{0,D_f^j}+\sum _{i=1}^{M_p}{\parallel {\delta }_i^p({\varphi }_h-{\varphi }^h)\parallel }_{1,D_p^i}\hfill \\ \hfill & \le \sum _{j=1}^{M_f}\parallel {\varphi }_j^f{\parallel }_{L^{\infty }\left({\Omega }_f\right)}\parallel {\mathit{u}}_h-{\mathit{u}}^h{\parallel }_{1,D_f^j}+\sum _{j=1}^{M_f}\parallel {\varphi }_j^f{\parallel }_{L^{\infty }\left({\Omega }_f\right)}\parallel p_h-p^h{\parallel }_{0,D_f^j}+\sum _{i=1}^{M_p}\parallel {\delta }_i^p{\parallel }_{L^{\infty }\left({\Omega }_p\right)}{\parallel {\varphi }_h-{\varphi }^h\parallel }_{1,D_p^i}\hfill \\ \hfill & \le c\sum _{j=1}^{M_f}(\parallel {\mathit{u}}_h-{\mathit{u}}^h{\parallel }_{1,D_f^j}+\parallel p_h-p^h{\parallel }_{0,D_f^j})+c\sum _{i=1}^{M_p}{\parallel {\varphi }_h-{\varphi }^h\parallel }_{1,D_p^i}\hfill \\ \hfill & \le c{H}^2.\hfill \end{array}$$

(61)

□

6. Numerical Results

In this section, two examples are provided to verify the theoretical results and the efficiency and effectiveness of the proposed two parallel algorithms. The first test is a convergence test with a manufactured solution. The second one is a modification of a classical lid-driven cavity flow problem. Since it is difficult to directly solve the coupled scheme (11) as the mesh size tends to 0, we compared our algorithms with two parallel algorithms with ${\mathbf{P}}_1\mathbf{b}$-$P_1$-$P_1$ finite element pairs presented in [18]. We list some abbreviations and symbols in the following.

• LPFEM—Local and parallel finite element method with ${\mathbf{P}}_1\mathbf{b}$-$P_1$-$P_1$ finite element pairs;
• LPPUFEM—Local and parallel partition of unity finite element method with ${\mathbf{P}}_1\mathbf{b}$-$P_1$-$P_1$ finite element pairs;
• LPSFEM—Local and parallel stabilized finite element method;
• LPPUSFEM—Local and parallel partition of unity stabilized finite element method;
• $({\tilde{\mathbf{u}}}^h,p^h,{\varphi }^h)$—Solution obtained by LPFEM;
• $({\tilde{\mathbf{u}}}_H^h,p_H^h,{\varphi }_H^h)$—Solution obtained by LPPUFEM;
• $({\mathbf{u}}^h,p^h,{\varphi }^h)$—Solution obtained by LPSFEM;
• $({\mathbf{u}}_H^h,p_H^h,{\varphi }_H^h)$—Solution obtained by LPPUSFEM;
• ${\mathcal{X}}_{\kappa }$—Implement the method $\mathcal{X}$ by dividing the domain ${\Omega }_f$ and ${\Omega }_p$ into $\kappa =L\times L$ sub-domains.

6.1. Test 1

In this test, one example with the analytical solution was considered to test the convergence order. Let ${\Omega }_f=[0,1]\times [1,2]$, ${\Omega }_p=[0,1]\times [0,1]$, and $\Gamma =[0,1]\times \left\{1\right\}$. The exact solution is

$$\left\{\begin{array}{c}{u}_1=(1-2x)(y-1),\hfill \\ u_2=x(x-1)+{(y-1)}^2,\hfill \\ p=x(1-x)(y-1)+\frac{y^3}{3}-y^2+y-0.5,\hfill \\ \varphi =x(1-x)(y-1)+\frac{y^3}{3}-y^2+y-0.5,\hfill \end{array}\right.$$

Then, ${\mathit{f}}_1$, $f_2$ can be derived by Equations (1) and (2), respectively. It is easy to verify that the solution satisfies the Beavers–Joseph–Saffman interface condition. For simplicity, let $\alpha ,\nu ,g,\rho =1$, $\mathbb{K}=\mathbb{I}$.

In the following, we introduce the details for the decomposition of ${\Omega }_f$ and ${\Omega }_p$ with $\kappa =2\times 2$ sub-domains. In the flow domain, divide ${\Omega }_f$ into four disjoint sub-regions

$$\begin{array}{cc}\hfill D_1=(0,0.5)\times (1,1.5)& D_2=(0.5,1)\times (1,1.5),\hfill \\ \hfill D_3=(0,0.5)\times (1.5,2)& D_4=(0.5,1)\times (1.5,2),\hfill \end{array}$$

and then enlarge them into

$$\begin{array}{cc}\hfill {\Omega }_1=(0,0.75)\times (1,1.75)& {\Omega }_2=(0.25,1)\times (1,1.75),\hfill \\ \hfill {\Omega }_3=(0,0.75)\times (1.25,2)& {\Omega }_4=(0.25,1)\times (1.25,2).\hfill \end{array}$$

In the porous media flow domain, divide ${\Omega }_p$ into four disjoint sub-regions

$$\begin{array}{cc}\hfill D^1=(0,0.5)\times (0,0.5)& D^2=(0.5,1)\times (0,0.5),\hfill \\ \hfill D^3=(0,0.5)\times (0.5,1)& D^4=(0.5,1)\times (0.5,1),\hfill \end{array}$$

and then enlarge them into

$$\begin{array}{cc}\hfill {\Omega }^1=(0,0.75)\times (0,0.75)& {\Omega }^2=(0.25,1)\times (0,0.75),\hfill \\ \hfill {\Omega }^3=(0,0.75)\times (0.25,1)& {\Omega }^4=(0.25,1)\times (0.25,1).\hfill \end{array}$$

Let us introduce the process to construct partition of unity functions for the fluid region. Let $H_{fix}=1/8$, and generate the uniform mesh triangulation $T_{H_{fix}}\left({\Omega }_f\right)$. Let $nd(nd=81)$ represent the number of nodes, $bf\left[i\right]$ stand for the piecewise linear basis function defined on the node i, and the first function is generated as follows:

$${\varphi }_1^f=\bigcup _{i\in G}bf\left[i\right],G=\{1\le i\le 36\mathrm{and}1\le \left\{i\mathrm{mod}9\right\}\le 4.\}$$

The other three functions could be obtained in the same way. We plot them in Figure 1 along with partition of unity functions for the porous media region.

The configuration between the coarse grid and the fine grid is $h=H^2$. In this case, the optimal error convergence rates of the proposed two algorithms could be derived. The uniform mesh is used, and the choice of $\lambda$ is $\lambda$ = 50 h.

The numerical results, including errors of the velocity, pressure, and piezometric head, obtained using four numerical methods are presented in

Table 1. The $H^1$-error of the velocity of ${\mathbf{LPSFEM}}_{2\times 2}$, ${\mathbf{LPFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	${\left|\left|\left|\mathit{u}-{\mathit{u}}^{\mathit{h}}\right|\right|\right|}_{1,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$	${\left|\left|\left|\mathit{u}-{\tilde{\mathit{u}}}^{\mathit{h}}\right|\right|\right|}_{1,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$
4	16	$7.48065\times {10}^{-2}$	−	$1.11563\times {10}^{-1}$	−
8	64	$1.79323\times {10}^{-2}$	$1.03030$	$2.65643\times {10}^{-2}$	$1.03515$
12	144	$8.01801\times {10}^{-3}$	$0.99258$	$1.18060\times {10}^{-2}$	$1.00004$

,

Table 2. The $L^2$-error of the pressure of ${\mathbf{LPSFEM}}_{2\times 2}$, ${\mathbf{LPFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	${\left|\left|\left|\mathit{p}-{\mathit{p}}^{\mathit{h}}\right|\right|\right|}_{0,{\Omega }_{\mathit{f}}}$	$\mathit{Rate}$	${\left|\left|\left|\mathit{p}-{\tilde{\mathit{p}}}^{\mathit{h}}\right|\right|\right|}_{0,{\Omega }_{\mathit{f}}}$	$\mathit{Rate}$
4	16	$5.32697\times {10}^{-3}$	−	$2.38948\times {10}^{-2}$	−
8	64	$1.05020\times {10}^{-3}$	$1.17133$	$4.58095\times {10}^{-3}$	$1.19149$
12	144	$4.42233\times {10}^{-4}$	$1.06655$	$2.14858\times {10}^{-3}$	$0.93361$

,

Table 3. The $H^1$-error of the piezometric head of ${\mathbf{LPSFEM}}_{2\times 2}$, ${\mathbf{LPFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	${\left|\left|\left|\mathit{\varphi }-{\mathit{\varphi }}^{\mathit{h}}\right|\right|\right|}_{1,{\mathbf{\Omega }}_{\mathbf{P}}}$	$\mathit{Rate}$	${\left|\left|\left|\mathit{\varphi }-{\tilde{\mathit{\varphi }}}^{\mathit{h}}\right|\right|\right|}_{1,{\mathbf{\Omega }}_{\mathbf{P}}}$	$\mathit{Rate}$
4	16	$3.88154\times {10}^{-2}$	−	$3.88539\times {10}^{-2}$	−
8	64	$9.29858\times {10}^{-3}$	$1.03077$	$9.30783\times {10}^{-3}$	$1.03077$
12	144	$4.11933\times {10}^{-3}$	$1.00400$	$4.12162\times {10}^{-3}$	$1.00454$

,

Table 4. The $H^1$-error of the velocity of ${\mathbf{LPPUSFEM}}_{2\times 2}$, ${\mathbf{LPPUFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	$\parallel \mathit{u}-{\mathit{u}}_{\mathit{H}}^{\mathit{h}}{\parallel }_{1,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$	$\parallel \mathit{u}-{\tilde{\mathit{u}}}_{\mathit{H}}^{\mathit{h}}{\parallel }_{1,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$
4	16	$7.37754\times {10}^{-2}$	−	$1.0415\times {10}^{-1}$	−
8	64	$1.79583\times {10}^{-2}$	$1.01924$	$2.5565\times {10}^{-2}$	$1.01319$
12	144	$7.91917\times {10}^{-3}$	$1.00966$	$1.1437\times {10}^{-2}$	$0.99195$

,

Table 5. The $L^2$-error of the pressure of ${\mathbf{LPPUSFEM}}_{2\times 2}$, ${\mathbf{LPPUFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	$\parallel \mathit{p}-{\mathit{p}}_{\mathit{H}}^{\mathit{h}}{\parallel }_{0,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$	$\parallel \mathit{p}-{\tilde{\mathit{p}}}_{\mathit{H}}^{\mathit{h}}{\parallel }_{0,{\mathbf{\Omega }}_{\mathit{f}}}$	$\mathit{Rate}$
4	16	$4.25241\times {10}^{-3}$	−	$2.45224\times {10}^{-2}$	−
8	64	$9.77831\times {10}^{-4}$	$1.06031$	$4.56773\times {10}^{-3}$	$1.21228$
12	144	$4.18785\times {10}^{-4}$	$1.04569$	$2.24674\times {10}^{-3}$	$0.87496$

and

Table 6. The $H^1$-error of the piezometric head of ${\mathbf{LPPUSFEM}}_{2\times 2}$, ${\mathbf{LPPUFEM}}_{2\times 2}$.

$1/\mathit{H}$	$1/\mathit{h}$	$\parallel \mathit{\varphi }-{\mathit{\varphi }}_{\mathit{H}}^{\mathit{h}}{\parallel }_{1,{\mathbf{\Omega }}_{\mathbf{P}}}$	$\mathit{Rate}$	$\parallel \mathit{\varphi }-{\tilde{\mathit{\varphi }}}_{\mathit{H}}^{\mathit{h}}{\parallel }_{1,{\mathbf{\Omega }}_{\mathbf{P}}}$	$\mathit{Rate}$
4	16	$3.75083\times {10}^{-2}$	−	$3.75503\times {10}^{-2}$	−
8	64	$9.28420\times {10}^{-3}$	$1.00718$	$9.29392\times {10}^{-3}$	$1.00723$
12	144	$4.08802\times {10}^{-3}$	$1.0115$	$4.09047\times {10}^{-3}$	$1.01205$

. As seen from these tables, we could derive the following conclusions:

(a)

Convergence orders (for the velocity, pressure, and piezometric head) of the four algorithm are all one with respect to the fine mesh size h, which agrees with the theoretical results;

(b)

LPSFEM derives a better approximation than LPFEM since the errors of LPSFEM are less than that of LPFEM. The same conclusion is suitable for the comparison of LPPUSFEM and LPPUFEM;

(c)

LPSFEM and LPPUSFEM exhibit almost the same errors, which indicates that the partition of unity functions scarcely ever affect the error accuracy. The same situation happens to LPFEM and LPPUFEM.

In

Table 7. The comparison of CPU time.

$1/\mathit{h}$	${\mathbf{LPSFEM}}_{2\times 2}$	${\mathbf{LPFEM}}_{2\times 2}$	${\mathbf{LPPUSFEM}}_{2\times 2}$	${\mathbf{LPPUFEM}}_{2\times 2}$
16	$0.042$	$0.064$	$0.044$	$0.066$
64	$0.406$	$0.547$	$0.410$	$0.55$
144	$2.078$	$2.672$	$2.085$	$2.675$
256	$6.186$	$9.279$	$6.193$	$9.385$
400	$17.012$	$33.258$	$17.02$	$33.271$

, we show the computational time of the four algorithms. As observed from this table, it is clear that ${\mathbf{LPSFEM}}_{2\times 2}$ and ${\mathbf{LPPUSFEM}}_{2\times 2}$ take less time than ${\mathbf{LPFEM}}_{2\times 2}$ and ${\mathbf{LPPUFEM}}_{2\times 2}$, namely, the two algorithms presented in this paper are more efficient.

To show the relation between the computational accuracy and CPU time with numbers of sub-domains, we ran Algorithms 1 and 2 by dividing ${\Omega }_f$ (${\Omega }_p$) into $2\times 2$, $3\times 3$, and $4\times 4$ sub-domains, respectively. The computing results and CPU time are plotted in Figure 2. It is not hard to see that the numerical results support the theoretical findings from Figure 2a–c. Furthermore, as seen from Figure 2d, as the number of sub-domains increases, the CPU time decreases, which is in accordance with our expectation.

6.2. Test 2

In this test, the computational domain and physical parameters were chosen as in test 1. The following modified lid-driven cavity flow problem was considered

$${\mathbf{u}|}_{{\Gamma }_f}=\left\{\begin{array}{cc}(1,0),\hfill & (x,y)\in [0,1]\times \left\{2\right\},\hfill \\ (0,0),\hfill & (x,y)\in \left\{0\right\}\times [1,2]\cup \left\{1\right\}\times [1,2],\hfill \end{array}\right.$$

(62)

and

$$\begin{array}{c}\hfill {\varphi |}_{{\Gamma }_p}=0.\end{array}$$

The external forces were set to 0, namely, ${\mathbf{f}}_1=\mathbf{0}$ and $f_2=0$. The mesh sizes were chosen as $h=H^2={(1/16)}^2$. Since the exact solution is unknown, we plot out the streamlines of four algorithms in Figure 3 for a comparison. As observed from Figure 3, the four algorithms derived similar results.

7. Conclusions

In this study, by utilizing the two-grid decoupled technique and the overlapping domain decomposition method, two local and parallel stabilized finite element algorithms are proposed and investigated for the mixed Stokes–Darcy model using the lowest equal-order finite element pairs. The algorithms were devised to circumvent the inf-sup condition by offsetting the discrete pressure space using the residual of the simple and symmetry term at the element level. The theoretical results indicate that the two proposed algorithms could arrive at the same error accuracy and convergence rates with the one-level method by properly choosing the configuration between the two mesh sizes. Some numerical results are reported to verify the theoretical findings and illustrate the efficiency and robustness of the proposed two algorithms.