1 Introduction

Many important applications require accurate solution of multi-domain, multi-physics coupling of groundwater and surface flows. The Stokes–Darcy fluid flow model is a coupling among surface and subsurface flows, which occurs in many natural and industrial events such as biomedicine, industrial processes, blood flow motion in the arteries, groundwater fluid flow in a karst aquifer, biofluid dynamics, hydrology, contaminant transport in a groundwater geometry through rivers, industrial filtration, petroleum engineering, spontaneous combustion of coal stockpiles, and so on [1–5]. A dynamical fluid model describes the free fluid flow in the conduit and porous medium fluid flow in the matrix separated by a shared interface.

Coupling two separate models via an interface generally needs some appropriate and effective interface conditions. For Stokes–Darcy model, the coupling conditions are well studied and usually modeled by three interface conditions. The first two consist of the continuity of the normal velocity across the interface, which is a consequence of the conservation of mass, and the balance of normal force across the interface. The third interface condition is a benchmark boundary condition invented by Beavers–Joseph in 1967 by conducting several experiments [6]. In the Beavers–Joseph interface condition, the tangential component of the stress force of the flow in the free medium at the interface is proportional to the jump of the tangential velocity across the interface. Saffman modified this interface condition in [7], and found that the velocity of the porous medium is smaller and can be dropped. Jones [8] reinterpreted this law for application to the curved boundaries and non-tangential flows.

After the rigorous investigation of Beavers and Joseph on the coupling condition between conduit and matrix, the study of Stokes–Darcy fluid flow model has become one of the most attractive research topics. The study of Stokes–Darcy fluid flow model started with intense numerical implementation [9, 10] while theoretical analysis begun with [11, 12]. The equilibrium Stokes–Darcy problem has only been understood recently in [11–14], and there has been an intense development of extensions and refinements thereafter. Over the last few decades, a great deal of effort has been devoted to developing an appropriate approximate solution to study the Stokes–Darcy model. Many different techniques and numerical methods were applied to investigate the Stokes–Darcy fluid flow model, such as coupled finite elements methods [15–21], two-grid/multi-grid methods [22–28], discontinuous Galerkin finite element methods [29–32], partitioned time stepping methods [33–36], least squares methods [37–39], domain decomposition methods [1, 2, 4, 40–48], local-parallel finite element methods [49], interface relaxation method [50, 51], motar finite element methods [52, 53], Lagrange multiplier methods [11, 54–59]. Among them, the theory and research literature about mixed spatial discretizations had been far less developed than that of non-mixed versions.

In [11], Layton, Schieweck and Yotov investigated a mixed variational formulation in both domains based on Beavers–Joseph–Saffman interface conditions and utilized a Lagrange multiplier on the interface to prove the well-posedness of the weak solution, which allowed us to decouple the coupled problem into two subproblems. Discacciati et al. studied the Navier/Stokes–Darcy fluid flow model and proposed an iterative subdomain method by using continuous finite elements in both regions with a second order elliptic problem in the Darcy domain and a standard mixed element method in the Stokes domain in [43, 47]. A locally conservative numerical method was applied to investigate the coupled free and porous medium flow by Rivière et al. in [30, 31], where a mixed finite element method was used for Darcy domain and a discontinuous Galerkin finite element method for Stokes region. A study of porous medium with small and large cavities using mixed finite element couple with the vuggy medium on the microscopic scale using Stokes equations was performed by Arbogast et al. in [15]. Mu and Xu studied mixed Stokes–Darcy fluid flow model using the two-grid method in [22]. A unified stabilized method was studied by Burman et al. [60], where they consider the lowest possible approximation order. A preconditioning technique was applied in [61] for mixed Stokes–Darcy fluid flow model by Cai et al. In [57–59], Gatica et al. extended the work of Layton [11] in a new dimension considering conforming mixed finite elements where the matrix subdomain is completely enclosed by the conduit region. The interface conditions allowed the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi–Raugel and Raviart–Thomas elements for the velocities, piecewise constants for the pressures and continuous piecewise-linear elements for the Lagrange multiplier. For more work on the mixed formulation, the reader can check [62–66].

In this paper, we investigate an approximate solution of the stationary Stokes–Darcy problem by developing a fully mixed stabilized finite element technique. The mixed formulation discussed in [11, 57–59, 62, 63] essentially needs a Lagrange multiplier, and the implementation is not easy in order to derive stability and convergence analysis. The fully mixed finite element scheme is proposed herein without introducing any Lagrange multiplier and computation is straightforward. A fully-discrete finite element algorithm is proposed, and we introduce a stabilized term essentially to ensure the stability of the temporal discretization. The stability of the finite element scheme is derived and the convergence analysis is discussed. To show the validity and efficiency of the numerical methods, we perform two numerical experiments which confirm the optimal convergence order considering an exact solution of the model problem. The effects of the stabilization parameter are investigated by considering different values of the stabilization parameter, which helps us to choose an accurate value of the stabilization parameter to obtain the optimal convergence order.

The rest of the paper is organized as follows. In Sect. 2, we describe the well known Stokes–Darcy fluid flow model with interface conditions. In Sect. 3, we present some notations, preliminaries, and variational formulation. The stabilized finite element method and its stability are discussed in Sect. 4. Section 5 contains a convergence analysis of the finite element scheme. In Sect. 6, we present two numerical tests to show the accuracy of the numerical methods. Finally, we conclude with a summary in Sect. 7.

2 The model problem

Let the two bounded domains be denoted by \(\Omega_{f}\), \(\Omega_{p}\subset R^{d}\) (\(d=2 \text{ or } 3\)) and lie across an interface Γ from each other. Here \(\Omega_{f}\cap \Omega _{p}={\emptyset }\), and \(\overline{\Omega }_{f}\cap \overline{\Omega }_{p}=\Gamma \), \(\overline{\Omega }=\overline{\Omega }_{f}\cup \overline{ \Omega }_{p}\), \(\mathbf{n}_{f}\) and \(\mathbf{n}_{p}\) are the unit outward normal vectors on \(\partial \Omega_{f}\) and \(\partial \Omega _{p}\), respectively, and \(\boldsymbol{\tau }_{i}\), \(i=1,\dots,d-1\), are the unit tangential vectors on the interface Γ, \(\Gamma_{f}= \partial \Omega_{f}\setminus \Gamma\), \(\Gamma_{p}=\partial \Omega_{p} \setminus \Gamma \). Note that \(\mathbf{n}_{p}=-\mathbf{n}_{f}\) on Γ. Figure 1 shows a sketch of the problem domain, its boundaries and some other notations.

Figure 1
figure 1

Global domain Ω consisting of the fluid region \(\Omega_{f}\) and the porous media region \(\Omega_{p}\) separated by the interface Γ

Full size image

The fluid velocity and pressure \(\mathbf{u}_{f}(x)\) and \(p(x)\) are governed by the Stokes equation in \(\Omega_{f}\):

$$\begin{aligned}& -\nabla \cdot \mathbb{T}=-2\nu \nabla \cdot \mathbb{D}(\mathbf{u}_{f})+ \nabla p =\mathbf{f}_{f}\quad \text{in } \Omega_{f}, \\ \end{aligned}$$
(2.1)
$$\begin{aligned}& \nabla \cdot \mathbf{u}_{f} =0\quad \text{in } \Omega_{f}, \end{aligned}$$
(2.2)

where \(\mathbb{T}=-p\mathbb{I}+2\nu \mathbb{D}(\mathbf{u}_{f})\) denotes the stress tensor, and \(\mathbb{D}(\mathbf{u}_{f})=\frac{1}{2}(\nabla \mathbf{u}_{f}+(\nabla \mathbf{u}_{f})^{T})\) represents the deformation tensor. The porous media flow is governed by the following Darcy equations on \(\Omega_{p}\) through the fluid velocity \(\mathbf{u}_{p}(x)\) and the piezometric head \(\phi (x)\):

$$\begin{aligned}& \mathbf{u}_{p} =-\mathbf{K}\nabla \phi \quad \text{in } \Omega_{p} , \end{aligned}$$
(2.3)
$$\begin{aligned}& \nabla \cdot \mathbf{u}_{p} =f_{p} \quad \text{in } \Omega_{p}. \end{aligned}$$
(2.4)

We impose impermeable boundary conditions, \(\mathbf{u}_{p}\cdot \mathbf{n}_{p}=0\) on \(\Gamma_{p}\), on the exterior boundary of the porous media region, and no slip conditions, \(\mathbf{u}_{f}=0\) on \(\Gamma_{f}\), in the Stokes region. Both selections of boundary conditions can be modified. On Γ the interface coupling conditions are conservation of mass, balance of forces and a tangential condition on the fluid region’s velocity on the interface. The correct tangential condition is not completely understood (possibly due to matching a pointwise velocity in the fluid region with an averaged or homogenized velocity in the porous region). In this paper, we take the Beavers–Joseph–Saffman (–Jones), see [6–8, 13], interfacial coupling

$$\begin{aligned}& \mathbf{u}_{f}\cdot \mathbf{n}_{f}+\mathbf{u}_{p} \cdot \mathbf{n}_{p} =0 \quad \text{on } \Gamma, \end{aligned}$$
(2.5)
$$\begin{aligned}& -\mathbf{n}_{f}\cdot \mathbb{T}\cdot \mathbf{n}_{f} =p-2 \nu \mathbf{n}_{f}\cdot \mathbb{D}(\mathbf{u}_{f})\cdot \mathbf{n}_{f} = \rho g\phi \quad \text{on } \Gamma, \end{aligned}$$
(2.6)
$$\begin{aligned}& -\mathbf{n}_{f}\cdot \mathbb{T}\cdot \tau_{i} = -2\mathbf{n}_{f} \cdot \mathbb{D}(\mathbf{u}_{f})\cdot \tau_{i} =\frac{\alpha }{\sqrt{ \tau_{i}\cdot \mathbf{K}\tau_{i}}}\mathbf{u}_{f}\cdot \tau_{i}, \quad 1\leq i \leq (d-1) \text{ on } \Gamma. \end{aligned}$$
(2.7)

This is a simplification of the original and more physically realistic Beavers–Joseph conditions (in which \(\mathbf{u}_{f}\cdot {\tau }_{i}\) in (2.8) is replaced by \((\mathbf{u}_{f}-\mathbf{u}_{p})\cdot {\tau }_{i}\)); see [6]. Here we denote

$$\begin{aligned}& \mathbf{f}_{f},f_{p} \text{ -- body forces in the fluid region and source in the porous region,} \\& \mathbf{K} \text{ -- symmetric positive definite (SPD) hydraulic conductivity tensor,} \\& \nu \text{ -- kinematic viscosity of fluid}, \\& \alpha \text{ -- constant parameter.} \end{aligned}$$

We shall also assume that all material and fluid parameters defined above are uniformly positive and bounded, i.e.,

$$ 0\leq k_{\min }\leq \lambda (\mathbf{K}\mathcal{)}\leq k_{\max }< \infty. $$
(2.8)

3 Notations and the variational formulation

In this part, we first introduce some Sobolev spaces [67] and norms. We denote the usual \(L^{2}\) norm by \(\|\cdot \|\) for square integrable scalar/vector/matrix-valued functions defined on domain \(\Omega_{f}\) or \(\Omega_{p}\), and the corresponding inner product by \((\cdot,\cdot)\), and similarly the \(L^{2}\)-norm in \(L^{2}(\Gamma)\) by \(\|\cdot \|_{\Gamma }\), for instance,

$$\begin{aligned}& \Vert p \Vert := \Biggl( \sum_{i=1}^{d} \int_{\Omega_{f}} \vert p \vert ^{2}\,\mathrm{d}x \Biggr) ^{1/2}, \quad \text{for } p\in L^{2}(\Omega_{f}), \\& \Vert \mathbf{v}_{f} \Vert := \Biggl( \sum _{i=1}^{d} \bigl\Vert v^{i}_{f} \bigr\Vert ^{2} \Biggr) ^{1/2}, \quad \text{for } \mathbf{v}_{f}= \bigl(v^{1}_{f}, \dots,v^{d} \bigr)\in L ^{2}(\Omega_{f})^{d}, \\& \Vert \nabla \mathbf{v}_{f} \Vert := \Biggl( \sum _{i=1}^{d} \bigl\Vert \nabla v^{i}_{f} \bigr\Vert ^{2} \Biggr) ^{1/2},\quad \text{for } \mathbf{v}_{f}=\in L^{2}(\Omega_{f})^{d}, \end{aligned}$$

and the inner product over Γ by

$$ (\phi_{1},\phi_{2})_{\Gamma }= \int_{\Gamma }\phi_{1}\phi_{2} \,\mathrm{d} \Gamma. $$

By setting the space

$$ H_{{\mathrm{div}}}=H(\mathrm{div};\Omega_{p}):= \bigl\{ \mathbf{v}_{p} \in L^{2}(\Omega_{p})^{d}: \nabla \cdot \mathbf{v}_{p}\in L^{2}(\Omega_{p}) \bigr\} , $$

we introduce the following spaces:

$$\begin{aligned} \mathbf{X}_{f} :=& \bigl\{ \mathbf{v}_{f}\in H^{1}(\Omega_{f})^{d}: \mathbf{v}_{f}=0 \text{ on } \Gamma_{f} \bigr\} , \\ Q_{f} :=&L^{2}(\Omega_{f}), \\ \mathbf{X}_{p} :=& \bigl\{ \mathbf{v}_{p}\in H(\mathrm{div}; \Omega_{p}):\mathbf{v} _{p}\cdot \mathbf{n}_{p}=0 \text{ on } \Gamma_{p} \bigr\} , \\ Q_{p} :=&L^{2}(\Omega_{p}). \end{aligned}$$

For the spaces \(\mathbf{X}_{f}\), \(\mathbf{X}_{p}\), we define the following norms:

$$\begin{aligned}& \Vert \mathbf{v}_{f} \Vert _{1} =\sqrt{ \Vert \mathbf{v}_{f} \Vert ^{2}+ \vert \mathbf{v} _{f} \vert _{1}^{2}}, \quad \text{with } \vert \mathbf{v}_{f} \vert _{1}= \Vert \nabla \mathbf{v}_{f} \Vert ~ \forall \mathbf{v}_{f}\in \mathbf{X}_{f}, \\& \Vert \mathbf{v}_{p} \Vert _{\mathrm{div}} =\sqrt{ \Vert \mathbf{v}_{p} \Vert ^{2}+ \Vert \nabla \cdot \mathbf{v}_{p} \Vert ^{2}} \quad \forall \mathbf{v}_{p}\in \mathbf{X}_{p}. \end{aligned}$$

The variational formulation of the steady-state Stokes–Darcy problem (2.1)–(2.7) reads as: Find \((\mathbf{u}_{f},p; \mathbf{u}_{p},\phi)\in (\mathbf{X}_{f}, Q_{f}; \mathbf{X}_{p},Q_{p})\) satisfying

$$\begin{aligned}& a_{f}(\mathbf{u}_{f},\mathbf{v}_{f})-b_{f}( \mathbf{v}_{f},p)+c_{ \Gamma }(\mathbf{v}_{f},\phi) =( \mathbf{f}_{f},\mathbf{v}_{f}) \quad \forall \mathbf{v}_{f}\in \mathbf{X}_{f}, \end{aligned}$$
(3.1)
$$\begin{aligned}& b_{f}(\mathbf{u}_{f},q) =0 \quad \forall q\in Q_{f}, \end{aligned}$$
(3.2)
$$\begin{aligned}& a_{p}(\mathbf{u}_{p},\mathbf{v}_{p})-b_{p}( \mathbf{v}_{p},\phi)-c _{\Gamma }(\mathbf{v}_{p},\phi) =0 \quad \forall \mathbf{v}_{p}\in \mathbf{X}_{p}, \end{aligned}$$
(3.3)
$$\begin{aligned}& b_{p}(\mathbf{u}_{p},\psi) =\rho g(f_{p},\psi) \quad \forall \psi \in Q_{p}, \end{aligned}$$
(3.4)

where the bilinear forms are defined as

$$\begin{aligned}& a_{f}(\mathbf{u}_{f},\mathbf{v}_{f}) :=2\nu \bigl(\mathbb{D}(\mathbf{u} _{f}),\mathbb{D}(\mathbf{v}_{f}) \bigr)+\sum_{i=1}^{d-1}\frac{\alpha }{\sqrt{ \tau_{i}\cdot \mathbf{K}\tau_{i}}}( \mathbf{u}_{f}\cdot \tau_{i}, \mathbf{v}_{f}\cdot \tau_{i})_{\Gamma }, \\& a_{p}(\mathbf{u}_{p},\mathbf{v}_{p}) :=\rho g \bigl(\mathbf{K}^{-1} \mathbf{u}_{p},\mathbf{v}_{p} \bigr), \\& b_{f}(\mathbf{v}_{f},p) :=(p,\nabla \cdot \mathbf{v}_{f}), \\& b_{p}(\mathbf{v}_{p},\phi) :=\rho g(\phi,\nabla \cdot \mathbf{v} _{p}), \\& c_{\Gamma }(\mathbf{v}_{f},\phi) :=\rho g (\phi, \mathbf{v}_{f} \cdot \mathbf{n}_{f})_{\Gamma }. \end{aligned}$$

After introducing

$$\begin{aligned}& \mathscr{L}(\mathbf{u}_{f},p,\mathbf{u}_{p},\phi; \mathbf{v}_{f},q, \mathbf{v}_{p},\psi):=a_{f}( \mathbf{u}_{f},\mathbf{v}_{f})-b_{f}( \mathbf{v}_{f},p)+b_{f}(\mathbf{u}_{f},q) \\& \quad {}+a_{p}(\mathbf{u}_{p},\mathbf{v}_{p})-b_{p}( \mathbf{v}_{p},\phi)+ b _{p}(\mathbf{u}_{p}, \psi) +c_{\Gamma }(\mathbf{v}_{f}-\mathbf{v}_{p}, \phi), \end{aligned}$$

the variational formulation (3.1)–(3.4) can be equivalently rewritten as follows: Find \((\mathbf{u}_{f},p;\mathbf{u} _{p},\phi) \in (\mathbf{X}_{f}, Q_{f}; \mathbf{X}_{p},Q_{p})\) satisfying

$$\begin{aligned} \mathscr{L}(\mathbf{u}_{f},p,\mathbf{u}_{p}, \phi;\mathbf{v}_{f},q, \mathbf{v}_{p},\psi)=( \mathbf{f}_{f},\mathbf{v}_{f})+\rho g(f_{p}, \psi) \end{aligned}$$
(3.5)

for all \((\mathbf{v}_{f},q;\mathbf{v}_{p},\psi) \in (\mathbf{X}_{f}, Q_{f}; \mathbf{X}_{p},Q_{p})\). It is easy to verify that this variational formulation is well-defined.

To end this section, we recall the following Poincaré, Korn’s and the trace inequalities, which will be used in the later analysis. There exist constants \(C_{P}\), \(C_{K}\), \(C_{\mathrm{v}}\), only depending on \(\Omega_{f}\), such that for all \(\mathbf{v}_{f}\in \mathbf{X}_{f}\),

$$ \Vert \mathbf{v}_{f} \Vert \leq C_{P} \vert \mathbf{v}_{f} \vert _{1},\qquad \vert \mathbf{v}_{f} \vert _{1} \leq C_{K} \bigl\Vert \mathbb{D}(\mathbf{v}_{f}) \bigr\Vert ,\qquad \Vert \mathbf{v}_{f} \Vert _{L^{2}( \Gamma)}\leq C_{\mathrm{v}} \Vert \mathbf{v}_{f} \Vert ^{1/2} \vert \mathbf{v}_{f} \vert _{1}^{1/2}. $$

Besides, there exists a constant \(\tilde{C}_{\mathrm{v}}\) that only depends on \(\Omega_{p}\) such that for all \(\psi \in Q_{p}\),

$$\begin{aligned} \Vert \psi \Vert _{L^{2}(\Gamma)}\leq \tilde{C}_{\mathrm{v}} \Vert \psi \Vert ^{1/2} \vert \psi \vert _{1} ^{1/2}. \end{aligned}$$
(3.6)

Hereafer, all the constants are positive unless otherwise specified.

4 The stabilized finite element method and its stability

First, we consider the family of triangulations \(T_{h}\) of Ω, consisting of \(T_{h}^{f}\) and \(T_{h}^{p}\), which are regular triangulations of \(\Omega_{f} \) and \(\Omega_{p} \), respectively, where \(h>0 \) is a positive parameter. We also assume that on the interface Γ the two meshes of \(T_{h}^{f}\) and \(T_{h}^{p}\), which form the regular triangulation \(T_{h}:=T_{h}^{f}\cup T_{h}^{p} \), coincide.

The domain of the uniformly regular triangulation \(\bar{\Omega }_{f} \cup \bar{\Omega }_{p} \) is such that \(\bar{\Omega } = \{\cup K : K \in T_{h}\}\) and \(h =\text{max}_{K\in T_{h}}h_{K}\). There exist positive constants \(c_{1} \) and \(c_{2} \) satisfying \(c_{1} h \leq h_{K}\leq c_{2} \rho_{K} \). To approximate the diameter \(h_{K} \) of the triangle (tetrahedral) K, \(\rho_{K} \) is the diameter of the greatest ball included in K. Based on the subdivisions \(T_{h}^{f} \) and \(T_{h}^{p} \), we can define finite element spaces \(\mathbf{X}_{fh} \subset \mathbf{X}_{f}\), \(Q^{h}_{fh} \subset Q_{f}\), \({\mathbf{X}_{ph} \subset \mathbf{X}_{p}}\), \(Q_{ph} \subset Q_{p}\). We consider the well-known MINI elements (\(P1b-P1 \)) to approximate the velocity and pressure in the conduit for Stokes equation [68]. To capture the fully mixed technique in the porous medium region linear Lagrangian elements, P1 are used for hydraulic (piezometric) head and Brezzi–Douglas–Marini (BDM1) piecewise constant finite elements are used for Darcy velocity [69]. In the fluid flow region, we select for the Stokes problem the finite element spaces \((\mathbf{X}_{fh},Q_{fh})\) that satisfy the velocity–pressure inf–sup condition:

There exists a constant \(\beta_{f} >0\), independent of h, such that

$$\begin{aligned}& \begin{aligned} &\mathbf{X}_{fh} \subset \mathbf{X}_{f},\qquad Q_{fh}\subset Q_{f}, \\ &\inf_{0\neq q^{h}\in Q_{fh}} \sup_{0\neq \mathbf{v}_{f}^{h}\in {\mathbf{X}}_{fh}}\frac{b_{f}( \mathbf{v}_{f}^{h},q^{h})}{ \vert \mathbf{v}_{f}^{h} \vert _{1} \Vert q^{h} \Vert } \geq \beta_{f}. \end{aligned} \end{aligned}$$
(4.1)

In the porous region, we use the finite element spaces \((\mathbf{X} _{ph},Q_{ph})\) that also satisfy a standard inf–sup condition:

There exists a \(\beta_{p}>0\) such that for all \(\phi^{h} \in Q _{ph}\),

$$\begin{aligned}& \begin{aligned}& \mathbf{X}_{ph} \subset \mathbf{X}_{p},\qquad Q_{ph}\subset Q_{p}, \\ &\beta_{p} \bigl\Vert \phi^{h} \bigr\Vert \leq \sup _{0\neq {\mathbf{v}_{p}^{h}}\in {\mathbf{X}}_{ph}}\frac{b_{p}( {\mathbf{v}}_{p}^{h},\phi^{h} )}{ \Vert {\mathbf{v}}_{p}^{h} \Vert _{\mathrm{div}}}. \end{aligned} \end{aligned}$$
(4.2)

From the inf–sup assumption (4.1), for a given arbitrary but fixed pressure \(p^{h}\in Q_{fh}\), we can get \(\mathbf{w}_{f}^{h} \in \mathbf{X}_{fh}\) such that

$$\begin{aligned} b_{f} \bigl({\mathbf{w}}_{f}^{h},p^{h} \bigr)\geq \widetilde{C_{1}} \bigl\Vert {\mathbf{w}}_{f}^{h} \bigr\Vert _{1} \bigl\Vert p^{h} \bigr\Vert . \end{aligned}$$
(4.3)

By normalizing \(\|{\mathbf{w}}_{f}^{h}\|_{1}=\lambda_{1} \|p^{h}\|\), thus

$$\begin{aligned} b_{f} \bigl({\mathbf{w}}_{f}^{h},p^{h} \bigr)\geq C_{1} \bigl\Vert p^{h} \bigr\Vert ^{2}. \end{aligned}$$
(4.4)

In a similar way, from the inf–sup assumption (4.2), for \(\phi^{h}\in Q_{ph}\), there exists a \(\mathbf{w}_{p}^{h}\in \mathbf{X}_{ph}\) such that

$$\begin{aligned} b_{p} \bigl({\mathbf{w}}_{p}^{h}, \phi^{h} \bigr)\geq \widetilde{C_{2}} \bigl\Vert { \mathbf{w}}_{p}^{h} \bigr\Vert _{\texttt{div}} \bigl\Vert \phi^{h} \bigr\Vert . \end{aligned}$$
(4.5)

Assume that \(\mathbf{w}_{p}^{h}\) is normalized so that \(\|{\mathbf{w}} _{p}^{h}\|_{\texttt{div}}=\lambda_{2} \|\phi^{h}\|\), thus

$$\begin{aligned} b_{p} \bigl({\mathbf{w}}_{p}^{h}, \phi^{h} \bigr)\geq C_{2} \bigl\Vert \phi^{h} \bigr\Vert ^{2}. \end{aligned}$$
(4.6)

Moreover, we need the inverse inequalities in both \(\mathbf{X}_{fh}\) and \(Q_{ph}\): there exist constants \(C_{\mathrm{inv}}\) and \(\tilde{C}_{\mathrm{inv}}\), which depend on the minimum angles in the finite element mesh used on \(\Omega_{f}\) and \(\Omega_{p}\), such that

$$\begin{aligned}& \bigl\vert \mathbf{v}_{f}^{h} \bigr\vert _{1} \leq C_{\mathrm{inv}}h^{-1} \bigl\Vert \mathbf{v}_{f}^{h} \bigr\Vert \quad \forall \mathbf{v}_{f}^{h}\in \mathbf{X}_{fh}, \end{aligned}$$
(4.7)
$$\begin{aligned}& \bigl\vert \psi^{h} \bigr\vert _{1} \leq \tilde{C}_{\mathrm{inv}}h^{-1} \bigl\Vert \psi^{h} \bigr\Vert \quad \forall \psi^{h}\in Q_{ph}. \end{aligned}$$
(4.8)

4.1 The stabilized finite element method

In this section, we present a stabilized finite element scheme for the Stokes–Darcy problem.

Algorithm 4.1

Find \((\mathbf{u}_{f}^{h},{p^{h}},\mathbf{u} _{p}^{h},{\phi^{h}})\in (\mathbf{X}_{fh}, Q_{fh}{,} \mathbf{X}_{ph},Q _{ph})\) satisfying

$$\begin{aligned} \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr)= \bigl( \mathbf{f}_{f}, \mathbf{v}_{f}^{h} \bigr)+\rho g \bigl(f_{p}, \psi^{h} \bigr), \end{aligned}$$
(4.9)

for any \((\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h}) \in (\mathbf{X}_{fh}, Q_{fh}, \mathbf{X}_{ph},Q_{ph})\), where

$$\begin{aligned} \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) =& {\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h}; \mathbf{v}_{f}^{h},q^{h}\mathbf{v}_{p}^{h}, \psi^{h} \bigr) \\ & {}+\frac{\delta }{h} \bigl( \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr) \cdot \mathbf{n}_{f}, \bigl(\mathbf{v}_{f}^{h}-\mathbf{v}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr) _{\Gamma }, \end{aligned}$$

and the term

$$ \frac{\delta }{h} \bigl( \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f}, \bigl(\mathbf{v}_{f}^{h}-\mathbf{v}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr) _{\Gamma }=\frac{\delta }{h} \int_{\Gamma } \bigl( \bigl( \mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr) \bigl( \bigl( \mathbf{v}_{f}^{h}- \mathbf{v}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr)\,d\Gamma, $$

is the stabilized term for the Stokes–Darcy problem.

4.2 The stability of the stabilized finite element method

In this section, we prove the stability of the stabilized finite element scheme.

Theorem 1

(Continuity of \(\widetilde{\mathscr{L}}\))

There exists a constant C such that

$$\begin{aligned} \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) \leq C \bigl(\big|\!\big|\!\big|\bigl( \mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h} \bigr)\big|\!\big|\!\big|\bigr) \bigl(\big|\!\big|\!\big|\bigl( \mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p}^{h},\psi^{h} \bigr)\big|\!\big|\!\big|\bigr), \end{aligned}$$

where

$$\begin{aligned} \big|\!\big|\!\big|\bigl(\mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p}^{h},\psi^{h} \bigr)\big|\!\big|\!\big|= \bigl\Vert \mathbf{v}_{f}^{h} \bigr\Vert _{1}+ \bigl\Vert q^{h} \bigr\Vert + \bigl\Vert \mathbf{v}_{p}^{h} \bigr\Vert _{ \texttt{div}}+ \bigl\Vert \psi^{h} \bigr\Vert +h^{-1/2} \bigl\Vert \bigl(\mathbf{v}_{f}^{h}- \mathbf{v} _{p}^{h} \bigr) \bigr\Vert _{\Gamma }. \end{aligned}$$

Proof

By the Schwarz inequality (3.6) and the inverse inequality (4.8), we have

$$\begin{aligned} c_{\Gamma } \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h}, \phi^{h} \bigr) \leq & \rho g \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{ \Gamma } \bigl\Vert \phi^{h} \bigr\Vert _{\Gamma } \\ \leq &\rho g\tilde{C}_{\mathrm{v}}\tilde{C}_{\mathrm{inv}} h^{-1/2} \bigl\Vert \bigl(\mathbf{u}_{f} ^{h}-\mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma } \bigl\Vert \phi^{h} \bigr\Vert \end{aligned}$$

and

$$\begin{aligned}& \frac{\delta }{h} \bigl( \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f}, \bigl(\mathbf{v}_{f}^{h}-\mathbf{v}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr)_{\Gamma } \\& \quad \leq \delta h^{-1/2} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma }h^{-1/2} \bigl\Vert \bigl( \mathbf{v}_{f}^{h}- \mathbf{v} _{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma }. \end{aligned}$$

Thus we can prove that \(\widetilde{\mathscr{L}}\) is continuous, i.e.,

$$\begin{aligned}& \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) \\& \quad =a_{f} \bigl(\mathbf{u}_{f}^{h}, \mathbf{v}_{f}^{h} \bigr)-b_{f} \bigl( \mathbf{v}_{f} ^{h},p^{h} \bigr)+b_{f} \bigl(\mathbf{u}_{f}^{h},q^{h} \bigr) \\& \qquad {}+a_{p} \bigl(\mathbf{u}_{p}^{h}, \mathbf{v}_{p}^{h} \bigr)-b_{p} \bigl( \mathbf{v}_{p} ^{h},\phi^{h} \bigr)+b_{p} \bigl(\mathbf{u}_{p}^{h}, \psi^{h} \bigr) \\& \qquad {}+\frac{\delta }{h} \bigl( \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f}, \bigl(\mathbf{v}_{f}^{h}-\mathbf{v}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr) +c_{\Gamma } \bigl( \mathbf{v}_{f}^{h}-\mathbf{u}_{p}^{h}, \phi^{h} \bigr) \\& \quad \leq C \bigl(\big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h} \bigr)\big|\!\big|\!\big|\bigr) \bigl(\big|\!\big|\!\big|\bigl( \mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p}^{h},\psi^{h} \bigr)\big|\!\big|\!\big|\bigr). \end{aligned}$$

The proof is complete. □

Theorem 2

(Coercivity of \(\widetilde{\mathscr{L}}\))

There exists a constant \(\beta >0\) such that the following inequality holds for all \({(\mathbf{u}_{f}^{h},p^{h},\mathbf{u}_{p}^{h},\phi^{h})} \in {( \mathbf{X}_{fh},Q_{fh},\mathbf{X}_{ph},Q_{ph})}\):

$$\begin{aligned}& \sup_{ (\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})\in (X _{fh}, Q_{fh}, X_{ph},Q_{ph})} \frac{\widetilde{\mathscr{L}}( \mathbf{u}_{f}^{h},p^{h},\mathbf{u}_{p}^{h},\phi^{h};\mathbf{v}_{f} ^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})}{|\!|\!|(\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})|\!|\!|} \geq \beta \big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h} \bigr)\big|\!\big|\!\big|. \end{aligned}$$
(4.10)

Proof

We will construct \((\widehat{\mathbf{v}_{f}^{h}},\widehat{q^{h}},\widehat{ \mathbf{v}_{p}^{h}},\widehat{\psi ^{h}})\) such that

$$\begin{aligned} \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h};\widehat{ \mathbf{v}_{f}^{h}},\widehat{q^{h}},\widehat{ \mathbf{v}_{p}^{h}},\widehat{\psi ^{h}} \bigr) \geq C \bigl(\big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p ^{h},\mathbf{u}_{p}^{h},\phi^{h} \bigr)\big|\!\big|\!\big|\bigr) \bigl( \big|\!\big|\!\big|\bigl( \widehat{ \mathbf{v}_{f}^{h}}, \widehat{q^{h}},\widehat{ \mathbf{v}_{p}^{h}}, \widehat{\psi ^{h}} \bigr) \big|\!\big|\!\big|\bigr) \end{aligned}$$

using the following three steps.

Step 1. By setting \(({\mathbf{v}_{f}^{h}},{q^{h}},{ \mathbf{v} _{p}^{h}},{\psi^{h}})=(\mathbf{u}_{f}^{h},p^{h},\mathbf{u}_{p}^{h}, \phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h})\), we derive

$$\begin{aligned}& \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{u}_{f}^{h},p^{h},\mathbf{u}_{p}^{h}, \phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h} \bigr) \\& \quad = \bigl\Vert \mathbf{u}_{f}^{h} \bigr\Vert _{1}^{2}+ \bigl\Vert \mathbf{u}_{p}^{h} \bigr\Vert _{\texttt{div}} ^{2}+\frac{\delta }{h} \bigl\Vert \bigl( \mathbf{u}_{f}^{h}-\mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma }^{2} +c_{\Gamma } \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h}, \phi^{h} \bigr). \end{aligned}$$

Note that

$$\begin{aligned} c_{\Gamma } \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h}, \phi^{h} \bigr) \geq & - \rho g\tilde{C}_{\mathrm{v}}\tilde{C}_{\mathrm{inv}} h^{-1/2} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma } \bigl\Vert \phi^{h} \bigr\Vert \\ \geq & -\frac{ \rho^{2} g^{2}\tilde{C}_{\mathrm{v}}^{2}\tilde{C}_{\mathrm{inv}}^{2}}{ \gamma h C_{2}} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert ^{2}_{\Gamma }-\frac{\gamma C_{2}}{4} \bigl\Vert \phi^{h} \bigr\Vert ^{2}, \end{aligned}$$

where γ is a real positive parameter, which will be determined later.

Step 2. By selecting \(({\mathbf{v}_{f}^{h}},{q^{h}},{ \mathbf{v}_{p}^{h}},{\psi^{h}})=(-\gamma \mathbf{w}_{f}^{h} ,0,- \gamma \mathbf{w}_{p}^{h},0)\) where \(\mathbf{w}_{f}^{h} \), \(\mathbf{w}_{p}^{h} \) satisfy (4.5) and (4.6), respectively, and normalizing \(\|(\mathbf{w}_{f}^{h}-\mathbf{w}_{p} ^{h})\cdot \mathbf{n}_{f}\|_{\Gamma }=\lambda_{3}\|(\mathbf{u}_{f} ^{h}-\mathbf{u}_{p}^{h})\cdot \mathbf{n}_{f}\|_{\Gamma }\), together with (4.3)–(4.6), we have

$$\begin{aligned}& \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h};-\gamma \mathbf{w}_{f}^{h} ,0,-\gamma \mathbf{w}_{p}^{h},0 \bigr) \\ & \quad =-\gamma a_{f} \bigl(\mathbf{u}_{f}^{h}, \mathbf{w}_{f}^{h} \bigr)-\gamma a_{p} \bigl( \mathbf{u}_{p}^{h},\mathbf{w}_{p}^{h} \bigr)+\gamma b_{f} \bigl(\mathbf{w}_{f} ^{h},p ^{h} \bigr)+\gamma b_{p} \bigl(\mathbf{w}_{p}^{h}, \phi^{h} \bigr) \\& \qquad {}-\frac{\gamma \delta }{h} \bigl( \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr) \cdot \mathbf{n}_{f}, \bigl(\mathbf{w}_{f}^{h}-\mathbf{w}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr)_{\Gamma }-\gamma c_{\Gamma } \bigl(\mathbf{w}_{f}^{h}- \mathbf{w}_{p}^{h}, \phi^{h} \bigr) \\& \quad \geq -\gamma \lambda_{1} \bigl\Vert \mathbf{u}_{f}^{h} \bigr\Vert _{1} \bigl\Vert p^{h} \bigr\Vert -\gamma \lambda_{2} \bigl\Vert \mathbf{u}_{p}^{h} \bigr\Vert _{\texttt{div}} \bigl\Vert {\phi }^{h} \bigr\Vert + \gamma C_{1} \bigl\Vert p^{h} \bigr\Vert ^{2}+\gamma C_{2} \bigl\Vert \phi^{h} \bigr\Vert ^{2} \\& \qquad {}-\frac{\gamma \delta \lambda_{3}}{h} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert ^{2}_{\Gamma }-\gamma \lambda _{3} \rho g \tilde{C}_{\mathrm{v}}\tilde{C}_{\mathrm{inv}} h^{-1/2} \bigl\Vert \bigl( \mathbf{u}_{f} ^{h}-\mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma } \bigl\Vert \phi^{h} \bigr\Vert \\& \quad \geq -\frac{\gamma \lambda_{1}^{2}}{2C_{1}} \bigl\Vert \mathbf{u}_{f}^{h} \bigr\Vert _{1} ^{2}-\frac{\gamma \lambda_{2}^{2}}{C_{2}} \bigl\Vert \mathbf{u}_{p}^{h} \bigr\Vert _{ \texttt{div}}^{2}+ \frac{\gamma C_{1}}{2} \bigl\Vert p^{h} \bigr\Vert ^{2}+ \frac{\gamma C _{2}}{2} \bigl\Vert \phi^{h} \bigr\Vert ^{2} \\& \qquad {}-\frac{\gamma \delta \lambda_{3}}{h} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert ^{2}_{\Gamma }-\frac{\gamma \lambda_{3}^{2} \rho^{2} g^{2}\tilde{C}_{\mathrm{v}}^{2}\tilde{C}_{\mathrm{inv}}^{2}}{ h C_{2}} \bigl\Vert \bigl( \mathbf{u}_{f}^{h}-\mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n} _{f} \bigr\Vert _{\Gamma }^{2}, \end{aligned}$$

here the following two main Young inequalities are used:

$$\begin{aligned}& \gamma \lambda_{2} \bigl\Vert \mathbf{u} _{p}^{h} \bigr\Vert _{\texttt{div}} \bigl\Vert {\phi }^{h} \bigr\Vert \le \frac{\gamma \lambda_{2}^{2}}{C_{2}} \bigl\Vert \mathbf{u} _{p}^{h} \bigr\Vert _{ \texttt{div}}^{2}+\frac{\gamma C_{2}}{4} \bigl\Vert {\phi }^{h} \bigr\Vert ^{2}, \\ & \gamma \lambda_{3} \rho g\tilde{C}_{\mathrm{v}} \tilde{C}_{\mathrm{inv}} h^{-1/2} \bigl\Vert \bigl( \mathbf{u}_{f}^{h}- \mathbf{u}_{p}^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma } \bigl\Vert \phi^{h} \bigr\Vert \\ & \quad \le \frac{\gamma \lambda_{3} ^{2}\rho^{2} g^{2}\tilde{C}_{\mathrm{v}}^{2} \tilde{C}_{\mathrm{inv}}^{2} }{hC_{2}} \bigl\Vert \bigl(\mathbf{u}_{f}^{h}- \mathbf{u}_{p} ^{h} \bigr)\cdot \mathbf{n}_{f} \bigr\Vert _{\Gamma }^{2} +\frac{\gamma C_{2}}{4} \bigl\Vert \phi^{h} \bigr\Vert ^{2}, \end{aligned}$$

where δ is another real positive parameter to be determined soon.

Step 3. Choosing \((\widehat{\mathbf{v}_{f}^{h}},\widehat{q^{h}},\widehat{ \mathbf{v}_{p}^{h}},\widehat{\psi ^{h}})=(\mathbf{u}_{f}^{h}-\gamma \mathbf{w}_{f}^{h} ,p^{h},\mathbf{u}_{p}^{h}-\gamma \mathbf{w}_{p} ^{h},\phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h})\), we can obtain by the above arguments the following inequality:

$$\begin{aligned}& \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{u}_{f}^{h}-\gamma \mathbf{w}_{f}^{h} ,p^{h}, \mathbf{u}_{p}^{h}-\gamma \mathbf{w}_{p}^{h},\phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h} \bigr) \\& \quad \geq \biggl(1-\frac{\gamma \lambda^{2}_{1}}{2C_{1}} \biggr) \Vert \mathbf{u}_{f} \Vert ^{2}+ \biggl(1-\frac{\gamma \lambda^{2}_{2}}{C_{2}} \biggr) \bigl\Vert \mathbf{u} ^{h}_{p} \bigr\Vert ^{2}_{\mathrm{div}}+ \frac{\gamma C_{1}}{2} \bigl\Vert p^{h} \bigr\Vert ^{2}+ \frac{\gamma C_{2}}{4} \bigl\Vert \phi^{h} \bigr\Vert ^{2} \\ & \quad \quad {}+ \biggl( \frac{ \delta (1-\gamma \lambda_{3})}{h}-\frac{\rho^{2} g ^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2}_{\mathrm{inv}}}{\gamma h C_{2}}-\frac{ \gamma \lambda^{2}_{3}\rho^{2} g^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2} _{\mathrm{inv}}}{hC_{2}} \biggr) \big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h} \bigr)\big|\!\big|\!\big|^{2}. \end{aligned}$$

Then we enforce the following conditions on γ and δ:

$$\begin{aligned}& 1-\frac{\gamma \lambda_{1}^{2}}{2C_{1}}\geq \frac{1}{2}, \qquad 1-\frac{ \gamma \lambda_{2}^{2}}{C_{2}}\geq \frac{1}{2},\qquad 1-\gamma \lambda _{3} \geq \frac{1}{4}, \\& \frac{ \delta (1-\gamma \lambda_{3})}{h}-\frac{\rho^{2} g^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2}_{\mathrm{inv}}}{\gamma h C_{2}}-\frac{\gamma \lambda^{2}_{3}\rho^{2} g^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2}_{\mathrm{inv}}}{hC _{2}}\geq \frac{\delta }{2h}. \end{aligned}$$

This encourages us to select sufficiently small γ and large δ as follows:

$$\begin{aligned}& \gamma \leq \min \biggl\{ \frac{C_{1}}{\lambda_{1}^{2}}, \frac{2C _{2}}{ \lambda_{2}^{2}}, \frac{1}{4\lambda_{3} } \biggr\} , \qquad \delta \geq \frac{4 \rho^{2} g^{2}\tilde{C}_{\mathrm{v}}^{2}\tilde{C}_{\mathrm{inv}} ^{2}(1+\gamma^{2}\lambda_{3}^{2} )}{ \gamma C_{2}}. \end{aligned}$$

To this end, we can obtain

$$\begin{aligned}& \widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{u}_{f}^{h}-\gamma \mathbf{w}_{f}^{h} ,p^{h}, \mathbf{u}_{p}^{h}-\gamma \mathbf{w}_{p}^{h},\phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h} \bigr) \\& \quad \geq \biggl(1-\frac{\gamma \lambda^{2}_{1}}{2C_{1}} \biggr) \Vert \mathbf{u}_{f} \Vert ^{2}+ \biggl(1-\frac{\gamma \lambda^{2}_{2}}{C_{2}} \biggr) \bigl\Vert \mathbf{u} ^{h}_{p} \bigr\Vert ^{2}_{\mathrm{div}}+ \frac{\gamma C_{1}}{2} \bigl\Vert p^{h} \bigr\Vert ^{2}+ \frac{\gamma C_{2}}{4} \bigl\Vert \phi^{h} \bigr\Vert ^{2} \\& \qquad {}+ \biggl( \frac{\delta }{h}-\frac{\gamma \delta \lambda_{3}}{h}-\frac{ \rho^{2} g^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2}_{\mathrm{inv}}}{\gamma h C_{2}}- \frac{ \gamma \lambda^{2}_{3}\rho^{2} g^{2} \tilde{C}^{2}_{\mathrm{v}}\tilde{C}^{2} _{\mathrm{inv}}}{hC_{2}} \biggr) \big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h} \bigr)\big|\!\big|\!\big|^{2} \\& \quad \geq C_{4} \bigl(\big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi ^{h} \bigr)\big|\!\big|\!\big|\bigr) \bigl(\big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h}-\gamma \mathbf{w}_{f}^{h} ,p^{h}, \mathbf{u}_{p}^{h}- \gamma \mathbf{w}_{p}^{h},\phi^{h}+ \nabla \cdot \mathbf{u}_{p}^{h} \bigr)\big|\!\big|\!\big|\bigr) \\& \quad = C \bigl(\big|\!\big|\!\big|\bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h} \bigr)\big|\!\big|\!\big|\bigr) \bigl(\big|\!\big|\!\big|\bigl( \widehat{\mathbf{v}_{f}^{h}}, \widehat{q^{h}},\widehat{ \mathbf{v}_{p}^{h}}, \widehat{\psi ^{h}} \bigr)\big|\!\big|\!\big|\bigr). \end{aligned}$$

The proof is complete. □

5 Error estimate for the stabilized finite element method

In this section, we prove the error estimate for the stabilized finite element method for the scheme.

Theorem 3

Assume that \((\mathbf{u}_{f},p;\mathbf{u}_{p},\phi)\) is an exact solution, the norms \(\|\mathbf{u}_{p}\|_{2}\), \(\|\mathbf{u}_{f}\|_{2}\), \(\|p\|_{1}\), and \(\|\phi \|_{1}\) are bounded from above, and, moreover, \((\mathbf{u}_{f}^{h},p^{h};\mathbf{u}_{p}^{h},\phi^{h})\) is the stabilized finite element solution, then we have

$$\begin{aligned} \bigl\Vert \mathbf{u}_{f}-\mathbf{u}_{f}^{h} \bigr\Vert _{1}+ \bigl\Vert p-p^{h} \bigr\Vert + \bigl\Vert \mathbf{u} _{p}-\mathbf{u}_{p}^{h} \bigr\Vert _{\mathrm{div}}+ \bigl\Vert \phi -\phi^{h} \bigr\Vert \leq Ch. \end{aligned}$$
(5.1)

Proof

First, by subtracting (4.9) from (3.5), thanks to the interface condition (2.5), we get the error equation as follows:

$$\begin{aligned}& {\mathscr{L}} \bigl(\mathbf{u}_{f},p,\mathbf{u}_{p},\phi; \mathbf{v}_{f} ^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr)-\widetilde{\mathscr{L}} \bigl( \mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h},\phi^{h}; \mathbf{v}_{f} ^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) \\& \quad =\widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f},p, \mathbf{u}_{p},\phi; \mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p}^{h},\psi^{h} \bigr)- \widetilde{ \mathscr{L}} \bigl(\mathbf{u}_{f}^{h},p^{h}, \mathbf{u}_{p}^{h}, \phi^{h}; \mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) \\& \quad =\widetilde{\mathscr{L}} \bigl(\mathbf{u}_{f}- \mathbf{u}_{f}^{h},p-p^{h}, \mathbf{u}_{p}- \mathbf{u}_{p}^{h},\phi -\phi^{h}; \mathbf{v}_{f}^{h},q ^{h},\mathbf{v}_{p}^{h}, \psi^{h} \bigr) =0. \end{aligned}$$
(5.2)

By introducing \((\bar{\mathbf{u}},\bar{p},\bar{\mathbf{u}}_{p},\bar{ \phi })\) as the interpolation of \((\mathbf{u}_{f},p,\mathbf{u}_{p}, \phi)\) from \((\mathbf{X}_{f},Q_{f},\mathbf{X}_{p},Q_{p})\) onto the finite element spaces \((\mathbf{X}_{fh},Q_{fh},\mathbf{X}_{ph},Q_{ph})\), we split the errors into two parts as

$$\begin{aligned}& \mathbf{u}_{f}-\mathbf{u}_{f}^{h} =( \mathbf{u}_{f}-\bar{\mathbf{u}} _{f})+ \bigl(\bar{ \mathbf{u}}_{f}-\mathbf{u}_{f}^{h} \bigr) \triangleq \bar{ \mathbf{e}}_{f}+\mathbf{e}_{f}^{h}, \\& p-p^{h} =(p-\bar{p})+ \bigl(\bar{p}-p^{h} \bigr)\triangleq \bar{\eta }+\eta^{h}, \\& \mathbf{u}_{p}-\mathbf{u}_{p}^{h} =( \mathbf{u}_{p}-\bar{\mathbf{u}} _{p})+ \bigl(\bar{ \mathbf{u}}_{p}-\mathbf{u}_{p}^{h} \bigr) \triangleq \bar{ \mathbf{e}}_{p}+\mathbf{e}_{p}^{h}, \\& \phi -\phi^{h} =(\phi -\bar{\phi })+ \bigl(\bar{\phi }- \phi^{h} \bigr)\triangleq \bar{\theta }+\theta^{h}, \end{aligned}$$

The interpolation error are listed below:

$$\begin{aligned}& \Vert \bar{\mathbf{e}}_{f} \Vert _{1}+ \Vert \bar{ \eta } \Vert \leq Ch \bigl( \Vert \mathbf{u} _{f} \Vert _{2}+ \Vert p \Vert _{1} \bigr), \qquad \Vert \bar{\mathbf{e}}_{p} \Vert _{\texttt{div}}+ \Vert \bar{ \theta } \Vert \leq Ch \bigl( \Vert \mathbf{u}_{p} \Vert _{2}+ \Vert \phi \Vert _{1} \bigr). \end{aligned}$$

Substituting them into the error equation (5.2), we get

$$\begin{aligned} \widetilde{\mathscr{L}} \bigl(\mathbf{e}_{f}^{h}, \eta^{h},\mathbf{e}_{p} ^{h}, \theta^{h};\mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p}^{h},\psi^{h} \bigr)=- \widetilde{ \mathscr{L}} \bigl(\bar{\mathbf{e}}_{f},\bar{\eta },\bar{ \mathbf{e}}_{p},\bar{\theta };\mathbf{v}_{f}^{h},q^{h}, \mathbf{v}_{p} ^{h},\psi^{h} \bigr). \end{aligned}$$

From the coercivity and the continuity of \(\widetilde{\mathscr{L}}\), trace and inverse inequalities, we obtain

$$\begin{aligned}& \beta \big|\!\big|\!\big|\bigl(\mathbf{e}_{f}^{h}, \eta^{h},\mathbf{e}_{p}^{h},\theta^{h} \bigr)\big|\!\big|\!\big|\\& \quad \leq \sup_{ (\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})}\frac{ \widetilde{\mathscr{L}}(\mathbf{e}_{f}^{h},\eta^{h},\mathbf{e}_{p} ^{h},\theta^{h};\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})}{ |\!|\!|(\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})|\!|\!|} \\& \quad = \sup_{ (\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})}\frac{- \widetilde{\mathscr{L}}(\bar{\mathbf{e}}_{f},\bar{\eta },\bar{ \mathbf{e}}_{p},\bar{\theta };\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p} ^{h},\psi^{h})}{|\!|\!|(\mathbf{v}_{f}^{h},q^{h},\mathbf{v}_{p}^{h},\psi^{h})|\!|\!|} \\& \quad \leq C\big|\!\big|\!\big|(\bar{\mathbf{e}}_{f},\bar{\eta },\bar{ \mathbf{e}}_{p},\bar{ \theta })\big|\!\big|\!\big|\leq C \bigl( \Vert \bar{\mathbf{e}}_{f} \Vert _{1}+ \Vert \bar{\eta } \Vert + \Vert \bar{ \mathbf{e}}_{p} \Vert _{\texttt{div}}+ \Vert \bar{\theta } \Vert +h^{-1/2} \bigl\Vert (\bar{ \mathbf{e}}_{f}-\bar{ \mathbf{e}}_{p}) \bigr\Vert _{\Gamma } \bigr) \\& \quad \leq C \bigl( \Vert \bar{\mathbf{e}}_{f} \Vert _{1}+ \Vert \bar{\eta } \Vert + \Vert \bar{ \mathbf{e}}_{p} \Vert _{\texttt{div}}+ \Vert \bar{\theta } \Vert +h^{-1} \bigl( \Vert \bar{ \mathbf{e}}_{f} \Vert + \Vert \bar{ \mathbf{e}}_{p} \Vert \bigr) \bigr) \\& \quad \leq Ch \bigl( \Vert \mathbf{u}_{f} \Vert _{2}+ \Vert p \Vert _{1}+ \Vert \mathbf{u}_{p} \Vert _{2}+ \Vert \phi \Vert _{1} \bigr). \end{aligned}$$

Finally, together with the interpolation error, we derive the error estimate (5.2). □

6 Numerical experiments

In this section, we present two numerical experiments to illustrate the accuracy and efficiency of the proposed stabilized fully mixed finite element method. The global domain Ω consists of two subdomains with free fluid flow region \(\Omega_{f}= [0,1]\times [1,2]\) and porous medium domain \(\Omega_{p}=[0,1]\times [0,1] \) for both numerical tests. The interface of the current computational domain is \(\Gamma =[0,1]\times \lbrace 1 \rbrace \).

The finite element spaces are constructed by the well-known MINI elements \((P1b-P1)\) for Stokes problem. To capture the fully mixed technique in the porous medium region, linear Lagrangian elements P1 are used for hydraulic (piezometric) head and Brezzi–Douglas–Marini (BDM1) piecewise constant finite elements are used for Darcy velocity. In the first numerical test, the influence of the stabilization parameter on the temporal discretization is examined by considering different values of δ and then the optimal convergence for the spacing h is checked. In the second numerical test, we focus on showing the effects of the hydraulic conductivity parameter K for gradual decreasing. All the numerical tests are executed by a specialized free domain software known as FreeFEM++ [70]. Graphs and figures are drawn by using MATLAB and Tecplot software.

6.1 Convergence test 1

As the first numerical test example, we choose the following exact solution, which satisfies the Beavers–Joseph–Saffman interface conditions (2.5)–(2.7):

{ u f = ( [ − 2 3 x ( y − 1 ) 3 ] + [ 2 − π sin ( π x ) ] ) , p = [ 2 − π sin ( π x ) ] sin ( 0.5 π y ) , ϕ = [ 2 − π sin ( π x ) ] [ 1 − y − cos ( π y ) ] , u p = − K ∇ ϕ .
(6.1)

The Dirichlet boundary condition and source term of the model problem are chosen in such a way that the above-listed functions are the exact solutions of the model problem. For computational convenience in the first numerical experiment, all the physical parameters ν, ρ, g, K, α are simply taken as 1.0.

To investigate the impact of the stabilization parameter on the convergence order for the finite element scheme, we perform several numerical tests from different aspects in the following. First, for the different values of \(\delta = 10^{k}\) (\(k=3, 2, 1, -1, -2, -3, -4, -5, 0\)), we use Fig. 2 to demonstrate the order of the convergence of \(\|\mathbf{u}_{f}^{h}-\mathbf{u}_{f}\|_{1,\Omega_{f}}\) and \(\|\mathbf{u}^{h}_{p}-\mathbf{u}_{p}\|_{0,\Omega_{p}}\). This study admits that the relatively large and small values of the stabilization parameter δ significantly affect the convergence orders of velocity and/or pressure. Without assuming the stabilization parameter δ (namely, \(\delta =0\)), the convergence order is also not accurate. We can obtain the accurate convergence order for the value of the stabilization parameter \(\delta =0.1 \). In Figs. 3 and 4, we present the velocity and pressure contours for the approximate and exact solutions with the different values of the stabilization parameter \(\delta =0.1, 1.0, 1000, 0.00001\) and 0. These figures also illustrate that the velocity and pressure contours are simultaneously regular for an approximate and exact solutions with the value of the stabilization parameter \(\delta =0.1 \). In fact, considering the small values of the stabilization parameter \(\delta =0.0001 \) and 0, the velocity contour changed dramatically for approximate solutions. The same illustration can be seen for the large values of \(\delta =1000 \) on the pressure contour. From the figures, we can conclude that for the relatively small and large values of the stabilization parameter δ cause a significant effect on the accuracy of the convergence order, velocity and pressure contours. The above discussion leads us to consider the value of the stabilization parameter \(\delta =0.1 \) to perform the numerical tests which ensure an optimal convergence order and generate velocity and pressure contours accurately.

Figure 2
figure 2

The order of convergence for the different values of the stabilization parameter δ for test 1. (Left) The order of the convergence of \(\|\mathbf{u}_{f}^{h}-\mathbf{u}_{f}\|_{1,\Omega_{f}}\); (Right) The order of the convergence of \(\|\mathbf{u}^{h}_{p}- \mathbf{u}_{p}\|_{0,\Omega_{p}}\)

Full size image
Figure 3
figure 3

An illustration of the velocity contour with different values of the stabilization parameter for test 1: true solution (top-left); \(\delta =0.1 \) (top-middle); \(\delta =1\) (top-right); \(\delta =1000\) (down-left); \(\delta =0.0001\) (down-middle); \(\delta =0\) (down-right)

Full size image
Figure 4
figure 4

An illustration of the pressure contour for the true solution and approximate solution with the different values of the stabilization parameter: true solution (top-left); \(\delta =0.1 \) (top-middle); \(\delta =1\) (top-right); \(\delta =1000\) (down-left); \(\delta =0.0001\) (down-middle); \(\delta =0\) (down-right)

Full size image

To demonstrate the order of convergence of the finite element stabilized scheme, we introduce Table 1, the errors between the computed and exact solutions with varying spacing \(h=1/4,1/8,1/16,1/32,1/64 \). Figure 5 is the log–log plot of the data in Table 1. We can observe from the figures that the optimal convergence order is obtained, which supports the theoretical analysis.

Figure 5
figure 5

The order of convergence of \(L^{2} \)-norm (left) and \(H^{1}\)-norm (right) for the change on the convergence order of the stabilized finite element scheme for test 1

Full size image
Table 1 Approximate error of the stabilized finite element scheme for test 1
Full size table

6.2 Convergence test 2

The main purpose of the second numerical is to show the influence of the physical parameter hydraulic conductivity K on the convergence order inspired by [33] for the stabilized finite element scheme. In this experiment, we set all the values of the physical parameters ν, ρ, g, K, α, δ the same as in the previous computation, except for different values of the hydraulic conductivity parameter K where \(\mathbf{K}= k\mathbb{I}\), \(k=0.1,0.01,0.001\).

The exact solution for the second numerical test satisfying the Beavers–Joseph–Saffman interface conditions is given by

{ u f = ( ( y 2 − 2 y + 1 ) ( x 2 − x ) ) , p = [ 2 ν ( x + y − 1 ) + g n 3 k ] , ϕ = [ n k ( x ( 1 − x ) ( y − 1 ) + 1 3 y 3 − y 2 + y ) + 2 ν g x ] , u p = − K ∇ ϕ .
(6.2)

The Dirichlet boundary condition and source terms of the model problem are chosen in such a way that the above-listed functions are the exact solutions of the model problem.

In Tables 2, 3 and 4, the approximation errors in differential norms are listed for the stabilized finite element scheme with varying hydraulic conductivity \(\mathbf{K}= 0.1\mathbb{I},0.01\mathbb{I}, 0.001\mathbb{I} \). From these tables we observe that the order of the magnitude of the Stokes and Darcy pressure p and ϕ gradually increases with the decrease of the value of the parameter K. From Fig. 6, we can see that the optimal convergence order is also obtained, which confirms the theoretical analysis.

Figure 6
figure 6

The order of convergence for stabilized finite element scheme using changing mesh for test 2: the error order of \(L^{2} \)-norm for \(\mathbf{K}=0.1\mathbb{I} \) (top-left), \(\mathbf{K}=0.01\mathbb{I} \) (top-middle), \(\mathbf{K}=0.001\mathbb{I} \) (top-right); the error order of \(H^{1} \)-norm for \(\mathbb{K}=0.1\mathbb{I} \) (down-left), \(\mathbf{K}=0.01\mathbb{I} \) (down-middle), \(\mathbf{K}=0.001 \mathbb{I} \) (down-right)

Full size image
Table 2 Approximate error of the stabilized finite scheme for test 2 with \(\mathbf{K}=0.1 \mathbb{I}\)
Full size table
Table 3 Approximate error of the stabilized finite element scheme for test 2 with \(\mathbf{K}=0.01 \mathbb{I} \)
Full size table
Table 4 Approximate error of the stabilized finite element scheme for test 2 with \(\mathbf{K}=0.001 \mathbb{I}\)
Full size table

7 Conclusion

In this contribution, we investigated a new fully mixed finite element method to solve the Stokes–Darcy fluid flow model without introducing any Lagrange multiplier. We proposed a stabilized finite element scheme and introduced a stabilization term to ensure the well-posedness of the temporal discretization. The desired stability and convergence analysis, including optimal error estimates, was derived for the proposed algorithm. To show the exclusive feature of the finite element scheme and numerical methods, we performed two numerical tests and illustrated the results of the experiments. The numerical test revealed the accuracy and efficiency of the proposed mixed finite element method.