A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem

Abstract

Combining the mixed discontinuous Galerkin method for the Darcy flow and the interior penalty discontinuous Galerkin methods for the Stokes problem, a locally conservative discrete scheme is proposed for numerically solving the coupled Stokes and Darcy problem. We prove the well-posedness of the solution of the proposed numerical scheme by boundedness, K-ellipticity and a discrete inf-sup condition. A priori error estimates, in proper norms are derived, and to verify the theoretical analysis, some numerical experiments are given.

1 Introduction

The coupled Stokes and Darcy model describes the interaction between free flow and porous media flow. Such systems arise, for example, in modeling the groundwater (aquifer) contamination through filtration and streams, and numerical modeling of this complicated interaction is a challenging work in both theoretical analysis and practical engineering applications. There are some related works of the coupled system. Based on the Beavers–Joseph–Saffman interface conditions [14] Layton, Schieweck, and Yotov [27] prove the existence and uniqueness of a weak solution of the coupled system and, present and analyze its numerical scheme by adopting continuous finite element methods to discretize the Stokes problem and mixed finite element methods (MFE) to discrete the Darcy problem. Rivière et al. [3, 4, 13] propose and analyze a locally conservative discrete scheme by employing discontinuous Galerkin (DG) methods and mixed finite element methods for the coupled Stokes and Darcy equations, and by utilizing the DG methods for the coupled Navier–Stokes and Darcy problem. In addition, based on DG methods and mixed finite element methods, a strongly conservative numerical scheme is given in [15] by this group. Fu and Lehrenfeld [16] propose a strongly conservative numerical scheme for the coupled system by considering hybrid discontinuous Galerkin methods (HDG) and mixed finite element methods. Based on a continuous trace approximation of velocity and a discontinuous trace approximation of pressure, Cesmelioglu et al. present a embedded-hybridized discontinuous Galerkin (EDG-HDG) finite element method [1] with strong mass conservation for the coupled Stokes–Darcy problem.

Mixed discontinuous Galerkin (MDG) method [12] and discontinuous Galerkin (DG) methods [2, 5, 6, 8, 11] are two kinds of locally mass conservative numerical methods. Mixed numerical formulations are popular for porous media problems and DG methods have many attractive properties such as being element-wise conservative, high-order methods, easily implementable on unstructured meshes. Then, we propose a locally conservative discrete scheme to numerically solving the coupled Darcy–Stokes problem, which is constructed by using DG methods to approximate the Stokes problem and MDG method to approximate the Darcy problem. The proposed scheme is different from the above mentioned numerical methods expect for the EDG-HDG finite element method, since we employ the MDG method to approximate the Darcy problem rather than the MFE methods, and the numerical scheme adopts the totally discontinuous polynomial spaces in both Stokes domain and Darcy region. Such choices of discontinuous polynomial spaces avoid the difficulty of the construction of conforming finite element space. It thus is more convenient for us to implement the algorithm in a unified framework of DG methods.

The EDG-HDG finite element method is an efficient and attractive numerical scheme with strong mass conservation, especially when the higher-order polynomial spaces are used. It also utilizes the element discontinuous polynomial spaces in both Stokes domain and Darcy region, even though a continuous approximation of trace of velocity is employed. Comparing to the proposed numerical scheme, from the point of degrees of freedom (DOF), the EDG-HDG finite element method needs fewer DOF in matching triangle (tetrahedra) meshes and quadrilateral (hexahedron) meshes if the lowest order finite element space is used, since a continuous approximation of trace of velocity is applied. However, if the meshes are polygonal and non-matching with hanging nodes, we can’t draw this conclusion. Furthermore, our scheme may be superior to this EDG-HDG finite element method if the lowest order finite element space is used and a complete HDG finite element method is utilized, which means the trace of velocity is discontinuous rather than continuous. Thus, our scheme has acceptable DOF for the lowest order finite element space. What’s more, our scheme requires less matrix assembly, storage, and is easier to code and implement. In a word, we think our scheme is more suitable for the lowest order polynomial space and the EDG-HDG finite element method is more attractive for higher-order polynomial spaces.

The features of the proposed numerical scheme are that the lowest order finite element space has acceptable DOF. Moreover, it is more convenient for us to implement the algorithm in a unified framework of DG methods and can be generalized to other porous media flow problems such as Stokes-Biot model [17] and Stokes-dual-porosity model [18, 21], since it is constructed by a straightforward combination of MDG method and DG methods. In addition, we present the numerical analysis for the proposed scheme in proper norms and show some numerical tests to verify the analysis. The novelty of the analysis mainly includes that we generalize the primal MDG method with Dirichlet boundary condition to an MDG method with Neumann boundary condition, and based on H(div)-like DG norm we also prove the K-ellipticity by using the local lift operator in the kernel space.

The outline of the article is given as follows: In Sect. 2, the coupled Stokes and Darcy equations, notation and numerical scheme are presented. Section 3 recalls some inequalities and approximation operators. In Sect. 4, the boundedness, K-ellipticity and a discrete inf-sup condition are derived. Section 5 proves the priori error estimates. In Sect. 6, some numerical experiments are used to validate the theoretical analysis.

2 Model Equations, Notation, and Scheme

Let $\Omega$ be a open bounded domain in ${\mathbb {R}}^d$, $d=2,3$, comprised of two subdomains $\Omega _1$ and $\Omega _2$. Let $\Gamma _{12}$ be the interface and $\Gamma _{12}=\partial \Omega _1\cap \partial \Omega _2$. Define $\Gamma _i=\partial \Omega _i\setminus \Gamma _{12}$, $i=1, 2$. Denote by ${\mathbf {n}}$ the unit outward normal vector to $\partial \Omega$. Let ${\mathbf {n}}_{12}$ (resp., $\tau _{12}^j$) be the unit normal (resp., tangential) vector to $\Gamma _{12}$ outward of $\Omega _1$, where $j=1, \dots , d-1$. Denote by ${\mathbf {u}}=({\mathbf {u}}_f, {\mathbf {u}}_s)$ the fluid velocity in $(\Omega _1, \Omega _2)$ and $p=(p_f, p_s)$ the fluid pressure in $(\Omega _1, \Omega _2)$. We assume the Stokes equations in $\Omega _1$, and there holds

$$\begin{aligned}&-\nabla \cdot {\mathbf {T}}({\mathbf {u}}_f, p_f)={\mathbf {f}} \qquad \text {in}\quad \Omega _1, \end{aligned}$$

(1)

$$\begin{aligned}&\nabla \cdot {\mathbf {u}}_f=0 \qquad \text {in}\quad \Omega _1, \end{aligned}$$

(2)

$$\begin{aligned}&{\mathbf {u}}_f=0 \qquad \text {on} \quad \Gamma _1. \end{aligned}$$

(3)

Here ${\mathbf {T}}$ is the stress tensor

$$\begin{aligned} {\mathbf {T}}({\mathbf {u}}_f, p_f)=-p_f{\mathbf {I}}+2\mu {\mathbf {D}}({\mathbf {u}}_f), \end{aligned}$$

where $\mu >0$ is the constant viscosity coefficient and the strain tensor is defined by

$$\begin{aligned} {\mathbf {D}}({\mathbf {u}}_f)=\frac{1}{2}(\nabla {\mathbf {u}}_f+\nabla {\mathbf {u}}_f^T). \end{aligned}$$

In region $\Omega _2$, the governing equations satisfy the Darcy equations

$$\begin{aligned}&\nabla \cdot {\mathbf {u}}_s=g \qquad \text {in} \quad \Omega _2, \end{aligned}$$

(4)

$$\begin{aligned}&{\mathbf {K}}^{-1}{\mathbf {u}}_s+\nabla p_s=0 \qquad \text {in} \quad \Omega _2, \end{aligned}$$

(5)

$$\begin{aligned}&{\mathbf {u}}_s \cdot {\mathbf {n}}=0 \qquad \text {on} \quad \Gamma _2, \end{aligned}$$

(6)

where the permeability tensor ${\mathbf {K}}$ is symmetric and positive definite, and satisfies for some $0< k_{min}\le k_{max}<\infty$,

$$\begin{aligned} k_{min} \xi ^T\xi \le \xi ^T{\mathbf {K}}({\mathbf {x}})\xi \le k_{max} \xi ^T\xi \quad \forall \xi \in {\mathbb {R}}^d. \end{aligned}$$

The physical quantities in $\Omega _1$ and $\Omega _2$ are coupled by the following interface conditions on $\Gamma _{12}$:

$$\begin{aligned} {\mathbf {u}}_f\cdot {\mathbf {n}}_{12}&={\mathbf {u}}_s\cdot {\mathbf {n}}_{12}, \end{aligned}$$

(7)

$$\begin{aligned} p_f-2\mu ({\mathbf {D}}({\mathbf {u}}_f){\mathbf {n}}_{12})\cdot {\mathbf {n}}_{12}&=p_s, \end{aligned}$$

(8)

$$\begin{aligned} {\mathbf {u}}_f\cdot \tau _{12}^j&=-2G({\mathbf {D}}({\mathbf {u}}_f){\mathbf {n}}_{12})\cdot \tau _{12}^j, \quad j=1, \dots , d-1. \end{aligned}$$

(9)

Note that interface condition (7) denotes the mass conservation, interface condition (8) stands for balance of forces, and interface (9) represents the Beaver–Joseph–Saffman law, where $G>0$ is friction coefficient determined by numerical experiments.

For $i=1, 2$, let $\varepsilon _h^i$ be a non-overlapping and quasi-uniform decomposition [22] of $\Omega _i$, let $\Gamma _h^i$ be the set of interior facets and let $h_i$ denote the maximum diameter of elements in $\varepsilon _h^i$. For any non-negative integer k and number $r\ge 1$, the classical Sobolev spaces [23] on a domain O is denoted by $W^{k,r}(O)=\{v \in L^r(O): D^m(v)\in L^r(O), \forall m\ge k\}$, where $D^m(v)$ are the partial derivatives of v of order m. The associated Sobolev norm (respectively, semi-norm) is denoted by $\Vert \cdot \Vert _{k,r,O}$ (respectively, $|\cdot |_{k,r,O}$), or by $\Vert \cdot \Vert _{k,O}$ (respectively, $|\cdot |_{k,O}$) if $r = 2$. We use the notation $H^k(O)$ for $W^{k,2}(O)$ and $L_0^2(O)$ for the space of square integrable functions with zero average. The $L^2$ inner-product will be denoted by $(\cdot ,\cdot )$. Moreover, let $H(div;\Omega _2)=\{{\mathbf {v}} \in (L^2(\Omega _2))^d, \nabla \cdot {\mathbf {v}}\in L^2(\Omega _2)\}$ with norm $\Vert {\mathbf {v}}\Vert _{H(div;\Omega _2)}^2=\Vert {\mathbf {v}}\Vert _{0,\Omega _2}^2+\Vert \nabla \cdot {\mathbf {v}}\Vert _{0,\Omega _2}^2$, and let $H_0(div;\Omega _2)=\{{\mathbf {v}} \in H(div;\Omega _2): {\mathbf {v}}\cdot {\mathbf {n}}|_{\Gamma _2}=0\}$. Throughout the paper, c will denote a generic positive constant whose value may vary with different equations but shall be independent of the mesh-sizes $h_1$ and $h_2$. Particularly, our scheme requires that the trace of the normal derivatives of ${\mathbf {u}}_f$ and the trace of ${\mathbf {u}}_f$, $p_f$, ${\mathbf {u}}_s$, $p_s$ are well defined, and are square-integrable, therefore, we define the following functional spaces:

$$\begin{aligned} {\mathbf {X}}^f&=\{{\mathbf {v}}_f\in (L^2(\Omega _1))^d, \quad \forall E \in \varepsilon _h^1, \quad {\mathbf {v}}_f|_{E}\in (H^2(E))^d\},\\ {\mathbf {X}}^s&=\{{\mathbf {v}}_s\in (L^2(\Omega _2))^d, \quad \forall E \in \varepsilon _h^2, \quad {\mathbf {v}}_s|_{E}\in (H^1(E))^d\},\\ M^f&=\{q_f\in L^2(\Omega _1), \quad \forall E \in \varepsilon _h^1, \quad q_f|_{E}\in H^1(E)\},\\ M^s&=\{q_s\in L^2(\Omega _2), \quad \forall E \in \varepsilon _h^2, \quad q_s|_{E}\in H^1(E)\}. \end{aligned}$$

Let w be any scalar or vector-valued function. Given a fixed unit normal vector ${\mathbf {n}}_e$ on each interior facet $e\in \partial E_1 \cap \partial E_2$, pointing from $E_1$ to $E_2$, the average $\{w\}$ and jump [w] of function w are uniquely defined

$$\begin{aligned} \{w\}=\frac{1}{2}(w|_{E_1}+w|_{E_2}), \quad [w]=w|_{E_1}-w|_{E_2}. \end{aligned}$$

In addition, if $e \in \partial \Omega$ and $e\in E_1$, then the average $\{w\}$ and jump [w] of function w are

$$\begin{aligned} \{w\}=w|_{E_1}, \quad [w]=w|_{E_1}. \end{aligned}$$

Define the general DG norms:

$$\begin{aligned} \Vert {\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}^2&=|||\nabla {\mathbf {v}}_f|||_{0,\Omega _1}^2+\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [{\mathbf {v}}_f]\Vert _{0,e}^2+\frac{\mu }{G}\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\Vert {\mathbf {v}}_f\cdot \tau _{12}^j\Vert _{0,e}^2,\\ \Vert q_f\Vert _{M^f}^2&=\Vert q_f\Vert _{0,\Omega _1}^2, ~~\quad ~~\Vert q_s\Vert _{M^s}^2=\Vert q_s\Vert _{0,\Omega _2}^2, \end{aligned}$$

and H(div)-like norm:

$$\begin{aligned} \Vert {\mathbf {v}}_s\Vert _{{\mathbf {X}}^s}^2&=||| {\mathbf {v}}_s|||_{0,\Omega _2}^2+|||\nabla \cdot {\mathbf {v}}_s|||_{0,\Omega _2}^2+\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [{\mathbf {v}}_s\cdot {\mathbf {n}}_e]\Vert _{0,e}^2, \end{aligned}$$

where |e| denotes the measure of facet e, the parameters $\sigma _{1,e}$ and $\sigma _{2,e}$ are positive penalty constants, and the $|||\cdot |||$ norm is the usual “broken” norm with $m=0$ or $m=1$

$$\begin{aligned} |||w|||_{m,\Omega _i}^2=\sum _{E\in \varepsilon _h^i}\Vert w\Vert _{m,E}^2 \quad \forall i=1, 2. \end{aligned}$$

Now we define

$$\begin{aligned} {\mathbf {X}}={\mathbf {X}}^f\times {\mathbf {X}}^s, ~~ M=\{q\in L_0^2(\Omega ): q|_{\Omega _1}\in M^f,q|_{\Omega _2}\in M^s\}, \end{aligned}$$

and the corresponding norms

$$\begin{aligned} \Vert {\mathbf {v}}\Vert _{{\mathbf {X}}}^2=\Vert {\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}^2+\Vert {\mathbf {v}}_s\Vert _{{\mathbf {X}}^s}^2, \quad \Vert q\Vert _{M}^2=\Vert q_f\Vert _{M^f}^2+\Vert q_s\Vert _{M^s}^2. \end{aligned}$$

(10)

Let $k_1$ and $k_2$ be positive integers. We consider the finite-dimensional approximation spaces ${\mathbf {X}}_h^f\subset {\mathbf {X}}^f$, ${\mathbf {X}}_h^s\subset {\mathbf {X}}^s$, $M_h^f \subset M^f$ and $M_h^s \subset M^s$, defined as follows:

$$\begin{aligned} {\mathbf {X}}_h^f&=\{{\mathbf {v}}_f\in {\mathbf {X}}^f, \quad \forall E \in \varepsilon _h^1, \quad {\mathbf {v}}_f|_{E}\in ({\mathbb {P}}_{k_1}(E))^d\},\\ {\mathbf {X}}_h^s&=\{{\mathbf {v}}_s\in {\mathbf {X}}^s, \quad \forall E \in \varepsilon _h^2, \quad {\mathbf {v}}_s|_{E}\in ({\mathbb {P}}_{k_2}(E))^d\},\\ M_h^f&=\{q_f\in M^f, \quad \forall E \in \varepsilon _h^1, \quad q_f|_{E}\in {\mathbb {P}}_{k_1-1}(E)\},\\ M_h^s&=\{q_s\in M^s, \quad \forall E \in \varepsilon _h^2, \quad q_s|_{E}\in {\mathbb {P}}_{k_2-1}(E)\}, \end{aligned}$$

where ${\mathbb {P}}_{k_1}(E)$, ${\mathbb {P}}_{k_2}(E)$ stand for polynomial spaces of degree less than or equal to $k_1$, $k_2$ respectively. Let ${\mathbf {X}}_h$ and $M_h$ be finite-dimensional subspaces and belong to ${\mathbf {X}}$ and M, respectively, such that

$$\begin{aligned} {\mathbf {X}}_h={\mathbf {X}}_h^f\times {\mathbf {X}}_h^s, \quad M_h=(M_h^f\times M_h^s)\cap L_0^2(\Omega ). \end{aligned}$$

Before giving the numerical scheme, some bilinear forms shall be introduced. For any ${\mathbf {u}}_f, {\mathbf {v}}_f \in {\mathbf {X}}^f$,

$$\begin{aligned} a_f({\mathbf {u}}_f,{\mathbf {v}}_f)&=2\mu \sum _{E\in \varepsilon _h^1}\int _E{{\mathbf {D}}({\mathbf {u}}_f):{\mathbf {D}}({\mathbf {v}}_f)d{\mathbf {x}}}+\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\frac{\sigma _{1,e}}{|e|}[{\mathbf {u}}_f]\cdot [{\mathbf {v}}_f]ds}\\&\quad -2 \mu \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}({\mathbf {u}}_f) {\mathbf {n}}_e\}\cdot [{\mathbf {v}}_f]ds}+2 \mu \epsilon \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}({\mathbf {v}}_f) {\mathbf {n}}_e\}\cdot [{\mathbf {u}}_f]ds}\\&\quad +\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\int _{e}{\frac{\mu }{G}{\mathbf {u}}_f\cdot \tau _{12}^j{\mathbf {v}}_f\cdot \tau _{12}^jds}, \end{aligned}$$

where $\epsilon =\pm 1$ and $\sigma _{1,e}>0$ is the penalty constant. For any ${\mathbf {u}}_s, {\mathbf {v}}_s \in {\mathbf {X}}^s$,

$$\begin{aligned} a_s({\mathbf {u}}_s,{\mathbf {v}}_s)=\sum _{E\in \varepsilon _h^2}\int _E{{\mathbf {K}}^{-1}{\mathbf {u}}_s\cdot {\mathbf {v}}_sd{\mathbf {x}}} +\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{\frac{\sigma _{2,e}}{|e|}[{\mathbf {u}}_s\cdot {\mathbf {n}}_e][{\mathbf {v}}_s\cdot {\mathbf {n}}_e]ds}, \end{aligned}$$

where the stability constant $\sigma _{2,e}>0$. For any ${\mathbf {v}}_f \in {\mathbf {X}}^f, p_f \in M^f$ and ${\mathbf {u}}_s \in {\mathbf {X}}^s, p_s \in M^s$,

$$\begin{aligned} b_f({\mathbf {v}}_f,p_f)&=-\sum _{E\in \varepsilon _h^1}\int _{E}{p_f\nabla \cdot {\mathbf {v}}_fd{\mathbf {x}}} +\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _{e}{[{\mathbf {v}}_f\cdot {\mathbf {n}}_e]\{p_f\}ds},\\ b_s({\mathbf {v}}_s,p_s)&=-\sum _{E\in \varepsilon _h^2}\int _{E}{p_s\nabla \cdot {\mathbf {v}}_sd{\mathbf {x}}}+\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _{e}{[{\mathbf {v}}_s\cdot {\mathbf {n}}_e]\{p_s\}ds}. \end{aligned}$$

Define the finite-dimensional space of functions $\Lambda _h={\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}$ on the interface and let

$$\begin{aligned} {\mathbf {V}}_h=\{({\mathbf {v}}_f,{\mathbf {v}}_s)\in {\mathbf {X}}_h:\sum _{e\in \Gamma _{12}}\int _{e}{\eta ({\mathbf {v}}_f-{\mathbf {v}}_s)\cdot {\mathbf {n}}_{12}ds=0}\quad \forall \eta \in \Lambda _h\}. \end{aligned}$$

Assumption 2.1

We assume $\Lambda _h=\{\eta \in L^2(\Gamma _{12}), \forall e \in \Gamma _{12},\eta |_{e}\in {\mathbb {P}}_{k_2}(e)\}$.

Indeed, the Assumption 2.1 holds true by choosing a proper basis of space ${\mathbf {X}}_h^s$.

Define $a=a_f+a_s$ and $b=b_f+b_s$, the numerical scheme reads as: Find $({\mathbf {u}}_h,p_h)\in {\mathbf {V}}_h\times M_h$ such that

$$\begin{aligned} a({\mathbf {u}}_h,{\mathbf {v}}_h)+b({\mathbf {v}}_h,p_h)&=\int _{\Omega _1}{{\mathbf {f}}\cdot {\mathbf {v}}_hd{\mathbf {x}}} \quad \forall {\mathbf {v}}_h \in {\mathbf {V}}_h, \end{aligned}$$

(11)

$$\begin{aligned} -b({\mathbf {u}}_h,q_h)&=\int _{\Omega _2}{g q_hd{\mathbf {x}}} \quad \forall q_h \in M_h&. \end{aligned}$$

(12)

Remark 2.1

We can check that the numerical scheme (11)–(12) is locally mass conservative. Indeed, taking the test function $q_h$ in (12) such that $q_h=1$ on element E and $q_h=0$ on the remaining elements E, we obtain

$$\begin{aligned} \int _{E}{\{{\mathbf {u}}_{h}\}\cdot {\mathbf {n}}_{E}d{\mathbf {x}}}=\int _{E}{\chi _{\Omega _2} g d{\mathbf {x}}} \quad \forall E \in \varepsilon _h^1\cup \varepsilon _h^2, \end{aligned}$$

where $\chi _{\Omega _2}$ is the characteristic function taking the value 0 in $\Omega _1$ and 1 in $\Omega _2$.

Remark 2.2

To facilitate the theoretical analysis, we introduce the space ${\mathbf {V}}_h$ of weakly-continuous-normal velocities on the interface. Clearly, it is difficult to construct this space, thus, an equivalent formulation to (11)–(12) is presented in Sect. 6. It only depends on the space ${\mathbf {X}}_h^s$, and it is more convenient for implementation. The space $\Lambda _h$, as a Lagrange multiplies space, is used to impose the continuity of the normal velocities. The choice $\Lambda _h={\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}$ is to ensure the well-posedness, stability and accuracy of the discrete scheme (11)–(12).

Next, we show the exact solution of the coupled Stokes and Darcy problem (1)–(9) satisfies the numerical scheme (11)–(12) up to an error term on the interface.

Lemma 2.1

Let $({\mathbf {u}}, p)$ satisfy the coupled Stokes–Darcy problem (1)–(9), such that ${\mathbf {u}}_f={\mathbf {u}}|_{\Omega _f}$, ${\mathbf {u}}_s={\mathbf {u}}|_{\Omega _s}$ and $p_f=p|_{\Omega _f}$, $p_s=p|_{\Omega _s}$, then $({\mathbf {u}},p)$ solves the variational problem

$$\begin{aligned} a({\mathbf {u}},{\mathbf {v}}_h)+b({\mathbf {v}}_h,p)&= \int _{\Omega _1}{{\mathbf {f}}\cdot {\mathbf {v}}_hd{\mathbf {x}}}-\sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}} \quad {\mathbf {v}}_h \in {\mathbf {V}}_h, \end{aligned}$$

(13)

$$\begin{aligned} -b({\mathbf {u}},q_h)&=\int _{\Omega _2}{g q_hd{\mathbf {x}}} \quad \forall q_h \in M_h. \end{aligned}$$

(14)

Proof

Multiplying the Stokes Eq. (1) by ${\mathbf {v}}_{f,h}\in {\mathbf {X}}_h^f$, integrating by parts over element E and summing over all elements E. From the regularity of the exact solution and the boundary condition, we can obtain

$$\begin{aligned}&\sum _{E\in \varepsilon _h^1}\int _{E}{(2\mu D({\mathbf {u}}_f):D({\mathbf {v}}_{f,h})-p_f\nabla \cdot {\mathbf {v}}_{f,h})d{\mathbf {x}}}+\sum _{e \in \Gamma _h^1\cup \Gamma _1}\int _{e}{\{p_f\}[{\mathbf {v}}_{f,h}\cdot {\mathbf {n}}_e]ds} \\&\qquad -2\mu \sum _{e \in \Gamma _h^1\cup \Gamma _1}\int _{e}{\{ D({\mathbf {u}}_f){\mathbf {n}}_e\}\cdot [{\mathbf {v}}_{f,h}]ds} +2\mu \epsilon \sum _{e \in \Gamma _h^1\cup \Gamma _1}\int _{e}{\{ D({\mathbf {v}}_{f,h}){\mathbf {n}}_e\}\cdot [{\mathbf {u}}_f]ds}\\&\qquad +\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _{e}{\frac{\sigma _{1,e}}{|e|}[{\mathbf {u}}_f]\cdot [{\mathbf {v}}_{f,h}]ds} -\sum _{e \in \Gamma _{12}}\int _{e}{(-p_f{\mathbf {I}}+2\mu D({\mathbf {u}}_f)){\mathbf {n}}_{12}\cdot {\mathbf {v}}_{f,h}ds}\\&\quad =\int _{\Omega _1}{{\mathbf {f}}\cdot {\mathbf {v}}_{f,h}d{\mathbf {x}}}. \end{aligned}$$

The interface term can be rewritten as

$$\begin{aligned} (-p_f{\mathbf {I}}+2\mu D({\mathbf {u}}_f)){\mathbf {n}}_{12}=-p_f{\mathbf {n}}_{12}+(2\mu (D({\mathbf {u}}_f){\mathbf {n}}_{12})\cdot {\mathbf {n}}_{12}){\mathbf {n}}_{12}+\sum _{j=1}^{d-1}(2\mu (D({\mathbf {u}}_f){\mathbf {n}}_{12})\cdot \tau _{12}^j)\tau _{12}^j, \end{aligned}$$

combining the interface conditions (8) and (9), we get

$$\begin{aligned} -\sum _{e \in \Gamma _{12}}\int _{e}{(-p_f{\mathbf {I}}+2\mu D({\mathbf {u}}_f)){\mathbf {n}}_{12}\cdot {\mathbf {v}}_{f,h}ds}=&\sum _{e \in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}\cdot {\mathbf {n}}_{12})ds}\\&+\frac{\mu }{G}\sum _{j=1}^{d-1}\sum _{e \in \Gamma _{12}}\int _{e}{({\mathbf {u}}_f\cdot \tau _{12}^j)({\mathbf {v}}_{f,h}\cdot \tau _{12}^j)ds}. \end{aligned}$$

Thus, we have

$$\begin{aligned} a_f({\mathbf {u}}_f,{\mathbf {v}}_{f,h})+b_f({\mathbf {v}}_{f,h},p_f)+\sum _{e \in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}\cdot {\mathbf {n}}_{12})ds}=({\mathbf {f}},{\mathbf {v}}_{f,h})_{\Omega _1} \quad \forall {\mathbf {v}}_{f,h} \in {\mathbf {X}}_h^f. \end{aligned}$$

(15)

Similarly, we obtain

$$\begin{aligned} -b_f({\mathbf {u}}_f,q_{f,h})&=0&\quad \forall q_{f,h} \in M_h^f, \end{aligned}$$

(16)

$$\begin{aligned} a_s({\mathbf {u}}_s,{\mathbf {v}}_{s,h})+b_s({\mathbf {v}}_{s,h},p_s)-\sum _{e \in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{s,h}\cdot {\mathbf {n}}_{12})ds}&=0&\quad \forall {\mathbf {v}}_{s,h} \in {\mathbf {X}}_h^s, \end{aligned}$$

(17)

$$\begin{aligned} -b_s({\mathbf {u}}_s,q_{s,h})&=(g,q_{s,h})_{\Omega _2}&\quad \forall q_{s,h} \in M_h^s. \end{aligned}$$

(18)

Adding (15)–(16) to (17)–(18), we complete the proof. $\square$

Remark 2.3

Note that, if $k_1=k_2$, the exact solution of the coupled system (1)–(9) satisfies the numerical scheme (11)–(12) without the interface error term appearing in (13).

3 Inequalities and Approximation Operators

Recall the standard trace inequalities [2], there holds on a given element E with diameter $h_E$

$$\begin{aligned} \forall \phi \in H^1(E),\quad \forall e \subset \partial E,\quad \Vert \phi \Vert _{0,e}^2&\le c(h_E^{-1}\Vert \phi \Vert _{0,E}^2+h_E|\phi |_{1,E}^2), \end{aligned}$$

(19)

$$\begin{aligned} \forall \phi \in H^2(E),\quad \forall e \subset \partial E,\quad \Vert \nabla \phi \cdot {\mathbf {n}}_e\Vert _{0,e}^2&\le c(h_E^{-1}\Vert \phi \Vert _{1,E}^2+h_E|\phi |_{2,E}^2), \end{aligned}$$

(20)

$$\begin{aligned} \forall \phi \in {\mathbb {P}}_k(E),\quad \forall e \subset \partial E,\quad \Vert \phi \Vert _{0,e}&\le ch_E^{-1/2}\Vert \phi \Vert _{0,E}, \end{aligned}$$

(21)

$$\begin{aligned} \forall \phi \in {\mathbb {P}}_k(E),\quad \forall e \subset \partial E,\quad \Vert \nabla \phi \cdot {\mathbf {n}}_e\Vert _{0,e}&\le ch_E^{-1/2}|\phi |_{1,E}. \end{aligned}$$

(22)

Also, recall the discrete Korn’s inequality [26]

$$\begin{aligned} \forall {\mathbf {v}}_{f,h} \in {\mathbf {X}}_h^f,\quad |||\nabla {\mathbf {v}}_{f,h}|||_{0,\Omega _1}^2 \le c\left( |||D( {\mathbf {v}}_{f,h})|||_{0,\Omega _1}^2+\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{1}{|e|}\Vert [ {\mathbf {v}}_{f,h}]\Vert _{0,e}^2\right) . \end{aligned}$$

(23)

Let $p\in L^2(\Omega )$, we denote by ${\tilde{p}}$ the $L^2$- projection of p in $M_h$ satisfying

$$\begin{aligned} \forall q_{f,h}\in {\mathbb {P}}_{k_1-1}(E), \quad \int _{E}{q_{f,h}(p-{\tilde{p}})}&=0 \quad \forall E \in \varepsilon _h^1, \end{aligned}$$

(24)

$$\begin{aligned} \forall q_{s,h}\in {\mathbb {P}}_{k_2-1}(E), \quad \int _{E}{q_{s,h}(p-{\tilde{p}})}&=0 \quad \forall E \in \varepsilon _h^2, \end{aligned}$$

(25)

and, if $p|_{\Omega _1}\in H^{k_1}(\Omega _1)$ and $p|_{\Omega _2}\in H^{k_2}(\Omega _2)$, then the following approximation properties hold

$$\begin{aligned} \Vert p-{\tilde{p}}\Vert _{m,E}&\le c h_{E}^{k_1-m}|p|_{k_1,E}, \quad E\in \varepsilon _h^1, \quad m=0,1, \end{aligned}$$

(26)

$$\begin{aligned} \Vert p-{\tilde{p}}\Vert _{m,E}&\le c h_{E}^{k_2-m}|p|_{k_2,E}, \quad E\in \varepsilon _h^2, \quad m=0,1. \end{aligned}$$

(27)

Let $\Pi _h^f : (H^1(\Omega _1))^d\rightarrow {\mathbf {X}}_h^f$ be the quasi-local interpolation [24], and the quasi-local interpolation satisfies for any $E\in \varepsilon _h^1$

$$\begin{aligned}&\forall {\mathbf {v}}_f\in (H^1(\Omega _1))^d, \quad \forall q_f \in {\mathbb {P}}_{k_1-1}(E),\quad \int _{E}{q_f \nabla \cdot (\Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f)d{\mathbf {x}}}=0, \end{aligned}$$

(28)

$$\begin{aligned}&\forall {\mathbf {v}}_f\in (H^1(\Omega _1))^d,\quad \forall e\in \Gamma _h^1, \quad \forall {\mathbf {q}}_f \in ({\mathbb {P}}_{k_1-1}(e))^d,\quad \int _{e}{{\mathbf {q}}_f\cdot [\Pi _h^f{\mathbf {v}}_f]ds}=0, \end{aligned}$$

(29)

$$\begin{aligned}&\forall {\mathbf {v}}_f\in (H_0^1(\Omega _1))^d,\quad \forall e\in \Gamma _1, \quad \forall {\mathbf {q}}_f \in ({\mathbb {P}}_{k_1-1}(e))^d,\quad \int _{e}{{\mathbf {q}}_f\cdot \Pi _h^f{\mathbf {v}}_fds}=0, \end{aligned}$$

(30)

$$\begin{aligned}&|||\Pi _h^f{\mathbf {v}}_f|||_{1,\Omega _1}\le c\Vert {\mathbf {v}}_f\Vert _{1,\Omega _1}. \end{aligned}$$

(31)

For any ${\mathbf {v}}_f\in (H_0^1(\Omega _1))^d$, by (28), (29) and (30) we have

$$\begin{aligned} b_f(\Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f,q_f)=0 \quad \forall q_f \in M_h^f. \end{aligned}$$

(32)

Moreover, the interpolation operator $\Pi _h^f$ satisfies the following approximation property

$$\begin{aligned} |\Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f|_{m,E}\le ch_E^{s-m}|{\mathbf {v}}_f|_{s,\delta (E)} \quad \forall 1 \le s\le k_1+1,\forall {\mathbf {v}}_f\in (H^s(\Omega _1))^d, m=0,1, \end{aligned}$$

(33)

where $\delta (E)$ is a macro-element containing E. Moreover, there exists at least one facet e of every element $E\in \varepsilon _h^1$ such that

$$\begin{aligned} \int _{e}{(\Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f) ds}=0 \quad \forall {\mathbf {v}}_f\in (H^s(\Omega _1))^d. \end{aligned}$$

(34)

Indeed, if $d=2$, when $k_1=1$ and $k_1=2$, (34) holds true for all facets, when $k_1=3$, it holds true for all facets of most practical mesh (see [24]), if $d=3$, when $k_1=1$, (34) holds true for all facets. Specially, for the interpolation operator $\Pi _h^f$, we have the following bounds.

Lemma 3.1

Let $1 \le s\le k_1+1$. For all ${\mathbf {v}}_f\in (H^s(\Omega _1))^d$ and ${\mathbf {v}}_f|_{\Gamma _1}=0$, there holds

$$\begin{aligned} \Vert \Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}&\le c h_1^{s-1}|{\mathbf {v}}_f|_{s,\Omega _1}, \end{aligned}$$

(35)

$$\begin{aligned} \Vert \Pi _h^f{\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}&\le c\Vert {\mathbf {v}}_f\Vert _{1,\Omega _1}. \end{aligned}$$

(36)

Proof

From the approximation property (33) and (34) (see Lemma 3.9 of [24]), we have

$$\begin{aligned} \Vert \Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}\le c|||\nabla (\Pi _h^f{\mathbf {v}}_f-{\mathbf {v}}_f)|||_{0,\Omega _1}\le c h_1^{s-1}|{\mathbf {v}}_f|_{s,\Omega _1}. \end{aligned}$$

Using the fact that $\Vert {\mathbf {v}}_f\Vert _{{\mathbf {X}}^f}\le c\Vert {\mathbf {v}}_f\Vert _{1,\Omega _1}$, for any ${\mathbf {v}}_f\in (H^1(\Omega _1))^d$, the bound (36) follows from (35) with $s=1$ and triangle inequality. $\square$

Let $\Pi _h^s : (H^{\theta }(\Omega _2))^d\cap H(div;\Omega _2)\rightarrow \tilde{{\mathbf {X}}}_h^s$ be the MFE interpolant [10] for any $\theta >0$, where $\tilde{{\mathbf {X}}}_h^s$ satisfies

$$\begin{aligned} {\tilde{\mathbf {X}}}_h^s\equiv H_0(div;\Omega _2)\cap {\mathbf {X}}_h^s, \end{aligned}$$

(37)

indeed, the space $\tilde{{\mathbf {X}}}_h^s$ is $BDM_{k_2}$ [25]. For any ${\mathbf {v}}_s\in (H^{\theta }(\Omega _2))^d\cap H_0(div;\Omega _2)$, it holds

$$\begin{aligned} b_s(\Pi _h^s{\mathbf {v}}_s-{\mathbf {v}}_s,q_s)&=0 \quad \forall q_s \in M_h^s, \end{aligned}$$

(38)

$$\begin{aligned} \int _e((\Pi _h^s{\mathbf {v}}_s-{\mathbf {v}}_s)\cdot {\mathbf {n}}_e){\mathbf {w}}_s\cdot {\mathbf {n}}_e&=0 \quad \forall e \in \Gamma _h^2\cup \Gamma _2\cup \Gamma _{12}, \forall {\mathbf {w}}_s \in {\mathbf {X}}_h^s. \end{aligned}$$

(39)

For any $E\in \varepsilon _h^2$, $\Pi _h^s$ satisfies the approximation properties

$$\begin{aligned} \Vert \Pi _h^s{\mathbf {v}}_s-{\mathbf {v}}_s\Vert _{m,E}&\le c h_E^s|{\mathbf {v}}_s|_{s-m,E} \quad 1\le s\le k_2+1, m=0,1, \end{aligned}$$

(40)

$$\begin{aligned} \Vert \nabla \cdot (\Pi _h^s{\mathbf {v}}_s-{\mathbf {v}}_s)\Vert _{0,E}&\le c h_E^s|\nabla \cdot {\mathbf {v}}_s|_{s,E} \quad 0\le s\le k_2. \end{aligned}$$

(41)

In addition, we have the following result [4, 25]

$$\begin{aligned} \Vert \Pi _h^s{\mathbf {v}}_s\Vert _{H(div;\Omega _2)}\le c (\Vert {\mathbf {v}}_s\Vert _{\theta ,\Omega _2}+\Vert \nabla \cdot {\mathbf {v}}_s\Vert _{0,\Omega _2}). \end{aligned}$$

(42)

Remark 3.1

Note that, the interpolation operator $\Pi _h^s$ holds in any dimension, However, the existence of interpolation operators $\Pi _h^f$, in three dimensions, for $k_1=1$ is presented in [10]. As for other $k_1$, we don’t know whether the interpolation operators $\Pi _h^f$ is exist.

4 Well-Posedness

In this section, we prove the boundedness of bilinear operators $a(\cdot ,\cdot )$ and $b(\cdot ,\cdot )$, K-ellipticity of bilinear operator $a(\cdot ,\cdot )$ and discrete inf-sup condition of bilinear operator $b(\cdot ,\cdot )$. Then, the well-posedness of the numerical scheme (11)–(12) is obtained by using the boundedness, K-ellipticity and discrete inf-sup condition.

The boundedness of bilinear operators $a(\cdot ,\cdot )$ and $b(\cdot ,\cdot )$ are proved in the following Lemma.

Lemma 4.1

There exists a constant c, independent of mesh-sizes $h_1$ and $h_2$ such that

$$\begin{aligned} a({\mathbf {u}}_h,{\mathbf {v}}_h)&\le c\Vert {\mathbf {u}}_h\Vert _{{\mathbf {X}}} \Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}} \quad \forall {\mathbf {u}}_h, {\mathbf {v}}_h \in {\mathbf {X}}_h,\\ b({\mathbf {v}}_h,p_h)&\le c \Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\Vert p_h\Vert _M \quad \forall {\mathbf {v}}_h \in {\mathbf {X}}_h, p_h\in M_h. \end{aligned}$$

Proof

By Cauchy–Schwarz inequality, trace inequalities (21) and (22), the bilinear operators $a_f(\cdot ,\cdot )$, $a_s(\cdot ,\cdot )$ and $b_f(\cdot ,\cdot )$, $b_s(\cdot ,\cdot )$ satisfy

$$\begin{aligned} a_f({\mathbf {u}}_{f,h},{\mathbf {v}}_{f,h})&\le c\Vert {\mathbf {u}}_{f,h}\Vert _{{\mathbf {X}}^f} \Vert {\mathbf {v}}_{f,h}\Vert _{{\mathbf {X}}^f} \quad \forall {\mathbf {u}}_{f,h}, {\mathbf {v}}_{f,h} \in {\mathbf {X}}_h^f,\\ a_s({\mathbf {u}}_{s,h},{\mathbf {v}}_{s,h})&\le c\Vert {\mathbf {u}}_{s,h}\Vert _{{\mathbf {X}}^s} \Vert {\mathbf {v}}_{s,h}\Vert _{{\mathbf {X}}^s} \quad \forall {\mathbf {u}}_{s,h}, {\mathbf {v}}_{s,h} \in {\mathbf {X}}_h^s,\\ b_f({\mathbf {v}}_{f,h},p_{f,h})&\le c \Vert {\mathbf {v}}_{f,h}\Vert _{{\mathbf {X}}^f}\Vert p_{f,h}\Vert _{M^f} \quad \forall {\mathbf {v}}_{f,h} \in {\mathbf {X}}_h^f, p_{f,h}\in M_h^f,\\ b_s({\mathbf {v}}_{s,h},p_{s,h})&\le c \Vert {\mathbf {v}}_{s,h}\Vert _{{\mathbf {X}}^s}\Vert p_{s,h}\Vert _{M^s} \quad \forall {\mathbf {v}}_{s,h} \in {\mathbf {X}}_h^s, p_{s,h}\in M_h^s. \end{aligned}$$

From the relations $a(\cdot ,\cdot )=a_f(\cdot ,\cdot )+a_s(\cdot ,\cdot )$ and $b(\cdot ,\cdot )=b_f(\cdot ,\cdot )+b_s(\cdot ,\cdot )$, immediately, we have the boundedness of bilinear operators $a(\cdot ,\cdot )$ and $b(\cdot ,\cdot )$. $\square$

Next, we present the K-ellipticity of bilinear operator $a(\cdot ,\cdot )$. To prove the K-ellipticity, the following conditions shall be given.

• 1. $\sigma _{1,e}\ge 1$ for all facets in $\Gamma _h^1\cup \Gamma _1$ if $\epsilon =1$, e.g., one may choose $\sigma _{1,e}=2$.

• 2. $\sigma _{1,e}\ge \sigma _0>0$ for $\sigma _0$ large enough if $\epsilon =-1$.

• 3. $\sigma _{2,e}\ge 1$ for all facets in $\Gamma _h^2\cup \Gamma _2$, e.g., one also may choose $\sigma _{2,e}=2$.

Specially, the local lifting operator [9] is introduced and used to prove the K-ellipticity.

Lemma 4.2

The local lifting operator $r_e$: $L^2(e)\rightarrow M_h^s$ is defined by

$$\begin{aligned} \int _{\Omega _2}{r_e(w)q_{s,h}d{\mathbf {x}}}=-\int _{e}{w\{q_{s,h}\}ds} \quad \forall w\in L^2(e), \forall q_{s,h} \in M_h^s. \end{aligned}$$

(43)

Then, for any $e \in \Gamma _h^2\cup \Gamma _2$, the following inequality holds

$$\begin{aligned} \Vert r_e(w)\Vert _{0,\Omega _2}\le c h_e^{-1/2}\Vert w\Vert _{0,e}. \end{aligned}$$

(44)

Proof

By taking $q_{s,h}=r_e(w)$ in (43) and using the trace inequality (21), we have

$$\begin{aligned} \forall e \in \Gamma _h^2, \quad \Vert r_e(w)\Vert _{0,\Omega _2}^2\le \frac{1}{2}\Vert w\Vert _{0,e}(\Vert r_e(w)^+\Vert _{0,e}+\Vert r_e(w)^-\Vert _{0,e})\le c h_e^{-1/2}\Vert w\Vert _{0,e} \Vert r_e(w)\Vert _{0,\Omega _2}, \end{aligned}$$

and

$$\begin{aligned} \forall e \in \Gamma _2, \quad \Vert r_e(w)\Vert _{0,\Omega _2}^2\le \Vert w\Vert _{0,e}\Vert r_e(w)\Vert _{0,e}\le c h_e^{-1/2}\Vert w\Vert _{0,e} \Vert r_e(w)\Vert _{0,\Omega _2}. \end{aligned}$$

$\square$

Note that $r_e(w)$ vanishes outside the union of the elements containing facet e.

Lemma 4.3

There exists a constant $C_{K}>0$, independent of mesh-sizes $h_1$ and $h_2$ such that

$$\begin{aligned} a({\mathbf {u}}_h,{\mathbf {u}}_h)>C_{K}\Vert {\mathbf {u}}_h\Vert _{{\mathbf {X}}}^2 \quad \forall {\mathbf {u}}_h \in {\mathbf {Z}}_h, \end{aligned}$$

(45)

where ${\mathbf {Z}}_h$ is the kernel space ${\mathbf {Z}}_h=\{{\mathbf {v}}_h \in {\mathbf {X}}_h;~~ b({\mathbf {v}}_h,q_h)=0 \quad \forall q_h\in M_h\}$.

Proof

Note that, if $\sigma _{1,e}$ is sufficiently large for $\epsilon =-1$ and if $\sigma _{1,e}=1$ for $\epsilon =1$, by discrete Korn’s inequality (23), we obtain the global coercivity in Stokes domain (see [2])

$$\begin{aligned} a_f({\mathbf {u}}_{f,h},{\mathbf {u}}_{f,h})>C_f\Vert {\mathbf {u}}_{f,h}\Vert _{{\mathbf {X}}^f}^2 \quad \forall {\mathbf {u}}_{f,h} \in {\mathbf {X}}_h^f, \end{aligned}$$

where $C_f$ is independent of mesh-size $h_1$. The proof is a trivial and not presented in this paper. We need to prove the remaining K-ellipticity of $a_s(\cdot ,\cdot )$. In light of the definition of the local lifting operator, for any ${\mathbf {u}}_{s,h}\in {\mathbf {Z}}_h$ and any $q_{s,h}\in M_h^s$, we have

$$\begin{aligned} b_s({\mathbf {u}}_{s,h},q_{s,h})&=-\sum _{E \in \varepsilon _h^2}\int _{E}{\nabla \cdot {\mathbf {u}}_{s,h}q_{s,h}d{\mathbf {x}}}-\int _{\Omega _2}{\sum _{e\in \Gamma _h^2\cup \Gamma _2}r_{e}([{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e])q_{s,h}d{\mathbf {x}}} \\&=-\sum _{E \in \varepsilon _h^2}\int _{E}{(\nabla \cdot {\mathbf {u}}_{s,h}+\sum _{e\subset \partial E\setminus \Gamma _{12}}r_{e}([{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e]))q_{s,h}d{\mathbf {x}}}. \end{aligned}$$

Due to ${\mathbf {u}}_{s,h}\in {\mathbf {Z}}_h$, it satisfies $b_s({\mathbf {u}}_{s,h},q_{s,h})=0$. Choosing

$$\begin{aligned} q_{s,h}=\nabla \cdot {\mathbf {u}}_{s,h}+\sum _{e\subset \partial E\setminus \Gamma _{12}}r_{e}([{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e]) \quad \forall E\in \varepsilon _h^2, \end{aligned}$$

yields

$$\begin{aligned} \nabla \cdot {\mathbf {u}}_{s,h}=-\sum _{e\subset \partial E\setminus \Gamma _{12}}r_{e}([{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e])\quad \forall E\in \varepsilon _h^2. \end{aligned}$$

By (44), we obtain

$$\begin{aligned} \Vert \nabla \cdot {\mathbf {u}}_{s,h}\Vert _{0,\Omega _2}\le c \sum _{e\in \Gamma _h^2\cup \Gamma _2} h_e^{-1/2}\Vert [{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e]\Vert _{0,e}. \end{aligned}$$

Note that $\sigma _{2,e}>1$, therefore

$$\begin{aligned} a_s({\mathbf {u}}_{s,h},{\mathbf {u}}_{s,h})=||| {\mathbf {u}}_{s,h}|||_{0,\Omega _2}^2+\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [{\mathbf {u}}_{s,h}\cdot {\mathbf {n}}_e]\Vert _{0,e}^2\ge C_s\Vert {\mathbf {u}}_{s,h}\Vert _{{\mathbf {X}}^s}^2 \quad \forall {\mathbf {u}}_{s,h} \in {\mathbf {Z}}_h, \end{aligned}$$

where $C_s$ is independent of mesh-size $h_2$. By combining the global coercivity of $a_f(\cdot ,\cdot )$ and the K-ellipticity of $a_s(\cdot ,\cdot )$, we finish the proof of the K-ellipticity by taking $C_{K}=\min (\frac{C_f}{2},\frac{C_s}{2})$. $\square$

Finally, a discrete inf-sup condition shall be derived.

Lemma 4.4

There exists a positive constant $\beta$, independent of mesh-sizes $h_1$ and $h_2$ such that

$$\begin{aligned} \inf _{q_h\in M_h}\sup _{{\mathbf {v}}_h \in {\mathbf {V}}_h}\frac{b({\mathbf {v}}_h,q_h)}{\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\Vert q_h\Vert _M}\ge \beta . \end{aligned}$$

(46)

Proof

To this end, we consider the space $\tilde{{\mathbf {X}}}_h={\mathbf {X}}_h^f\times \tilde{{\mathbf {X}}}_h^s$ and $\tilde{{\mathbf {X}}}_h\subset {\mathbf {X}}_h$, where the space $\tilde{{\mathbf {X}}}_h^s$ is introduced in (37). Define

$$\begin{aligned} \tilde{{\mathbf {V}}}_h=\left\{ ({\mathbf {v}}_f,{\mathbf {v}}_s)\in \tilde{{\mathbf {X}}}_h:\sum _{e\in \Gamma _{12}}\int _{e}{\eta ({\mathbf {v}}_f-{\mathbf {v}}_s)\cdot {\mathbf {n}}_{12}=0}\quad \forall \eta \in \Lambda _h\right\} . \end{aligned}$$

If the following inf-sup condition holds

$$\begin{aligned} \inf _{q_h\in M_h}\sup _{{\mathbf {v}}_h \in {\tilde{\mathbf {V}}}_h}\frac{b({\mathbf {v}}_h,q_h)}{\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\Vert q_h\Vert _M}\ge \beta , \end{aligned}$$

(47)

immediately, we finish the proof of this Lemma. Let $q_h\in M_h$ be fixed, then there exists a ${\mathbf {v}} \in (H^1(\Omega ))^d$ such that

$$\begin{aligned} \nabla \cdot {\mathbf {v}}=-q_h \quad \text {in} \quad \Omega , \qquad {\mathbf {v}}=0\quad \text {on} \quad \partial \Omega , \end{aligned}$$

satisfying

$$\begin{aligned} \Vert {\mathbf {v}}\Vert _{1,\Omega }\le c\Vert q_h\Vert _{0,\Omega }. \end{aligned}$$

Note that, by ${\mathbf {v}} \in (H^1(\Omega ))^d$,

$$\begin{aligned} b({\mathbf {v}},q_h)=-\int _{\Omega }{(\nabla \cdot {\mathbf {v}})q_hd{\mathbf {x}}}=\Vert q_h\Vert _{0,\Omega }^2, \end{aligned}$$

which, combining with the given priori bound, yields

$$\begin{aligned} b({\mathbf {v}},q_h)\ge \frac{1}{c}\Vert q_h\Vert _{0,\Omega }\Vert {\mathbf {v}}\Vert _{1,\Omega }. \end{aligned}$$

The idea of the proof of the inf-sup condition is that we construct a $\pi _h{\mathbf {v}}\in \tilde{{\mathbf {V}}}_h$ such that the inf-sup condition (47) holds. To this end, let $\pi _h : {\mathbf {X}}^f \times ({\mathbf {X}}^s \cap (H^1(\Omega _2))^2)\rightarrow \tilde{{\mathbf {V}}}_h$ satisfying

$$\begin{aligned} b(\pi _h{\mathbf {v}}-{\mathbf {v}},q_h)=0 \quad \forall q_h \in M_h, \quad \text {and} \quad \Vert \pi _h{\mathbf {v}}\Vert _{{\mathbf {X}}}\le c \Vert {\mathbf {v}}\Vert _{1,\Omega }. \end{aligned}$$

(48)

Let $\pi _h{\mathbf {v}}=(\pi _h^f{\mathbf {v}},\pi _h^s{\mathbf {v}})\in {\mathbf {X}}_h^f\times \tilde{{\mathbf {X}}}_h^s$. We take $\pi _h^f{\mathbf {v}}=\Pi _h^f{\mathbf {v}}$ where $\pi _h^f : {\mathbf {X}}^f\rightarrow {\mathbf {X}}_h^f$ is the quasi-local interpolation defined in (28)–(30). Clearly, by (30) and (36) we have

$$\begin{aligned} b_f(\pi _h^f{\mathbf {v}}-{\mathbf {v}},q_h)=0\quad \forall q_h \in M_h, \quad \text {and} \quad \Vert \pi _h^1{\mathbf {v}}\Vert _{{\mathbf {X}}^f}\le c \Vert {\mathbf {v}}\Vert _{1,\Omega _1}. \end{aligned}$$

(49)

To define $\pi _h^s$, we consider the auxiliary problem

$$\begin{aligned} \nabla \cdot \nabla \phi =0 ~~ \text {in} \quad \Omega _2,\\ \nabla \phi \cdot {\mathbf {n}}=0 ~~ \text {on} \quad \Gamma _2,\\ \nabla \phi \cdot {\mathbf {n}}_{12}=(\pi _h^f{\mathbf {v}}-{\mathbf {v}})\cdot {\mathbf {n}}_{12} ~~~\text {on} \quad \Gamma _{12}. \end{aligned}$$

The auxiliary problem is well-defined, since

$$\begin{aligned} \int _{\Gamma _{12}}{(\pi _h^f{\mathbf {v}}-{\mathbf {v}})\cdot {\mathbf {n}}_{12}ds}=0, \end{aligned}$$

due to (34). Let ${\mathbf {z}}=\nabla \phi$. Note that, the piecewise smooth function $\pi _h^f{\mathbf {v}}\cdot {\mathbf {n}}_{12}\in H^{\theta }(\Gamma _{12})$ for any $0<\theta <1/2$. By elliptic regularity [20], we can obtain

$$\begin{aligned} \Vert {\mathbf {z}}\Vert _{\theta ,\Omega _2}\le c\Vert (\pi _h^f{\mathbf {v}}-{\mathbf {v}})\cdot {\mathbf {n}}_{12}\Vert _{\theta -1/2,\Gamma _{12}}\quad 0\le \theta \le 1/2. \end{aligned}$$

(50)

Let ${\mathbf {w}}={\mathbf {v}}+{\mathbf {z}}$, the auxiliary problem implies $\nabla \cdot {\mathbf {w}}= \nabla \cdot {\mathbf {v}}$ in $\Omega _2$ and ${\mathbf {w}}\cdot {\mathbf {n}}_{12}=\pi _h^f{\mathbf {v}}\cdot {\mathbf {n}}_{12}$ on $\Gamma _{12}$. We now define $\pi _h^s{\mathbf {v}}=\Pi _h^s{\mathbf {w}}$, where $\Pi _h^s : (H^{\theta }(\Omega _2))^d\cap H_0(div;\Omega _2)\rightarrow \tilde{{\mathbf {X}}}_h^s$ is the MFE interpolation defined in (38). Employing (38), it holds

$$\begin{aligned} b_s(\pi _h^s{\mathbf {v}},q_{s,h})&=b_s(\Pi _h^s{\mathbf {w}},q_{s,h})=b_s({\mathbf {w}},q_{s,h})\\&=-\int _{\Omega _2}{\nabla \cdot {\mathbf {w}}q_{s,h} d{\mathbf {x}}}=-\int _{\Omega _2}{\nabla \cdot {\mathbf {v}}q_{s,h} d{\mathbf {x}}}=b_s({\mathbf {v}},q_{s,h}) \quad \forall q_{s,h} \in M_h^s, \end{aligned}$$

due to the regularity ${\mathbf {w}} \in H_0(div;\Omega _2)$. Thus, $\pi _h{\mathbf {v}}=(\pi _h^f{\mathbf {v}},\pi _h^s{\mathbf {v}})$ satisfies

$$\begin{aligned} b(\pi _h{\mathbf {v}},q_h)=0 \quad \forall q_h \in M_h. \end{aligned}$$

We can check that $\pi _h{\mathbf {v}}\in \tilde{{\mathbf {V}}}_h$. Indeed, for every $e\in \Gamma _{12}$ and $\eta \in \Lambda _h$, by (39) and the fact that $\Lambda _h={\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}$,

$$\begin{aligned} \int _{e}{\pi _h^s{\mathbf {v}}\cdot {\mathbf {n}}_{12}\eta ds}=\int _{e}{\Pi _h^s{\mathbf {w}}\cdot {\mathbf {n}}_{12}\eta ds}=\int _{e}{{\mathbf {w}}\cdot {\mathbf {n}}_{12}\eta ds}=\int _{e}{\pi _h^f{\mathbf {v}}\cdot {\mathbf {n}}_{12}\eta ds}. \end{aligned}$$

Using (40), (42) and (50), we have

$$\begin{aligned} \Vert \pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s}&=\Vert \Pi _h^s{\mathbf {w}}\Vert _{{\mathbf {X}}^s}\\&\le \Vert \Pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s}+\Vert \Pi _h^s{\mathbf {z}}\Vert _{{\mathbf {X}}^s}\\&\le c(\Vert {\mathbf {v}}\Vert _{1,\Omega }+\Vert {\mathbf {z}}\Vert _{\theta ,\Omega _2})\\&\le c (\Vert {\mathbf {v}}\Vert _{1,\Omega }+\Vert (\pi _h^f{\mathbf {v}}-{\mathbf {v}})\cdot {\mathbf {n}}_{12}\Vert _{0,\Gamma _{12}}). \end{aligned}$$

It remains to bound the last term. For every $e\in \Gamma _{12}$, and facet of $E\in \varepsilon _h^1$, using (19) and (33)

$$\begin{aligned} \Vert (\pi _h^f{\mathbf {v}}-{\mathbf {v}})\cdot {\mathbf {n}}_{12}\Vert _{0,e}\le ch_E^{-1/2}(\Vert \pi _h^f{\mathbf {v}}-{\mathbf {v}}\Vert _{0,E}+h_K\Vert \pi _h^f{\mathbf {v}}-{\mathbf {v}}\Vert _{1,E}) \le ch_K^{1/2}|{\mathbf {v}}|_{1,\delta (E)}. \end{aligned}$$

(51)

Therefore

$$\begin{aligned} \Vert \pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s}\le c \Vert {\mathbf {v}}\Vert _{1,\Omega }, \end{aligned}$$

combining with (49), which proves (48). Now using (48) we have

$$\begin{aligned} \frac{1}{c}\Vert q_h\Vert _M\le \frac{b({\mathbf {v}},q_h)}{\Vert {\mathbf {v}}\Vert _{1,\Omega }}=\frac{b(\pi _h{\mathbf {v}},q_h)}{\Vert {\mathbf {v}}\Vert _{1,\Omega }}\le \frac{b(\pi _h{\mathbf {v}},q_h)}{\frac{1}{c}\Vert \pi _h{\mathbf {v}}\Vert _{{\mathbf {X}}}} \quad \forall q_h\in M_h, \end{aligned}$$

and finish the proof of inf-sup condition (47). $\square$

Now, in light of boundedness, K-ellipticity and the discrete inf-sup condition, we analyze the existence and uniqueness, and stability of solution of discrete scheme (11)–(12). However, the stability is a direct result of saddle point problem [10]. Thus, we only present the existence and uniqueness of solution.

Theorem 4.1

The numerical scheme (11)–(12) has a unique solution.

Proof

Since the scheme (11)–(12) is square and finite-dimensional system, it is equivalent to the uniqueness of homogeneous system. The homogeneous system is obtain by setting ${\mathbf {f}}=0$ and $g=0$. Thus, we have

$$\begin{aligned} b({\mathbf {u}}_h,q_h)=0 \quad \forall q_h\in M_h, \end{aligned}$$

which implies ${\mathbf {u}}_h\in {\mathbf {Z}}_h$. Taking ${\mathbf {v}}_h={\mathbf {u}}_h$ and $q_h=p_h$ in (11) and (12), respectively, we can obtain $a({\mathbf {u}}_h,{\mathbf {u}}_h)=0$. The K-ellipticity (45), immediately, yields ${\mathbf {u}}_h=0$. In light of the discrete inf-sup condition (47), we have $p_h=0$ and finish the proof. $\square$

5 A Priori Error Estimates

In this section, a priori error estimates under proper norms are obtained for both velocity field and pressure field. Before giving the error estimates, an approximation conclusion is obtained in the space ${\mathbf {V}}_h$.

Lemma 5.1

Let ${\mathbf {v}}\in (H^1(\Omega ))^d$ such that ${\mathbf {v}}|_{\Omega _1}\in (H^{k_1+1}(\Omega _1))^d$ and ${\mathbf {v}}|_{\Omega _2}\in (H^{k_2+1}(\Omega _2))^d$, there exists $\tilde{{\mathbf {v}}}_h\in \tilde{{\mathbf {V}}}_h\subset {\mathbf {V}}_h$ such that

$$\begin{aligned}&b({\mathbf {v}}-\tilde{{\mathbf {v}}}_h,q_h)=0 \quad \forall q_h \in M_h, \end{aligned}$$

(52)

$$\begin{aligned}&\forall e \in \Gamma _h^1\cup \Gamma _1, \quad \int _{e}[\tilde{{\mathbf {v}}}_h]\cdot {\mathbf {q}}_h ds=0 \quad \forall {\mathbf {q}}_h \in ({\mathbb {P}}_{k_1}(e))^d, \end{aligned}$$

(53)

$$\begin{aligned}&\Vert {\mathbf {v}}-\tilde{{\mathbf {v}}}_h\Vert _{{\mathbf {X}}}\le c(h_1^{k_1}|{\mathbf {v}}|_{k_1+1,\Omega _1}+h_2^{k_2+1}|{\mathbf {v}}|_{k_2+1,\Omega _2}+h_2^{k_2}|\nabla \cdot {\mathbf {v}}|_{k_2,\Omega _2}). \end{aligned}$$

(54)

Proof

Let $\tilde{{\mathbf {v}}}_h=\pi _h{\mathbf {v}}$, by the construction of $\pi _h{\mathbf {v}}$ in Lemma 4.4, we can easily obtain (52) and (53). To show approximation (54), we first note that (35) implies that

$$\begin{aligned} \Vert {\mathbf {v}}-\pi _h^f{\mathbf {v}}\Vert _{{\mathbf {X}}^f}\le c h_1^{k_1}|{\mathbf {v}}|_{k_1+1,\Omega _1}. \end{aligned}$$

(55)

Next,

$$\begin{aligned} \Vert {\mathbf {v}}-\pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s}=\Vert {\mathbf {v}}-\Pi _h^s{\mathbf {w}}\Vert _{{\mathbf {X}}^s}\le \Vert {\mathbf {v}}-\Pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s} +\Vert \Pi _h^s({\mathbf {w}}-{\mathbf {v}})\Vert _{{\mathbf {X}}^s}. \end{aligned}$$

(56)

Using (40) and (41), there holds

$$\begin{aligned} \Vert {\mathbf {v}}-\Pi _h^s{\mathbf {v}}\Vert _{{\mathbf {X}}^s}\le c h_2^{k_2+1}|{\mathbf {v}}|_{k_2+1,\Omega _2}+h_2^{k_2}|\nabla \cdot {\mathbf {v}}|_{k_2,\Omega _2}. \end{aligned}$$

(57)

The last term in (56) can be bounded by using (42), (50), (19) and (33)

$$\begin{aligned} \Vert \Pi _h^s({\mathbf {w}}-{\mathbf {v}})\Vert _{{\mathbf {X}}^s}&=\Vert \Pi _h^s{\mathbf {z}}\Vert _{{\mathbf {X}}^s}\le \Vert {\mathbf {z}}\Vert _{\theta ,\Omega _2}\nonumber \\&\le c\Vert ({\mathbf {v}}-\pi _h^f{\mathbf {v}})\cdot {\mathbf {n}}_{12}\Vert _{0,\Gamma _{12}}\le c h_1^{k_1+1/2}|{\mathbf {v}}|_{k_1+1,\Omega _1}. \end{aligned}$$

(58)

Combing (55)–(58), we finish the proof. $\square$

Theorem 5.1

Let $({\mathbf {u}},p)$ be the solution of the coupled Stokes and Darcy problem (1)–(9). Assume that ${\mathbf {u}}|_{\Omega _i}\in (H^{k_i+1}(\Omega _i))^d$, $p|_{\Omega _i}\in H^{k_i}(\Omega _i)$ for $i=1, 2$. Let $({\mathbf {u}}_h,p_h)$ be the numerical solution of discrete scheme (11)–(12). Then, we have the following estimate

$$\begin{aligned} \Vert {\mathbf {u}}-{\mathbf {u}}_h\Vert _{{\mathbf {X}}}\le&ch_1^{k_1}(|{\mathbf {u}}|_{k_1+1,\Omega _1}+|p|_{k_1,\Omega _1})\\&+ch_2^{k_2}(|{\mathbf {u}}|_{k_2+1,\Omega _2}+|p|_{k_2,\Omega _2}) +ch_2^{k_2-1/2}h_1^{1/2}|p|_{k_2,\Omega _2}. \end{aligned}$$

Proof

Subtracting (13)–(14) from (11)–(12), the error equations are

$$\begin{aligned} a({\mathbf {u}}_h-{\mathbf {u}},{\mathbf {v}}_h)+b({\mathbf {v}}_h,p_h-p)-\sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}ds}=0 \quad \forall {\mathbf {v}}_h \in {\mathbf {V}}_h, \end{aligned}$$

(59)

$$\begin{aligned} b({\mathbf {u}}_h-{\mathbf {u}},q_h)=0 \quad \forall q_h \in M_h. \end{aligned}$$

(60)

Let $\tilde{{\mathbf {u}}}_h$ be the interpolation of ${\mathbf {u}}$ defined in Lemma 5.1 and let ${\tilde{p}}_h$ be the $L^2$- projection of p, satisfying (24) and (25), we then introduce the following notions

$$\begin{aligned} \chi&={\mathbf {u}}_h-\tilde{{\mathbf {u}}}_h,\quad&\theta ={\mathbf {u}}-\tilde{{\mathbf {u}}}_h&, \\ \xi&=p_h-{\tilde{p}}_h, \quad&\zeta =p-{\tilde{p}}_h&. \end{aligned}$$

Based on the above notions, the error Eqs. (59)–(60) can be rewritten as

$$\begin{aligned} a(\chi ,{\mathbf {v}}_h)+b({\mathbf {v}}_h,\xi )&=a(\theta ,{\mathbf {v}}_h)+b({\mathbf {v}}_h,\zeta )+\sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}ds} \quad \forall {\mathbf {v}}_h \in {\mathbf {V}}_h, \end{aligned}$$

(61)

$$\begin{aligned} b(\chi ,q_h)&=b(\theta ,q_h) \quad \forall q_h \in M_h. \end{aligned}$$

(62)

By (52) in Lemma 5.1, we have $b(\theta ,q_h)=0$, thus

$$\begin{aligned} b(\chi ,q_h)=0 \quad \forall q_h \in M_h, \end{aligned}$$

which implies $\chi \in {\mathbf {Z}}_h$. Choosing ${\mathbf {v}}_h=\chi$ and $q_h=\xi$ in (61) and (62) yields

$$\begin{aligned} a(\chi ,\chi )+b(\chi ,\xi )&=a(\theta ,\chi )+b(\chi ,\zeta )+\sum _{e\in \Gamma _{12}}\int _{e}{p_s(\chi _f-\chi _s)\cdot {\mathbf {n}}_{12}ds} ,\\ b(\chi ,\xi )&=0. \end{aligned}$$

equivalently,

$$\begin{aligned} a(\chi ,\chi )=a(\theta ,\chi )+b(\chi ,\zeta )+\sum _{e\in \Gamma _{12}}\int _{e}{p_s(\chi _f-\chi _s)\cdot {\mathbf {n}}_{12}ds}. \end{aligned}$$

(63)

By $\chi \in {\mathbf {Z}}_h$, the K-ellipticity (45) yields $a(\chi ,\chi )\ge C_{K} \Vert \chi \Vert _{{\mathbf {X}}}^2$. We only to bound the right hand sides of (63). The first term can be bounded as follows:

$$\begin{aligned} a_f(\theta ,\chi )&=2\mu \sum _{E\in \varepsilon _h^1}\int _E{{\mathbf {D}}(\theta ):{\mathbf {D}}(\chi )d{\mathbf {x}}}\\&\quad +\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\frac{\sigma _{1,e}}{|e|}[\theta ]\cdot [\chi ]ds} -2 \mu \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}(\theta ) {\mathbf {n}}_e\}\cdot [\chi ]ds}\\&\quad +2 \mu \epsilon \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}(\chi ) {\mathbf {n}}_e\}\cdot [\theta ]ds} +\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\int _{e}{\frac{\mu }{G}\theta \cdot \tau _{12}^j\chi \cdot \tau _{12}^jds}\\&=T_1+T_2+T_3+T_4+T_5. \end{aligned}$$

Using Cauchy–Schwarz inequality, Young inequality and the approximation property (33),

$$\begin{aligned} T_1&\le 2\mu \sum _{E\in \varepsilon _h^1}\Vert \nabla \theta \Vert _{0,E}\Vert \nabla \chi \Vert _{0,E}\le \frac{C_{K}}{8}|||\nabla \chi |||_{0,\Omega _1}^2+c|||\nabla \theta |||_{0,\Omega _1}^2\\&\le \frac{C_{K}}{8}|||\nabla \chi |||_{0,\Omega _1}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2. \end{aligned}$$

By Cauchy–Schwarz inequality, Young inequality, trace inequality (19) and the approximation property (33)

$$\begin{aligned} T_2&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2 +c\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\theta ]\Vert _{0,e}^2\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2. \end{aligned}$$

Let $L_h({\mathbf {u}})$, defined in $\Omega _1$, stand for the classic Lagrange interpolation of degree $k_1$, and note that $L_h({\mathbf {u}})$ satisfies the optimal approximation, for any $E\in \varepsilon _h^1$

$$\begin{aligned} |L_h({\mathbf {u}})-{\mathbf {u}}|_{m,E}\le c h_1^{s-m}|{\mathbf {u}}|_{s,E}\quad \forall 2\le s\le k_1+1,\quad m=0,1,2. \end{aligned}$$

(64)

For a fixed $e\in \Gamma _h^1\cup \Gamma _1$, using the Lagrange interpolation in $T_3$, we have

$$\begin{aligned} \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}(\theta ) {\mathbf {n}}_e\}\cdot [\chi ]ds}=&\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}({\mathbf {u}}-L_h({\mathbf {u}})) {\mathbf {n}}_e\}\cdot [\chi ]ds}\\&+\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}(L_h({\mathbf {u}})-\tilde{{\mathbf {u}}}_h) {\mathbf {n}}_e\}\cdot [\chi ]ds}. \end{aligned}$$

The first part can be bounded by using trace inequality (20) and the approximation property of the Lagrange interpolation (64)

$$\begin{aligned} \sum _{e\in \Gamma _h^1\cup \Gamma _1}&\int _e{\{{\mathbf {D}}({\mathbf {u}}-L_h({\mathbf {u}})) {\mathbf {n}}_e\}\cdot [\chi ]ds}\\&\le \sum _{e\in \Gamma _h^1\cup \Gamma _1} \frac{\sigma _{1,e}^{1/2}}{|e|^{1/2}}\Vert [\chi ]\Vert _{0,e}\frac{|e|^{1/2}}{\sigma _{1,e}^{1/2}}\Vert \{{\mathbf {D}}({\mathbf {u}}-L_h({\mathbf {u}})) {\mathbf {n}}_e\}\Vert _{0,e}\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2 +c\frac{|e|}{\sigma _{1,e}}(h_e^{-1}|L_h({\mathbf {u}})-\tilde{{\mathbf {u}}}_h|_{1,E_e^{12}}^2+h_e|L_h({\mathbf {u}})-\tilde{{\mathbf {u}}}_h|_{2,E_e^{12}}^2) \\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2, \end{aligned}$$

where $E_e^{12}$ represents the union of $E_e^{1}$ and $E_e^{2}$ $(e=E_e^{1}\cap E_e^{2})$. Similarly, by the trace inequality (21), triangle inequality, and the approximation (33), we have

$$\begin{aligned} \sum _{e\in \Gamma _h^1\cup \Gamma _1}&\int _e{\{{\mathbf {D}}(L_h({\mathbf {u}})-\tilde{{\mathbf {u}}}_h) {\mathbf {n}}_e\}\cdot [\chi ]ds}\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2 +c\sum _{e\in \Gamma _h^1\cup \Gamma _1}|L_h({\mathbf {u}})-\tilde{{\mathbf {u}}}_h|_{1,E_e^{12}}^2\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2. \end{aligned}$$

Therefore,

$$\begin{aligned} T_3\le \frac{C_{K}}{4}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2. \end{aligned}$$

The fourth term vanishes due to the continuity of ${\mathbf {u}}$ and the property (53) of $\tilde{{\mathbf {u}}}_h$,

$$\begin{aligned} T_4=0. \end{aligned}$$

The last term can be estimated by using the trace inequality (19),

$$\begin{aligned} T_5&\le \frac{\mu }{G}\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\Vert \theta \Vert _{0,e}\Vert \chi \cdot \tau _{12}^j\Vert _{0,e}\\&\le \frac{\mu }{2G}\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\Vert \chi \cdot \tau _{12}^j\Vert _{0,e}^2+C\sum _{e\in \Gamma _{12}}(h_e^{-1}\Vert \theta \Vert _{0,E}+h_e|\theta |_{1,E})\\&\le \frac{\mu }{2G}\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\Vert \chi \cdot \tau _{12}^j\Vert _{0,e}^2+ch_1^{2k_1+1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2. \end{aligned}$$

Let us now estimate $a_s(\theta ,\chi )$,

$$\begin{aligned} a_s(\theta ,\chi )=\sum _{E\in \varepsilon _h^2}\int _E{{\mathbf {K}}^{-1}\theta \cdot \chi d{\mathbf {x}}} +\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{\frac{\sigma _{2,e}}{|e|}[\theta \cdot {\mathbf {n}}_e][\chi \cdot {\mathbf {n}}_e]ds}. \end{aligned}$$

Using the Cauchy–Schwarz inequality, Young inequality and the approximation property (40) to the first part, we have

$$\begin{aligned} \sum _{E\in \varepsilon _h^2}\int _E{{\mathbf {K}}^{-1}\theta \cdot \chi d{\mathbf {x}}}\le \frac{C_{K}}{8}\Vert \chi \Vert _{0,\Omega _2}^2+ch_2^{2k_2+2}|{\mathbf {u}}|_{k_2+1,\Omega _2}^2. \end{aligned}$$

The second part is bounded by using trace inequality (19) and the approximation (40)

$$\begin{aligned} \sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{\frac{\sigma _{2,e}}{|e|}[\theta \cdot {\mathbf {n}}_e][\chi \cdot {\mathbf {n}}_e]ds}&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [\chi \cdot {\mathbf {n}}_e]\Vert _{0,e}^2 +c\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [\theta \cdot {\mathbf {n}}_e]\Vert _{0,e}^2\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [\chi \cdot {\mathbf {n}}_e]\Vert _{0,e}^2+ch_2^{2k_2}|{\mathbf {u}}|_{k_2+1,\Omega _2}^2. \end{aligned}$$

Next, we estimate $b_f(\chi ,\zeta )$, by the trace inequality (19), and properties (24) and (26),

$$\begin{aligned} b_f(\chi ,\zeta )&=-\sum _{E\in \varepsilon _h^1}\int _{E}{\zeta \nabla \cdot \chi d{\mathbf {x}}}+\sum _{e\in \Gamma _1\cup \Gamma _h^1}\int _e{[\chi \cdot {\mathbf {n}}_e]\{\zeta \}ds}\\&=\sum _{e\in \Gamma _1\cup \Gamma _h^1}\int _e{[\chi \cdot {\mathbf {n}}_e]\{\zeta \}ds}\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2+ch_1^{2k_1}|p|_{k_1,\Omega _1}^2. \end{aligned}$$

Similarly, by the trace inequality (19), and properties (25) and (27),

$$\begin{aligned} b_s(\chi ,\zeta )&=-\sum _{E\in \varepsilon _h^2}\int _{E}{\zeta \nabla \cdot \chi d{\mathbf {x}}}+\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{[\chi \cdot {\mathbf {n}}_e]\{\zeta \}ds}\\&=\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{[\chi \cdot {\mathbf {n}}_e]\{\zeta \}ds}\\&\le \frac{C_{K}}{8}\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [\chi \cdot {\mathbf {n}}_e]\Vert _{0,e}^2+ch_2^{2k_2}|p|_{k_2,\Omega _2}^2. \end{aligned}$$

It remains to estimate the last term in (63). Since $\chi$ belongs to ${\mathbf {V}}_h$, we obtain

$$\begin{aligned} \sum _{e\in \Gamma _{12}}\int _{e}{p_s(\chi _f-\chi _s)\cdot {\mathbf {n}}_{12}ds}=\sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)(\chi _f-\chi _s)\cdot {\mathbf {n}}_{12}ds}, \end{aligned}$$

where ${\tilde{p}}_h^s\in \Lambda _h$ is the $L^2$- projection of $p_s$ with respect to $L^2$ inner product on the interface. Thus, from the definition of the Lagrange multiplier space $\Lambda _h$, we have

$$\begin{aligned} \sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)\chi _s\cdot {\mathbf {n}}_{12}ds}=0. \end{aligned}$$

For any interface facet e and any piecewise vector-valued constant ${\mathbf {c}}_e$, there holds

$$\begin{aligned} \sum _{e\in \Gamma _{12}}&\int _{e}{p_s(\chi _f-\chi _s)\cdot {\mathbf {n}}_{12}ds}=\sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)\chi _f\cdot {\mathbf {n}}_{12}ds}\\ =&\sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)(\chi _f-{\mathbf {c}}_e)\cdot {\mathbf {n}}_{12}ds}= \sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)(\chi _f-{\mathbf {c}}_e)\cdot {\mathbf {n}}_{12}ds}. \end{aligned}$$

Assume that each interface facet e is shared by the element $E_e^2\in \varepsilon _h^2$ and parts of the elements $E_{e,i}^1 \in \varepsilon _h^1$, $i=1,k_e$. Then, by the trace inequality (19) and approximation property of $L^2$- projection, we have (see [4])

$$\begin{aligned} \int _{e}{(p_s-{\tilde{p}}_h^s)(\chi _f-{\mathbf {c}}_e)\cdot {\mathbf {n}}_{12}ds}\le ch_2^{k_2-1/2}|p|_{k_2,E_e^2}\sum _{i=1}^{k_e}(h_1^{-1/2}\Vert \chi _f-{\mathbf {c}}_e\Vert _{0,E_{e,i}^1}+h_1^{1/2}\Vert \nabla \chi _f\Vert _{0,E_{e,i}^1}), \end{aligned}$$

therefore

$$\begin{aligned} \sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s)(\chi _f-{\mathbf {c}}_e)\cdot {\mathbf {n}}_{12}ds}&\le c\sum _{e\in \Gamma _{12}}\left( h_2^{k_2-1/2}|p|_{k_2,E_e^2}\sum _{i=1}^{k_e}h_1^{1/2}\Vert \nabla \chi _f\Vert _{0,E_{e,i}^1}\right) \\&\le \frac{C_{K}}{8}|||\nabla \chi |||_{0,\Omega _1}^2+ch_2^{2k_2-1}h_1|p|_{k_2,\Omega _2}^2. \end{aligned}$$

Indeed, we can estimate the interface term by using the discrete Poincar$\mathrm {\acute{e}}$ inequality [2] and not introduce the piecewise vector-valued constant ${\mathbf {c}}_e$ if p has sufficient smoothness ($p|_{\Omega _2}\in H^{k_2+1}(\Omega _2)$). Then, based on the above estimates we obtain

$$\begin{aligned} a(\chi ,\chi )\le&\frac{C_{K}}{4}|||\nabla \chi |||_{0,\Omega _1}^2+\frac{C_{K}}{2}\sum _{e\in \Gamma _h^1\cup \Gamma _1}\frac{\sigma _{1,e}}{|e|}\Vert [\chi ]\Vert _{0,e}^2 +\frac{\mu }{2G}\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\Vert \chi \cdot \tau _{12}^j\Vert _{0,e}^2\\&+\frac{C_{K}}{4}|||\chi |||_{0,\Omega _2}^2+\frac{C_{K}}{4}\sum _{e\in \Gamma _h^2\cup \Gamma _2}\frac{\sigma _{2,e}}{|e|}\Vert [\chi \cdot {\mathbf {n}}_e]\Vert _{0,e}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2 \\&+ch_2^{2k_2}|{\mathbf {u}}|_{k_2+1,\Omega _2}^2+ch_1^{2k_1}|p|_{k_1,\Omega _1}^2+ch_2^{2k_2}|p|_{k_2,\Omega _2}^2+ch_2^{2k_2-1}h_1|p|_{k_2,\Omega _2}^2. \end{aligned}$$

Combing the K-ellipticity, we have

$$\begin{aligned} \Vert \chi \Vert _{{\mathbf {X}}}^2&\le ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2 +ch_2^{2k_2}|{\mathbf {u}}|_{k_2+1,\Omega _2}^2\\&\quad +ch_1^{2k_1}|p|_{k_1,\Omega _1}^2+ch_2^{2k_2}|p|_{k_2,\Omega _2}^2+ch_2^{2k_2-1}h_1|p|_{k_2,\Omega _2}^2, \end{aligned}$$

which complete the proof by using (54) and the triangle inequality. $\square$

Theorem 5.2

Under the same assumptions and notions of Theorem 5.1, we obtain

$$\begin{aligned} \Vert p_h-p\Vert _{0,\Omega }&\le ch_1^{k_1}(|{\mathbf {u}}|_{k_1+1,\Omega _1}+|p|_{k_1,\Omega _1})\\&\quad +ch_2^{k_2}(|{\mathbf {u}}|_{k_2+1,\Omega _2}+|p|_{k_2,\Omega _2}) +ch_2^{k_2-1/2}h_1^{1/2}|p|_{k_2,\Omega _2}. \end{aligned}$$

Proof

The error equation (59) can be written as

$$\begin{aligned} a({\mathbf {u}}_h-u,{\mathbf {v}}_h)+b({\mathbf {v}}_h,p_h-{\tilde{p}}_h) =b({\mathbf {v}}_h,p-{\tilde{p}}_h)+\sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}ds} \quad \forall {\mathbf {v}}_h \in {\mathbf {V}}_h. \end{aligned}$$

(65)

From the discrete inf-sup condition (46),

$$\begin{aligned} \Vert p_h-{\tilde{p}}_h\Vert _{0,\Omega }\le \frac{1}{\beta } \sup _{{\mathbf {v}}_h \in {\mathbf {V}}_h}\frac{b({\mathbf {v}}_h,p_h-{\tilde{p}}_h)}{\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}}. \end{aligned}$$

(66)

For any ${\mathbf {v}}_h \in {\mathbf {V}}_h$, we assume that $p_h-{\tilde{p}}_h$ and ${\mathbf {v}}_h$ satisfy (66). From (65), it holds

$$\begin{aligned} b({\mathbf {v}}_h,p_h-{\tilde{p}}_h)= -a({\mathbf {u}}_h-{\mathbf {u}},{\mathbf {v}}_h)+b({\mathbf {v}}_h,p-{\tilde{p}}_h)+\sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}ds} . \end{aligned}$$

(67)

To bound the term $a({\mathbf {u}}_h-{\mathbf {u}},{\mathbf {v}}_h)$ in (67),

$$\begin{aligned}&a({\mathbf {u}}_h-{\mathbf {u}},{\mathbf {v}}_h)= 2\mu \sum _{E\in \varepsilon _h^1}\int _E{{\mathbf {D}}({\mathbf {u}}_h-{\mathbf {u}}):{\mathbf {D}}({\mathbf {v}}_h)d{\mathbf {x}}} +\sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\frac{\sigma _{1,e}}{|e|}[{\mathbf {u}}_h-{\mathbf {u}}]\cdot [{\mathbf {v}}_h]ds}\\&\qquad -2 \mu \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}({\mathbf {u}}_h-{\mathbf {u}}) {\mathbf {n}}_e\}\cdot [{\mathbf {v}}_h]ds}+2 \mu \epsilon \sum _{e\in \Gamma _h^1\cup \Gamma _1}\int _e{\{{\mathbf {D}}({\mathbf {v}}_h) {\mathbf {n}}_e\}\cdot [{\mathbf {u}}_h-{\mathbf {u}}]ds}\\&\qquad +\sum _{j=1}^{d-1}\sum _{e\in \Gamma _{12}}\int _{e}{\frac{\mu }{G}({\mathbf {u}}_h-{\mathbf {u}})\cdot \tau _{12}^j{\mathbf {v}}_h\cdot \tau _{12}^jds} +\sum _{E\in \varepsilon _h^2}\int _E{({\mathbf {u}}_h-{\mathbf {u}})\cdot {\mathbf {v}}_hd{\mathbf {x}}}\\&\qquad +\sum _{e\in \Gamma _h^2\cup \Gamma _2}\int _e{\frac{\sigma _{2,e}}{|e|}[({\mathbf {u}}_h-{\mathbf {u}})\cdot {\mathbf {n}}_e][{\mathbf {v}}_h\cdot {\mathbf {n}}_e]ds}\\&\quad =Q_1+Q_2+Q_3+Q_4+Q_5+Q_6+Q_7. \end{aligned}$$

We now estimate each $Q_i$ terms for $i=1,7$. The terms $Q_1$, $Q_2$, $Q_5$, $Q_6$ and $Q_7$ are bounded by Cauchy-Schwarz inequality,

$$\begin{aligned} Q_1+ Q_2+ Q_5+ Q_6+ Q_7\le c\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\Vert {\mathbf {u}}_h-{\mathbf {u}}\Vert _{{\mathbf {X}}}. \end{aligned}$$

$Q_3$ is estimated by utilizing the Lagrange interpolation

$$\begin{aligned} Q_3&\le c \sum _{e\in \Gamma _{h}^1\cup \Gamma _1}\left( \frac{|e|}{\sigma _{1,e}}\right) ^{1/2}\Vert \nabla ({\mathbf {u}}_h-{\mathbf {u}})\Vert _{0,e}\left( \frac{\sigma _{1,e}}{|e|}\right) ^{1/2}\Vert [{\mathbf {v}}_h]\Vert _{0,e}\\&\le c \Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\left( \sum _{e\in \Gamma _{h}^1\cup \Gamma _1}(h_1\Vert \nabla ({\mathbf {u}}_h-\tilde{{\mathbf {u}}}_h)\Vert _{0,e}^2+h_1\Vert \nabla ({\mathbf {u}}-\tilde{{\mathbf {u}}}_h)\Vert _{0,e}^2)\right) ^{1/2}\\&\le c\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}(\Vert {\mathbf {u}}_h-\tilde{{\mathbf {u}}}_h\Vert _{{\mathbf {X}}}^2+ch_1^{2k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}^2)^{1/2}. \end{aligned}$$

By using the trace inequality (21), there holds

$$\begin{aligned} Q_4&\le c\sum _{e\in \Gamma _{h}^1\cup \Gamma _1}\Vert \{D({\mathbf {v}}_h){\mathbf {n}}_e\}\Vert _{0,e}\Vert [{\mathbf {u}}_h-{\mathbf {u}}]\Vert _{0,e}\\&\le c \sum _{e\in \Gamma _{h}^1\cup \Gamma _1}h_1^{-1/2}\Vert \nabla {\mathbf {v}}_h\Vert _{0,E_e^{12}}\left( \frac{\sigma _{1,e}}{|e|}\right) ^{1/2-1/2}\Vert [{\mathbf {u}}_h-{\mathbf {u}}]\Vert _{0,e}\\&\le c\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}\Vert {\mathbf {u}}_h-{\mathbf {u}}\Vert _{{\mathbf {X}}}. \end{aligned}$$

For the term $b({\mathbf {v}}_h,p-{\tilde{p}}_h)$ in (67), by the properties (24) and (26), we have

$$\begin{aligned} b_f({\mathbf {v}}_h,p-{\tilde{p}}_h)&=\sum _{e\in \Gamma _{h}^1\cup \Gamma _1}\int _{e}{\{p-{\tilde{p}}_h\}[{\mathbf {v}}_h\cdot {\mathbf {n}}_e]ds}\\&\le \sum _{e\in \Gamma _{h}^1\cup \Gamma _1}\left( \frac{\sigma _{1,e}}{|e|}\right) ^{1/2}\Vert [{\mathbf {v}}_h]\Vert _{0,e}\left( \frac{|e|}{\sigma _{1,e}}\right) ^{1/2}\Vert \{p-{\tilde{p}}_h\}\Vert _{0,e}\\&\le c h_1^{k_1}|p|_{k_1,\Omega _1} \Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}. \end{aligned}$$

Similarly,

$$\begin{aligned} b_s({\mathbf {v}}_h,p-{\tilde{p}}_h)&=\sum _{e\in \Gamma _{h}^2\cup \Gamma _2}\int _{e}{\{p-{\tilde{p}}_h\}[{\mathbf {v}}_h\cdot {\mathbf {n}}_e]ds}\\&\le \sum _{e\in \Gamma _{h}^2\cup \Gamma _2}\left( \frac{\sigma _{2,e}}{|e|}\right) ^{1/2}\Vert [{\mathbf {v}}_h\cdot {\mathbf {n}}_e]\Vert _{0,e}\left( \frac{|e|}{\sigma _{2,e}}\right) ^{1/2}\Vert \{p-{\tilde{p}}_h\}\Vert _{0,e}\\&\le c h_2^{k_2}|p|_{k_2,\Omega _2}\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}. \end{aligned}$$

Thus,

$$\begin{aligned} b({\mathbf {v}}_h,p-{\tilde{p}}_h)\le c \Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}(h_1^{k_1}|p|_{k_1,\Omega _1}+h_2^{k_2}|p|_{k_2,\Omega _2}). \end{aligned}$$

Similar to the proof in Theorem 5.1, the last interface integral term in (67) is bounded by

$$\begin{aligned} \sum _{e\in \Gamma _{12}}\int _{e}{p_s({\mathbf {v}}_{f,h}-{\mathbf {v}}_{s,h})\cdot {\mathbf {n}}_{12}ds}&=\sum _{e\in \Gamma _{12}}\int _{e}{(p_s-{\tilde{p}}_h^s){\mathbf {v}}_{f,h}\cdot {\mathbf {n}}_{12}ds}\\&\le c h_2^{k_2-1/2}h_1^{1/2}|p|_{k_2,\Omega _2}\Vert {\mathbf {v}}_h\Vert _{{\mathbf {X}}}. \end{aligned}$$

Combing the above bounds and the discrete inf-sup condition (66), we have

$$\begin{aligned} \Vert p_h-{\tilde{p}}\Vert _{0,\Omega }&\le c(\Vert {\mathbf {u}}_h-{\mathbf {u}}\Vert _{{\mathbf {X}}}+\Vert {\mathbf {u}}_h-\tilde{{\mathbf {u}}}_h\Vert _{{\mathbf {X}}}+h_1^{k_1}|{\mathbf {u}}|_{k_1+1,\Omega _1}\\&+h_1^{k_1}|p|_{k_1,\Omega _1}+h_2^{k_2}|p|_{k_2,\Omega _2}+h_2^{k_2-1/2}h_1^{1/2}|p|_{k_2,\Omega _2}). \end{aligned}$$

In light of Theorem 5.1 and triangle inequality, we complete the proof. $\square$

6 Implementation and Numerical Experiments

6.1 Implementation

In this section, an equivalent discrete scheme (see [19]) is given because it is hard to directly construct the space of function ${\mathbf {V}}_h$. Defining the following bilinear forms

$$\begin{aligned} \Lambda _f(\eta ,{\mathbf {v}}_{f,h})=\sum _{e \in \Gamma _{12}}\int _{e}{\eta {\mathbf {v}}_{s,h}\cdot {\mathbf {n}}_{12}ds}\quad \forall \eta \in \Lambda _h, \forall {\mathbf {v}}_{f,h}\in {\mathbf {X}}_h^f,\\ \Lambda _s(\eta ,{\mathbf {v}}_{s,h})=\sum _{e \in \Gamma _{12}}\int _{e}{\eta {\mathbf {v}}_{s,h}\cdot {\mathbf {n}}_{12}ds}\quad \forall \eta \in \Lambda _h, \forall {\mathbf {v}}_{s,h}\in {\mathbf {X}}_h^s. \end{aligned}$$

The numerical scheme (11)-(12) can be rewritten as: Find $({\mathbf {u}}_h,p_h,\lambda _h)\in {\mathbf {X}}_h\times M_h\times \Lambda _h$ such that ${\mathbf {u}}_{f,h}={\mathbf {u}}_h|_{\Omega _1}$, ${\mathbf {u}}_{s,h}={\mathbf {u}}_h|_{\Omega _2}$ and $p_{f,h}=p_h|_{\Omega _1}$, $p_{s,h}=p_h|_{\Omega _2}$ satisfy

$$\begin{aligned} a_f({\mathbf {u}}_{f,h},{\mathbf {v}}_{f,h})+b_f({\mathbf {v}}_{f,h},p_{f,h})+\Lambda _f(\lambda _h,{\mathbf {v}}_{f,h})&=({\mathbf {f}},{\mathbf {v}}_{f,h})_{\Omega _1}\quad&\forall {\mathbf {v}}_{f,h} \in {\mathbf {X}}_h^f, \end{aligned}$$

(68)

$$\begin{aligned} -b_f({\mathbf {u}}_{f,h},q_{f,h})&=0 \quad&\forall q_{f,h} \in M_h^f, \end{aligned}$$

(69)

$$\begin{aligned} a_s({\mathbf {u}}_{s,h},{\mathbf {v}}_{s,h})+b_f({\mathbf {v}}_{s,h},p_{s,h})-\Lambda _s(\lambda _h,{\mathbf {v}}_{s,h})&=0\quad&\forall {\mathbf {v}}_{s,h} \in {\mathbf {X}}_h^s, \end{aligned}$$

(70)

$$\begin{aligned} -b_s({\mathbf {u}}_{s,h},q_{s,h})&=(g,q_{s,h})_{\Omega _2} \quad&\forall q_{s,h} \in M_h^s, \end{aligned}$$

(71)

$$\begin{aligned} \Lambda _f(\eta _h,{\mathbf {u}}_{f,h})-\Lambda _s(\eta _h,{\mathbf {u}}_{s,h})&=0 \quad&\forall \eta _h \in \Lambda _h, \end{aligned}$$

(72)

$$\begin{aligned} \int _{\Omega _1}{p_{f,h}d{\mathbf {x}}}+\int _{\Omega _2}{p_{s,h}d{\mathbf {x}}}&=0. \end{aligned}$$

(73)

We can easily verify that the numerical schemes (11)-(12) and (68)-(73) are equivalent. In the following numerical examples, the discrete scheme (68)-(73) is applied.

For simplicity, we show how to choose a suitable basis for ${\mathbf {X}}_h^s$ such that the Assumption 2.1 always holds true for $d=2$. In general, for any element E and for $i+j\le k_2$, we use the following basis

$$\begin{aligned} \begin{pmatrix} span\{1&{}x&{}y&{}xy&{}x^2&{}y^2&{}\ldots &{}x^iy^j\}\\ span\{1&{}x&{}y&{}xy&{}x^2&{}y^2&{}\ldots &{}x^iy^j\} \end{pmatrix} . \end{aligned}$$

However, ${\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}$ doesn’t contain constant if the interface edge $e\subset \{(x,y): y=x+\text {constant}\}$, thus, it doesn’t belong to discontinuous piecewise polynomials of degree $k_2$. To avoid this problem, let constants $a>0$ and $b>0$, and $a\ne b$, then the Assumption 2.1 always holds true by taking the following basis

$$\begin{aligned} \begin{pmatrix} span\{a&{}x&{}y&{}xy&{}x^2&{}y^2&{}\ldots &{}x^iy^j\}\\ span\{b&{}x&{}y&{}xy&{}x^2&{}y^2&{}\ldots &{}x^iy^j\} \end{pmatrix} . \end{aligned}$$

This conclusion is trivial, thus we don’t present the concrete proof here. Similarly, let $i+j+m\le k_2$ and let constants $a>0$, $b>0$, $c>0$ and $a\ne b$, $a\ne c$, $b\ne c$, then the Assumption 2.1 always holds true for $d=3$ by taking the following basis

$$\begin{aligned} \begin{pmatrix} span\{a&{}x&{}y&{}z&{}xy&{}xz&{}yz&{}\ldots &{}x^iy^jz^m\}\\ span\{b&{}x&{}y&{}z&{}xy&{}xz&{}yz&{}\ldots &{}x^iy^jz^m\}\\ span\{c&{}x&{}y&{}z&{}xy&{}xz&{}yz&{}\ldots &{}x^iy^jz^m\} \end{pmatrix} . \end{aligned}$$

6.2 Numerical Experiments

In this section, under uniformly matching mesh, the convergence analysis of the coupled system shall be reported by some numerical tests. In these numerical examples, the domain $\Omega =[0,1]\times [0,1]$, Stokes domain $\Omega _1=[0,1]\times [0.5,1]$, Darcy domain $\Omega _2=[0,1]\times [0,0.5]$, the interface $\Gamma _{12}=[0,1]\times \{0.5\}$. In addition, we consider the stability constants $\sigma _{1,e}=30\mu$, $\sigma _{2,e}=1$ and $\epsilon =\pm 1$.

6.2.1 Rates of Convergence

In this part, some tests are given to verify the rates of convergence. Let the permeability tensor ${\mathbf {K}}={\tilde{k}}{\mathbf {I}}$, we consider the coupled system with the following exact solution [1]

$$\begin{aligned} u_{1,f}&= -sin(\pi x)exp(y/2)/(2\pi ^2), \quad u_{2,f}=cos(\pi x)exp(y/2)/\pi ,\\ p_f&= \frac{{\tilde{k}}\mu -2}{{\tilde{k}}\pi }cos(\pi x)exp(y/2),\\ u_{1,s}&= -2sin(\pi x)exp(y/2), \quad ~~~~~~~u_{2,s}= cos(\pi x)exp(y/2)/\pi ,\\ p_s&=-\frac{2}{{\tilde{k}}\pi }cos(\pi x)exp(y/2), \end{aligned}$$

with $G=2/(1+4\pi ^2)$. Then, using the exact solution, the source terms ${\mathbf {f}}$ and g are determined by the coupled Stokes-Darcy system (1) and (4), respectively, and the boundary conditions are obtained by restricting the corresponding true solution to boundary $\partial \Omega$. To fully verify our analysis, we consider the coupled system with different $\mu$ and ${\tilde{k}}$ in the following tests.

Table 1 The convergence rates under $k_1=k_2=1$, $\mu ={\tilde{k}}=1$ and $\epsilon =-1$

Table 2 The convergence rates under $k_1=k_2=1$, $\mu ={\tilde{k}}=1$ and $\epsilon =1$

Table 3 The convergence rates under $k_1=k_2=2$, $\mu ={\tilde{k}}=1$ and $\epsilon =-1$

Table 4 The convergence rates under $k_1=k_2=2$, $\mu ={\tilde{k}}=1$ and $\epsilon =1$

Table 5 The convergence rates under $k_1=k_2=1$, $\mu =10^{-3}$, ${\tilde{k}}=1$ and $\epsilon =-1$

Table 6 The convergence rates under $k_1=k_2=1$, $\mu =1$, ${\tilde{k}}=10^{3}$ and $\epsilon =-1$

Table 7 The convergence rates under $k_1=k_2=1$, $\mu =10^{-6}$, ${\tilde{k}}=10^{3}$ and $\epsilon =-1$

Table 8 The convergence rates under $k_1=k_2=1$, $\mu =10^{-3}$, ${\tilde{k}}=10^{-3}$ and $\epsilon =-1$

Let $\mu =1$ and ${\tilde{k}}=1$, we report the numerical results given in Tables 1, 2, 3, 4 for different $\epsilon$ and finite element spaces, which are optimal and predicted by the analysis. To adequately verify our analysis, let $k_1=k_2=1$ and $\epsilon =-1$, some numerical results with different $\mu$ and ${\tilde{k}}$ are presented in Tables 5, 6, 7, 8. Note that the exact solution pressure changes with different viscosity $\mu$ and permeability ${\tilde{k}}$, the errors of the pressure and velocity will also change since the pressure depends on the permeability ${\tilde{k}}$ and the velocity error is related to the pressure (see Theorem 5.1). Particularly, it is obvious when ${\tilde{k}}$ is relatively small and pressure is relatively large (see Table 8). However, these numerical results with different $\mu$ and ${\tilde{k}}$ are consistent with our convergence analysis. From Tables 1, 2, 3, 4,5, 6, 7, 8, we can conclude that the numerical results support the theoretical analysis derived in Theorems 5.1 and 5.2. In addition, the $L^2$ error of velocity in both Stokes and Darcy regions are optimal, even though the $L^2$ optimal convergence can not be proved in this paper.

Fig. 1

The numerical result of Sect. 6.2.2 including the permeability, velocity field and pressure field

6.2.2 Coupled Surface and Subsurface Flow

In this part, we consider an example proposed in [1, Example 6.2]. This example is representative of surface flow coupled to subsurface flow. Let the boundary of Darcy region be partitioned as $\Gamma _2=\Gamma _2^a\cup \Gamma _2^b$, where $\Gamma _2^a=\{x=0~or~x=1\}$ and $\Gamma _2^b=\{y=0\}$. Similarly, let the boundary of Stokes region $\Gamma _1=\Gamma _1^a\cup \Gamma _1^b\cup \Gamma _1^c$, where $\Gamma _1^a=\{x=0\}$, $\Gamma _1^b=\{x=1\}$ and $\Gamma _1^c=\{y=1\}$. Then, we consider the following boundary conditions:

$$\begin{aligned} {\mathbf {u}}_f&=(y(1.5-y)/5,0),~~&\text {on} ~~ \Gamma _1^a,\\ {\mathbf {T}}({\mathbf {u}}_f, p_f)&=0,~~&\text {on} ~~ \Gamma _1^b,\\ {\mathbf {u}}_f\cdot {\mathbf {n}}&=0, ~~\text {and}~~{\mathbf {T}}({\mathbf {u}}_f, p_f)^t=0,~~&\text {on} ~~ \Gamma _1^c,\\ {\mathbf {u}}_s\cdot {\mathbf {n}}&=0,~~&\text {on} ~~ \Gamma _2^a,\\ p_s&=-0.05,~~&\text {on} ~~ \Gamma _2^b, \end{aligned}$$

where ${\mathbf {T}}({\mathbf {u}}_f, p_f)^t$ stands for tangential stress (see [1, Example 6.2] and [7, Example 7.2]). Let $\epsilon =-1$, $k_1=k_2=1$, $\mu =0.1$, $5G={\mathbf {K}}^{-1/2}$, ${\mathbf {f}}=0$, $g=0$ and the permeability

$$\begin{aligned} {\tilde{k}}= 700(1+0.5(\sin (10\pi x)\cos (20\pi y^2) + \cos ^2(6.4\pi x)\sin (9.2\pi y))) + 100. \end{aligned}$$

Based on these choices we numerically solve this coupled system on a mesh with $h_1=h_2=1/128$.

The numerical results are given in Fig. 1, which shows the permeability field, velocity field and pressure field. As shown in Fig. 1b, the fluid flow from inlet into interface and then Darcy region, which is similar with the one presented in [1, Example 6.2] and [7, Example 7.2]. The tangential velocity of flow is discontinuous along the interface and the flow field has relatively small velocity at low permeability in the Darcy region $\Omega _2$. The pressure field given in Fig. 1c is highest at the entrance (around the inlet $\Gamma _1^a$) and discontinuous across the interface. In summary, the proposed scheme can deal with the physical problem and capture the discontinuity of velocity field and pressure field on the interface.