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.
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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
Here \({\mathbf {T}}\) is the stress tensor
where \(\mu >0\) is the constant viscosity coefficient and the strain tensor is defined by
In region \(\Omega _2\), the governing equations satisfy the Darcy equations
where the permeability tensor \({\mathbf {K}}\) is symmetric and positive definite, and satisfies for some \(0< k_{min}\le k_{max}<\infty \),
The physical quantities in \(\Omega _1\) and \(\Omega _2\) are coupled by the following interface conditions on \(\Gamma _{12}\):
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:
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
In addition, if \(e \in \partial \Omega \) and \(e\in E_1\), then the average \(\{w\}\) and jump [w] of function w are
Define the general DG norms:
and H(div)-like norm:
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 \(m=1\)
Now we define
and the corresponding norms
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:
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
Before giving the numerical scheme, some bilinear forms shall be introduced. For any \({\mathbf {u}}_f, {\mathbf {v}}_f \in {\mathbf {X}}^f\),
where \(\epsilon =\pm 1\) and \(\sigma _{1,e}>0\) is the penalty constant. For any \({\mathbf {u}}_s, {\mathbf {v}}_s \in {\mathbf {X}}^s\),
where the stability constant \(\sigma _{2,e}>0\). For any \({\mathbf {v}}_f \in {\mathbf {X}}^f, p_f \in M^f\) and \({\mathbf {u}}_s \in {\mathbf {X}}^s, p_s \in M^s\),
Define the finite-dimensional space of functions \(\Lambda _h={\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}\) on the interface and let
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
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
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
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
The interface term can be rewritten as
combining the interface conditions (8) and (9), we get
Thus, we have
Similarly, we obtain
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 \(h_E\)
Also, recall the discrete Korn’s inequality [26]
Let \(p\in L^2(\Omega )\), we denote by \({\tilde{p}}\) the \(L^2\)- projection of p in \(M_h\) satisfying
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
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 \(E\in \varepsilon _h^1\)
For any \({\mathbf {v}}_f\in (H_0^1(\Omega _1))^d\), by (28), (29) and (30) we have
Moreover, the interpolation operator \(\Pi _h^f\) satisfies the following approximation property
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
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
Proof
From the approximation property (33) and (34) (see Lemma 3.9 of [24]), we have
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
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
For any \(E\in \varepsilon _h^2\), \(\Pi _h^s\) satisfies the approximation properties
In addition, we have the following result [4, 25]
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
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
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
Then, for any \(e \in \Gamma _h^2\cup \Gamma _2\), the following inequality holds
Proof
By taking \(q_{s,h}=r_e(w)\) in (43) and using the trace inequality (21), we have
and
\(\square \)
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
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])
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
Due to \({\mathbf {u}}_{s,h}\in {\mathbf {Z}}_h\), it satisfies \(b_s({\mathbf {u}}_{s,h},q_{s,h})=0\). Choosing
yields
By (44), we obtain
Note that \(\sigma _{2,e}>1\), therefore
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
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
If the following inf-sup condition holds
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
satisfying
Note that, by \({\mathbf {v}} \in (H^1(\Omega ))^d\),
which, combining with the given priori bound, yields
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
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
To define \(\pi _h^s\), we consider the auxiliary problem
The auxiliary problem is well-defined, since
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
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
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
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 \(\Lambda _h={\mathbf {X}}_h^s\cdot {\mathbf {n}}_{12}\),
Using (40), (42) and (50), we have
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)
Therefore
combining with (49), which proves (48). Now using (48) we have
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
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
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
Next,
Using (40) and (41), there holds
The last term in (56) can be bounded by using (42), (50), (19) and (33)
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
Proof
Subtracting (13)–(14) from (11)–(12), the error equations are
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
Based on the above notions, the error Eqs. (59)–(60) can be rewritten as
By (52) in Lemma 5.1, we have \(b(\theta ,q_h)=0\), thus
which implies \(\chi \in {\mathbf {Z}}_h\). Choosing \({\mathbf {v}}_h=\chi \) and \(q_h=\xi \) in (61) and (62) yields
equivalently,
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:
Using Cauchy–Schwarz inequality, Young inequality and the approximation property (33),
By Cauchy–Schwarz inequality, Young inequality, trace inequality (19) and the approximation property (33)
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 \(E\in \varepsilon _h^1\)
For a fixed \(e\in \Gamma _h^1\cup \Gamma _1\), using the Lagrange interpolation in \(T_3\), we have
The first part can be bounded by using trace inequality (20) and the approximation property of the Lagrange interpolation (64)
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
Therefore,
The fourth term vanishes due to the continuity of \({\mathbf {u}}\) and the property (53) of \(\tilde{{\mathbf {u}}}_h\),
The last term can be estimated by using the trace inequality (19),
Let us now estimate \(a_s(\theta ,\chi )\),
Using the Cauchy–Schwarz inequality, Young inequality and the approximation property (40) to the first part, we have
The second part is bounded by using trace inequality (19) and the approximation (40)
Next, we estimate \(b_f(\chi ,\zeta )\), by the trace inequality (19), and properties (24) and (26),
Similarly, by the trace inequality (19), and properties (25) and (27),
It remains to estimate the last term in (63). Since \(\chi \) belongs to \({\mathbf {V}}_h\), we obtain
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
For any interface facet e and any piecewise vector-valued constant \({\mathbf {c}}_e\), there holds
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])
therefore
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
Combing the K-ellipticity, we have
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
Proof
The error equation (59) can be written as
From the discrete inf-sup condition (46),
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
To bound the term \(a({\mathbf {u}}_h-{\mathbf {u}},{\mathbf {v}}_h)\) in (67),
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,
\(Q_3\) is estimated by utilizing the Lagrange interpolation
By using the trace inequality (21), there holds
For the term \(b({\mathbf {v}}_h,p-{\tilde{p}}_h)\) in (67), by the properties (24) and (26), we have
Similarly,
Thus,
Similar to the proof in Theorem 5.1, the last interface integral term in (67) is bounded by
Combing the above bounds and the discrete inf-sup condition (66), we have
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
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
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
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
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
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]
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.
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.
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:
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
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.
Data Availability Statement
The data used to support the findings of this study are available from the corresponding author upon request.
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We would like to acknowledge the financial support from the National Natural Science Foundation of China Grant No.11771348, 11971378, 51876170.
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Wen, J., Su, J., He, Y. et al. A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem. J Sci Comput 85, 26 (2020). https://doi.org/10.1007/s10915-020-01342-6
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DOI: https://doi.org/10.1007/s10915-020-01342-6