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Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

  • Hui Xie
  • Xuejun XuEmail author
Original Paper
  • 80 Downloads

Abstract

In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale \(H(\mathrm {div})\) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this \(H(\mathrm {div})\) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale \(H(\mathrm {div})\) system is bounded by \(C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))\), where \(\hat{\delta }\) denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.

Keywords

High contrast Mixed FEM DD preconditioner Spectral coarse space 

Mathematics Subject Classification

65N30 65N22 65N55 

1 Introduction

Any practical problem in applied sciences occurs in the media which contains multiple scales. For example, in reservoir simulation [7], porous media [10], and material sciences, high-contrast media properties always exist.

In this paper, we will consider the following linear second-order multiscale elliptic PDE:
$$\begin{aligned} -\mathrm {div}(\kappa (x)\nabla p)=f \quad \text { in } \quad \Omega \end{aligned}$$
(1.1)
in a bounded domain \(\Omega \subset {\mathbb {R}}^d\), with \(d=2,3\), subject to the appropriate boundary condition. Here, the highly heterogeneous properties of the medium are built into the permeability \(\kappa (x)\), that means \(\max _{x\in \Omega }\kappa (x)\gg \min _{x\in \Omega }\kappa (x)\). Without loss of generality, we assume that \(\min _{x\in \Omega }\kappa (x)\ge 1\) and \(\kappa \) is a piecewise constant on the fine mesh. The heterogeneity has adverse effects upon the traditional numerical techniques. On one hand, it brings small scales into the solution, so that the standard numerical method may not have high resolution unless the mesh size is fine enough. On the other hand, the resulting linear system is more ill-conditioned due to high-contrast \(\kappa (x)\), see [17]. Given these issues, some approaches intend to solve the problem on a coarse grid without sacrificing the accuracy, which include GFEM [5], MsFEM [22], MsFV [24], MHM [4], GMsFEM [14], and some multiscale methods based on localized orthogonal decomposition (LOD) [19, 25]. For the mixed variational formulation, some variants were also developed, which include mixed MsFEM [9], mixed VMM [1, 2], mixed GMsFEM [8], etc.
An alternative way to remedy these issues is to construct effective preconditioners for the large ill-conditioned linear system discretized by a fine mesh. Our goal of this paper belongs to the latter one. It is well known that, in many applications, the velocity \(\mathbf {u}:=-\kappa (x)\nabla p\) rather than p is the interesting variable. Hence, it is great interest to investigate the saddle-point problem that arises from (1.1) when we introduce the velocity variable \({\mathbf {u}}\). However, the corresponding discrete matrix is indefinite. This makes it harder for designing efficient preconditioners than ones for the symmetric and positive definite system. There exist several known techniques for solving the saddle-point problems in the literature, e.g., some block-diagonal preconditioners [30] for the MINRES method [29], and Uzawa algorithm [12]. In [26, 27], the original saddle problem was reduced to a symmetric and positive definite velocity equation by working in the divergence-free velocity space, and then developed some optimal multilevel and DD preconditioners for it, but the construction of divergence-free basis in three dimension is quite complicated. In [35], by introducing a penalty term, the velocity of this penaltied PDE is equivalent to the \(H(\mathrm {div})\) system, which includes a penalty parameter \(\delta >0\). They developed some optimal hierarchical and multigrid methods, which are shown to be insensitive with respect to the parameter \(\delta \) and possible jumps in the permeability \(\kappa (x)\) as long as the discontinuities occur only across the initial nonoverlapping subdomain interfaces. In this paper, we use an equivalent transformation initialized by [34] to get a new saddle-point problem. For this new saddle-point problem, in [34], a block-diagonal matrix is constructed and the inverse of this block-diagonal matrix is applied as the preconditioner for the new saddle-point system. The uniform eigenvalue estimates of the preconditioned system were derived. We observe that there are two blocks on the diagonal part of this block-diagonal matrix. The first one just corresponds to the following bilinear form:
$$\begin{aligned} \int _{\Omega }\left( \kappa (x)^{-1}\mathbf {u}\cdot \mathbf {v}+\frac{1}{\delta }\nabla \cdot {\mathbf {u}}\nabla \cdot {\mathbf {v}}\right) ,\quad \forall \ \mathbf {u},\mathbf {v}\in V_h, \end{aligned}$$
(1.2)
where \(V_h\) is the discretized velocity space adopted in the mixed finite-element method and we do not need \(\delta \rightarrow 0\), because the fully equivalence of this transformation, which is the distinct difference from the above penalty method. The second block is a constant times the identity matrix. In the large-scale computation, the direct inverse of the first block is infeasible and unstable. The core of our method is how to design an efficient preconditioner for this first block. Namely, we focus on the construction of the effective preconditioner for the \(H(\mathrm {div})\) system (1.2). One class of highly effective and popular preconditioners, which have been studied and used widely, is DD methods [33, 37]. First, we mention some earlier DD methods related to ours. To the best of our knowledge, the first studies on DD methods for Raviart–Thomas approximation of (1.2) are due to Arnold et al. [3], where the 2D case was studied with the two-level overlapping Schwarz method and was continued by [20], where the 3D case was studied by a multilevel method. These pioneering works introduced a key idea that stable decomposition of Raviart–Thomas function should treat the kernel of divergence operator and its orthogonal complement separately. Often, either Helmholtz decomposition [3] or regular decomposition [21] is applied for this purpose. In these works, their preconditioner is shown to be optimal under the assumption that the coefficient is constant on the whole domain. Some iterative substructuring methods [36] also have been studied for this problem, but their condition number estimate of the preconditioned system depends on the possible jumps within subdomains. Recently, in [28], discrete harmonic extension from the interface of coarse grid was introduced to form the coarse space for two-level overlapping DD method. Under the assumption that discontinuities occur only across the initial nonoverlapping subdomains interfaces, their DD method is proved to be optimal. It is well known that coarse space is crucial for the DD method. Some different spectral coarse spaces [11, 15, 16, 31, 32] have been developed and proven to be optimal for the scalar multiscale elliptic problem. This inspires us to develop one spectral coarse space for our vector-valued multiscale \(H(\mathrm {div})\) problem. More precisely, we construct a spectral coarse space to make our two-level overlapping DD preconditioner still optimal with respect to more relaxed assumptions on the discontinuities of coefficient \(\kappa (x)\) than the ones of [28]. The condition number estimate of the preconditioned system is independent of possible jumps of \(\kappa (x)\) within subdomains under a mild restriction. Note that the solution of (1.2) is equivalent to the velocity of some high-contrast elliptic problems. Motivated by local eigenvalue coarse space initialized by [15], we solve similar local eigenvalue problems on face supporting patches with the mixed finite-element method. We may find that some near-vanishing eigenvalues correspond to the inclusions and channels in \(\kappa (x)\). We choose the velocities corresponding to these “bad” eigenvalues, and multiply them with the partition of unity functions belonging to its face supporting patches. These weighted velocities, together with trivial Raviart–Thomas basis on the coarse grid, form our newly developed coarse space. We use the direct solvers on overlapping subdomains as local components in our DD method. For the local stability, using similar ideas in [23], we may derive a weighted Helmholtz decomposition and then obtain the condition number estimate by a Schwarz framework [33], which is independent of the possible jumps in \(\kappa (x)\) under a mild assumption. By [34], we know that this optimality implies that we can replace the first block of the block-diagonal preconditioner with our newly developed preconditioner for (1.2), and then, the optimality will inherit. This means that our new block-diagonal preconditioner may serve as a part of preconditioner for the MINRES iteration for mixed finite-element discretization.

The organization of the paper is as follows. In Sect. 2, we present the mixed finite-element discretization and the equivalent transformation to another saddle-point problem. An abstract optimal block-diagonal preconditioner for this saddle-point problem is given in this section. Section 3 is devoted to the construction of our spectral coarse space and local space based on DD method; the main result is also shown in this section. In Sect. 4, the detailed proof of the stability of coarse component and local component is derived, respectively. In the final section, we give some numerical results to confirm our theoretical findings. We use notation \(A\lesssim B\) for the inequality \(A\le C B\), where C is a generic positive constant independent of the mesh size, but the constant of which may take different values in different occurrences.

2 The Problem and an Abstract Preconditioner

In this section, we consider mixed element discretization of problem (1.1) with the Neumann boundary condition:
$$\begin{aligned} -\kappa (x)\nabla p\cdot {\mathbf {n}}=0 \quad \text { on}\quad \partial \Omega . \end{aligned}$$
For simplicity of presentation, here, we only consider the Neumann boundary condition, and our result may be generalized to the other boundary conditions. The source function f satisfies the following compatibility condition:
$$\begin{aligned} \int _{\Omega }f\mathrm{d}x=0. \end{aligned}$$
By introducing the velocity unknown \(\mathbf {u}:=-\kappa (x)\nabla p\), we can rewrite the problem (1.1) into the following mixed variational formulation: we find \((\mathbf {u},p)\in V\times W\), such that
$$\begin{aligned}&\int _{\Omega }\kappa (x)^{-1}\mathbf {u}\cdot \mathbf {v}-\int _{\Omega } \nabla \cdot \mathbf {v}~p=0,\\&\int _{\Omega }\nabla \cdot \mathbf {u}~ q=\int _{\Omega }f~q, \end{aligned}$$
for all \((\mathbf {v}, q)\in V\times W\), where V and W are the spaces \(H_0(\mathrm {div};\Omega )\) and \(L^2(\Omega )/{\mathbb {R}}\), respectively. Here:
$$\begin{aligned} H_0(\mathrm {div};\Omega ):=\left\{ \mathbf {v}\in L^2(\Omega )^d: \ \nabla \cdot \mathbf {v}\in L^2(\Omega ),\ \mathbf {v}\cdot {\mathbf {n}}=0 \quad \text { on }\quad \partial \Omega \right\} , \end{aligned}$$
and all \(q\in L^2(\Omega )/{\mathbb {R}}\) means q is in the quotient space \(L^2(\Omega )\) over the constant space.

The mixed finite-element discretization is obtained by restricting the trial and test functions to the finite dimensional subspace \(V_h\subset V\) and \(W_h\subset W\). As is well known, the suitable pair of finite-element spaces \(V_h\) and \(W_h\) have been proposed by many authors (cf. [6]). For simplicity, we choose the pair of the lowest order Raviart–Thomas element space and piecewise constant space as \(V_h\) and \(W_h\), which is stable and satisfies the known inf-sup condition.

The corresponding discrete problem is to find \((\mathbf {u}_h,p_h)\in V_h \times W_h\), such that
$$\begin{aligned}&\int _{\Omega }\kappa (x)^{-1}\mathbf {u}_h\cdot \mathbf {v}_h-\int _{\Omega } \nabla \cdot \mathbf {v}_h p_h=0, \end{aligned}$$
(2.1)
$$\begin{aligned}&\int _{\Omega }\nabla \cdot \mathbf {u}_h q_h=\int _{\Omega }fq_h, \end{aligned}$$
(2.2)
for all \((\mathbf {v}_h,q_h)\in V_h\times W_h\). Note that \(\nabla \cdot \mathbf {v}_h\) belongs to \(W_h\), so we may choose \(q_h=\frac{1}{\delta }\nabla \cdot \mathbf {v}_h\) in Eq. (2.2) and add Eqs. (2.1)–(2.2). Then, we get equivalent discrete saddle-point problems:
$$\begin{aligned}&\int _{\Omega }\left[ \kappa (x)^{-1}\mathbf {u}_h\cdot \mathbf {v}_h+\frac{1}{\delta }\nabla \cdot \mathbf {u}_h \nabla \cdot \mathbf {v}_h\right] -\int _{\Omega } \nabla \cdot \mathbf {v}_h p_h= \frac{1}{\delta }\int _{\Omega }f\nabla \cdot \mathbf {v}_h, \end{aligned}$$
(2.3)
$$\begin{aligned}&\int _{\Omega }\nabla \cdot \mathbf {u}_h q_h=\int _{\Omega }fq_h. \end{aligned}$$
(2.4)
If we define three operators \(M:V_h\rightarrow V_h\), \(B:V_h\rightarrow W_h\) and \(B^{\mathrm{T}}:W_h\rightarrow V_h\) as follows:
$$\begin{aligned} (M\mathbf {v}_h,\mathbf {w}_h)= & {} \int _{\Omega }\alpha (x)^{-1}\mathbf {v}_h\cdot \mathbf {w}_h,\\ (B\mathbf {v}_h,q_h)= & {} -\int _{\Omega }\nabla \cdot \mathbf {v}_h q_h,\\ (B^{\mathrm{T}}q_h,\mathbf {w}_h)= & {} -\int _{\Omega }\nabla \cdot \mathbf {w}_h q_h, \end{aligned}$$
where\((\cdot ,\cdot )\) is the \(L^2(\Omega )\) inner product. Thus, this new saddle-point problem in operator formulation reads the following:
$$\begin{aligned} {\mathcal {A}}_{\delta } \left[ \begin{array}{c} \mathbf {u}_h\\ p_h \end{array} \right] := \left[ \begin{array}{cc} M+\frac{1}{\delta }B^{\mathrm{T}}B &{} B^{\mathrm{T}}\\ B &{} 0 \end{array} \right] \left[ \begin{array}{c} \mathbf {u}_h\\ p_h \end{array} \right] =- \left[ \begin{array}{c} \frac{1}{\delta }B^{\mathrm{T}}\phi \\ \phi \end{array} \right] , \end{aligned}$$
(2.5)
where \(\phi \) is the \(L^2\) projection of the function f on \(W_h\), and B and \(B^{\mathrm{T}}\) may be regarded as the discrete divergence operator and gradient operator, respectively.

Next, we state an important lemma from [34], which shows that the transformed operator \({\mathcal {A}}_{\delta }\) is equivalent to a block-diagonal symmetric and positive definite one, and this lemma motivates the construction of our DD preconditioner.

Lemma 2.1

We define\(M_{\delta }:=M+\frac{1}{\delta }B^{\mathrm{T}}B\). If\(M_{\delta }\)is spectrally equivalent to a symmetric and positive definite operator\(B_{\delta }\), more precisely, there exist two positive constants\(\gamma _1\)and\(\gamma _2\), such that the following relation holds:
$$\begin{aligned} \gamma _1(M_{\delta }\mathbf {v},\mathbf {v})\le (B_{\delta }\mathbf {v},\mathbf {v})\le \gamma _2(M_{\delta }\mathbf {v}, \mathbf {v}) \quad \text { for all } \mathbf {v}\in V_h. \end{aligned}$$
Then, for the block-diagonal operator\({\mathcal {D}}:=\mathrm {diag}(B_{\delta }, \delta _1{\mathcal {I}})\), \({\mathcal {I}}\)is the identity operator, the eigenvalues of\({\mathcal {D}}^{-1}{\mathcal {A}}_{\delta }\)are contained in the following union of intervals\(\triangle ^-\cup \triangle ^+\), where
$$\begin{aligned} \triangle ^-=\left[ \frac{1}{2\gamma _1}\left( 1-\sqrt{1+4\frac{\gamma _1}{\delta _1}\delta }\right) ,\ \frac{1}{2\gamma _2}\left( 1-\sqrt{1+4\frac{\gamma _2\lambda _0}{\delta _1(\delta +\lambda _0)}\delta }\right) \right] \end{aligned}$$
(2.6)
contains the negative eigenvalues, and
$$\begin{aligned}&\triangle ^+=[\gamma _2^{-1},\gamma _1^{-1}]\cup \left[ \frac{1}{2\gamma _2} \left( 1+\sqrt{1+4\frac{\lambda _0\gamma _2}{\delta _1(\delta +\lambda _0)}\delta }\right) ,\ \frac{1}{2\gamma _1}\left( 1+\sqrt{1+4\frac{\gamma _1}{\delta _1}\delta }\right) \right] \end{aligned}$$
(2.7)
contains the positive ones. The constants\(\delta \)and\(\delta _1\)are artificial positive parameters, which may be adjusted to accelerate the convergence. The constant\(\lambda _0\)is related to the inf-sup condition constant and the minimal value of\(\kappa (x)\).

Remark 2.1

From (2.6) and (2.7), note that, if \(\delta \rightarrow 0\) and \(\delta _1=O(1)\), the ends of \(\triangle ^-\) approach zero, and in such a limit case, the convergence behavior of MINRES is not as clear. Therefore, there is no reason to choose \(\delta \) too small. This is different from the requirement of the penalty method. In addition, the convergence rate of MINRES is independent of the high-contrast \(\kappa (x)\) if the ends of \(\triangle ^-\) and \(\triangle ^+\) are independent of the high-contrast \(\kappa (x)\), and details about this may refer to [13].

By Lemma 2.1, note that a good preconditioner \(B_{\delta }^{-1}\) for \(M_{\delta }\) will lead to a good block-diagonal preconditioner \({\mathcal {D}}^{-1}\) for \({\mathcal {A}}_{\delta }.\) Therefore, the rest of our paper is concentrating on the construction of a robust preconditioner \(B_{\delta }^{-1}\) for the \(H(\mathrm {div})\) system (1.2) with the DD method. To this end, we first introduce the Schwarz framework for the DD method, then give our constructions of the coarse space and local space, and then show the main result of our DD preconditioner corresponding to abstract Lemma 2.1 in the next section.

3 A Preconditioner Based on DD Method

3.1 Schwarz Framework

Recall that the domain \(\Omega \) is a bounded polyhedron in \({\mathbb {R}}^d(d=2,3)\). We decompose \(\Omega \) into N nonoverlapping subdomains \(\Omega _i\) of diameter \(H_i\). We introduce two triangulations \({\mathcal {T}}_H\) and \({\mathcal {T}}_h\) of the domain \(\Omega \), which are shape regular with maximum diameters H and h, respectively. The fine triangulation \({\mathcal {T}}_h\) is a refinement of \({\mathcal {T}}_H\). For each subdomain, \(h_i\) denotes the minimum diameter of elements in \({\mathcal {T}}_h\). We assume that each \(\Omega _i\) is a union of elements in \({\mathcal {T}}_h\). We also consider the overlapping subdomains \(\Omega _{i}^{'}\) obtained from \(\Omega _i\) by adding layers of elements in \({\mathcal {T}}_h\). In addition, we define the interface \(\Gamma \) by the following:
$$\begin{aligned} \Gamma =({\mathop \cup \limits_{i=1}^{N}} \partial \Omega _i)\backslash \partial \Omega . \end{aligned}$$
Next, we state some classical assumptions as [33] for the presentation of our analysis.

Assumption 3.1

For \(i=1,\ldots ,N\), there exists \(\delta _i>0\), such that, if x belongs to \(\Omega _{i}^{'}\), then
$$\begin{aligned} \mathrm {dist}(x,\partial \Omega _{j}^{'}\backslash \partial \Omega )\ge \delta _i \end{aligned}$$
for a suitable \(j=j(x)\), possibly equal to i, with \(x\in \Omega _{j}^{'}\).

Assumption 3.2

(finite covering) Any point \(x\in {\bar{\Omega }}\) is contained in at most \(N_c\) overlapping subdomains of \(\{\Omega _{i}^{'}\}\).

Assumption 3.3

For \(i=1,\ldots ,N\),
$$\begin{aligned} H_K\le CH_i \end{aligned}$$
for all \(K\in {\mathcal {T}}_H\), such that \(K\cap \Omega _{i}^{'}\ne \emptyset \). Here, the positive constant C does not depend on \(\Omega _{i}^{'}\) and \({\mathcal {T}}_H\).

If \(V_i\), \(i=0,\ldots ,N\) are auxiliary spaces, such that \(\begin{aligned}V_h=\sum _{i=0}^{N}V_i\end{aligned}\), where \(V_0\) and \( V_i\) are spaces defined on the coarse grid \({\mathcal {T}}_H\) and the overlapping subdomain \(\Omega _i^{'}\), respectively.

For \(i=1,\ldots ,N\), we define the operator \(T_i:V_h \rightarrow V_i\) by the following:
$$\begin{aligned} a_i(T_i\mathbf {u}, \mathbf {v})=a(\mathbf {u},\mathbf {v}), \quad \forall \mathbf {u}\in V_h, \quad \mathbf {v}\in V_i^h, \end{aligned}$$
and for \(i=0\), we define the operator \(T_0: V_h\rightarrow V_0\) by the following:
$$\begin{aligned} a_0(T_0\mathbf {u},\mathbf {v})=a(\mathbf {u},\mathbf {v}), \quad \forall \mathbf {u}\in V_h, \quad \mathbf {v}\in V_0, \end{aligned}$$
where \(a(\mathbf {u},\mathbf {v})=\int _{\Omega }(\kappa (x)^{-1}\mathbf {u}\cdot \mathbf {v}+\frac{1}{\delta }\nabla \cdot {\mathbf {u}}\nabla \cdot {\mathbf {v}}), \forall \ \mathbf {u},\mathbf {v}\in V_h\), and \(a_i(\cdot ,\cdot )\) and \(a_0(\cdot ,\cdot )\) are variational forms of local problems and coarse problems, respectively. If we introduce an abstract right-hand side term for our \(H(\mathrm {div})\) system \(a(\mathbf {u},\mathbf {v})\) as
$$\begin{aligned} a(\mathbf {u},\mathbf {v})=F(\mathbf {v}), \quad \forall \ \mathbf {v}\in V_h, \end{aligned}$$
then the preconditioned system reads
$$\begin{aligned} T\mathbf {u}=g, \end{aligned}$$
(3.1)
where \(\begin{aligned}T:=\sum _{i=0}^{N}T_i\end{aligned}\), \(\begin{aligned}g:=\sum _{i=0}^{N}g_i,\end{aligned}\) where \(g_i\in V_i\) is the solution of
$$\begin{aligned} a_i(g_i,v)=F(v), \quad \forall v\in V_i, i=0,\ldots ,N. \end{aligned}$$
We state the abstract Schwarz framework (Section 2.3 of [33]) by giving two abstract lemmas, which concern the minimal and maximal eigenvalues of the preconditioned system (3.1).

Lemma 3.1

(Stable decomposition) There exists a constant\(C_0\), such that every\(\mathbf {u}\in V_h\)admits a decomposition\(\begin{aligned}\mathbf {u}=\sum _{i=0}^{N}\mathbf {u}_i\end{aligned}\), where\(\mathbf {u}_i\in V_i\), such that
$$\begin{aligned} \sum _{i=0}^{N}a(\mathbf {u}_i,\mathbf {u}_i)\le C_0^2a(\mathbf {u},\mathbf {u}); \end{aligned}$$
then
$$\begin{aligned} a(T\mathbf {u},\mathbf {u})\ge C_0^{-2}a(\mathbf {u},\mathbf {u}). \end{aligned}$$

Lemma 3.2

If Assumption 3.2 is satisfied and the exact coarse bilinear form and local bilinear forms are used, namely:
$$\begin{aligned} a_0(\mathbf {u},\mathbf {v})=a(\mathbf {u},\mathbf {v}), \quad \forall \ \mathbf {u},\mathbf {v}\in V_0 \end{aligned}$$
and
$$\begin{aligned} a_i(\mathbf {u}_i,\mathbf {v}_i)=a_{\Omega _i^{'}}(\mathbf {u}_i,\mathbf {v}_i), \quad \forall \ \mathbf {u}_i,\mathbf {v}_i \in V_i, i=1,\dots ,N, \end{aligned}$$
where\(a_{\Omega _i^{'}}(\mathbf {u}_i,\mathbf {v}_i):=a(\mathbf {u}_i,\mathbf {v}_i)|_{\Omega _i^{'}}\), then the maximal eigenvalue of the preconditioned operator Tis bounded by \((N_c+1)\).

Lemma 3.3

From Lemmas  3.1 and 3.2 , we know any DD preconditioner for \(M_{\delta }\) , denoted by \(M_{DD}^{-1}\) , has \(\gamma _1=\frac{1}{N_c+1}\) and \(\gamma _2=C_0^2\) , such that
$$\begin{aligned} \gamma _1(M_{\delta }\mathbf {v},\mathbf {v})\le (M_{DD}\mathbf {v},\mathbf {v})\le \gamma _2(M_{\delta }\mathbf {v},\mathbf {v}) \text { for all } \mathbf {v}\in V_h. \end{aligned}$$

Now, we are in a position to give our spectral coarse space \(V_0\), local space \(V_i\), and the estimate of \(C_0^2\) for our DD preconditioner.

3.2 Coarse Space

It is known that the coarse space is very crucial for the scalability of the DD method. In [28], a discrete harmonic extension from each segment of the interface \(\Gamma \) is used, and it is shown to be optimal under the assumption that the coefficients in the \(H(\mathrm {div})\) system are constant in each subdomain \(\Omega _i\). In many applications, jumps of permeability \(\kappa (x)\) are impossible or difficult to align the subdomain decomposition \(\{\Omega _i\}_{i=1}^{N}\). Therefore, the more sophisticated coarse space is needed. In this section, we will introduce a new coarse space.

An important observation which was made by several authors [3, 28] is that the velocity \(\mathbf {v}:=\beta ^{-1}\nabla w\) of the following elliptic problem:
$$\begin{aligned} -\mathrm {div}(\beta ^{-1}\nabla w)+\alpha ^{-1}w=g \quad \text { in } \quad \Omega , \end{aligned}$$
which is subject to Neumann(or Dirichlet) boundary condition is the solution of the boundary value problem
$$\begin{aligned} -\nabla (\alpha \mathrm {div}\mathbf {u})+\beta \mathbf {u}={\mathbf {f}} \quad \text { in} \quad \Omega , \end{aligned}$$
subject to some appropriate essential (or nature) boundary condition, and \({\mathbf {f}}=\nabla (\alpha g)\). This equivalence also motivates the construction of our coarse space.
We give some notations for the constructions of the coarse space. The interior nodes of Raviart–Thomas element in \({\mathcal {T}}_H\) are denoted by \(\{{\mathbf {x}}_i\}_{i=1}^{\hat{N}}\), where \(\hat{N}\) is the number of interior nodes. For each \({\mathbf {x}}_i\), it must lie in the common edge (resp. face) of two coarse elements of \({\mathcal {T}}_H\) in 2D (resp. 3D), and the corresponding union set of these two coarse elements is denoted by \(\omega _i\) (see Fig. 1).
Fig. 1

The local domain \(\omega _i\) associated with each \({\mathbf {x}}_i\); here, the thick and thin solid elements are elements of \({\mathcal {T}}_H\) and \({\mathcal {T}}_h\) in 2D, respectively

On each \(\omega _i,i=1,\ldots ,{\hat{N}}\), we solve the following local eigenvalue problem:
$$\begin{aligned} -\mathrm {div}(\kappa (x)\nabla p)+ \delta p=\, & {} \lambda \kappa (x) p \quad \text { in } \quad \omega _i,\nonumber \\ p=\, & {} 0 \text { on } \partial \omega _i. \end{aligned}$$
(3.2)
To obtain the velocity of problem (3.2), mixed finite-element method is used, i.e., find \((\mathbf {u}_h,p_h,\lambda _h)\in V_h^{\omega _i}\times W_h^{\omega _i}\times {\mathbb {R}}\) satisfies
$$\begin{aligned} (\kappa ^{-1}\mathbf {u}_h,\mathbf {v}_h)-(\mathrm {div}\mathbf {v}_h, p_h)= & \, 0, \quad \forall \ \mathbf {v}_h\in V_h^{\omega _i}, \end{aligned}$$
(3.3)
$$\begin{aligned} (\mathrm {div}\mathbf {u}_h, q_h)+\delta (p_h,q_h)= & {} \lambda _h(\kappa p_h, q_h), \quad \forall \ q_h \in W_h^{\omega _i}, \end{aligned}$$
(3.4)
where \(V_h^{\omega _i}:=\{\mathbf {u}\ |\ \mathbf {u}_{|K}\in \mathcal {RT}(K), K\in {\mathcal {T}}_h\cap \omega _i \text { and } \mathbf {u}\in H(\mathrm {div};\omega _i)\}\) and \(W_h^{\omega _i}:=\{p\ |\ p_{|K}\in {\mathbb {R}}, K\in {\mathcal {T}}_h\cap \omega _i \text { and } p\in L^2(\omega _i)\}\).
We note that \(\{\omega _i\}_{i=1}^{{\hat{N}}}\) is an overlapping covering of \(\Omega \). Define the set of coarse basis functions:
$$\begin{aligned} \varvec{\Phi }_{i,l}:=\pi _{RT}^h(\chi _i \mathbf {u}_h^{i,l}), \quad \text { for } \quad 1\le i\le {\hat{N}} \quad \text { and } \quad 1\le l \le L_i, \end{aligned}$$
where \(\pi _{RT}^h\) is the Raviart–Thomas interpolation on \({\mathcal {T}}_h\) and \(\chi _i\) is the partition of unity function subordinate to \(\omega _i\). Here, we choose \(\chi _i\) as the two-side nonconforming linear finite-element basis function subordinate to \({\mathbf {x}}_i\). \(L_i\) is an integer number for each \(\omega _i\), and \(\mathbf {u}_h^{i,l}\) is lth velocity for \(\omega _i\) (the eigenvalue is in increasing order). Note that, for each \(\omega _i\), there are possibly several coarse basis functions. Denote by \(V_0\) as our spectral coarse space:
$$\begin{aligned} V_0:=\text {span}\left\{ {\pi }_{RT}^h(\varvec{\Psi }_i),\quad 1\le i\le {\hat{N}}\right\} \cup \text {span}\{\varvec{\Phi }_{i,l}, \ 1\le i\le {\hat{N}} \quad \text { and } \quad 1\le l \le L_i\}, \end{aligned}$$
(3.5)
where \(\varvec{\Psi }_i\) is the standard Raviart–Thomas basis for each \(\omega _i\) subordinated to the coarse grid \({\mathcal {T}}_H\). \(L_i\) is the number of our chosen increased eigenvalues, such that \(L_i\) is larger than the number of inclusions and channels in \(\omega _i\).

Remark 3.1

Different from the eigenvalue problem in [15], we have an additional zero-order term in our eigenvalue problem. By the similar analysis, we still have near-vanishing eigenvalues with respect to high contrast, since the high-order derivative term is dominated, but zero eigenvalues will not appear due to the zero-order term and local Dirichlet boundary condition.

Remark 3.2

In our numerical experiments, local Neumann boundary condition for pressure is also tried, and it is also robust with possible jumps. In addition, local Neumann problems are used to identify the number of inclusion and channel contained in local domains \(\{\omega _i\}\); here, the term “channel” means high conductivity region which intersects with \(\partial \omega _i\), and the term “inclusion” is used for the high conductivity region which does not intersect with \(\partial \omega _i\).

Remark 3.3

Note that the dimension of local eigenvalue space \(V_h^{\omega _i}\) is proportional to \(O((H/h)^d)\), and d is the space dimension. Although the dimension of \(V_h^{\omega _i}\) may become large with the large ratio H / h, we only need and calculate the eigenfunctions related to those near-zero eigenvalues. We calculate the eigenvalues from the smallest one, and if there is an obvious gap between the calculated eigenvalues, the calculation is terminated. Therefore, the possible large dimension of \(V_h^{\omega _i}\) has little effects on the efficiency.

3.3 Local Spaces

In this section, we give the local spaces for our \(H(\mathrm {div})\) problems. They are the local spaces which are obtained by restricting the global space \(V_h\) to the local overlapping subdomains \(\{\Omega _i^{'}\}\):
$$\begin{aligned} V_i=V_h^{\Omega _i^{'}}:=\{\mathbf {u}\ |\ \mathbf {u}_{|K}\in \mathcal {RT}(K), K\in {\mathcal {T}}_h\cap \Omega _i^{'} \quad \text { and} \quad \mathbf {u}\in H(\mathrm {div};\Omega _i^{'})\}. \end{aligned}$$

3.4 Main Result

We will show the stability estimates for the coarse space (Lemma 4.3) and local spaces (Lemma 4.7) in Sect. 4, respectively. In advance, by these estimates, the following theorems are shown as our main result.

Theorem 3.1

If coarse space (3.5) is adopted and exact local solvers on the local subdomains are used, from Lemma3.1, our DD preconditioner for\(M_{\delta }\), denoted by\(M_{DD}^{-1}\), is quasi-optimal with\(C_0^2=C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))\), whereCis a uniformly bounded constant which is independent of jumps of\(\kappa \)and the mesh sizes\(H_i\), \(h_i\). Moreover, we have the resulting block-diagonal preconditioner\(D_{\delta }^{-1}=\mathrm {diag}(M_{DD}^{-1},\delta _1^{-1}I)\)for\({\mathcal {A}}_{\delta }\)is optimal. Namely, the intervals of eigenvalue bounds\(\triangle ^-\)and\(\triangle ^+\)are independent of jumps of\(\kappa \).

Proof

For any \(\mathbf {u}\in V_h\), we will find coarse component \(\mathbf {u}_0\) and local components \(\mathbf {u}_i:=\mathbf {curl}\ \mathbf {w}_i+\mathbf {v}_i\) in Sect. 4. By Lemmas 4.34.7, and weighted Helmholtz decomposition in Sect. 4, we obtain the following:
$$\begin{aligned} \sum _{i=0}^N a(\mathbf {u}_i,\mathbf {u}_i)\le \,& {} a(\mathbf {u}_0,\mathbf {u}_0)+C\sum _{i=1}^N({\hat{a}}_i(\mathbf {v}_i,\mathbf {v}_i)+{\hat{a}}_i(\mathbf {curl}\ \mathbf {w}_i,\mathbf {curl}\ \mathbf {w}_i))\\\le\, & {} C \left( 1+\frac{H^2}{{\hat{\delta }}^2}\right) \left( 1+\log ^4\left( \frac{H}{h}\right) \right) a(\mathbf {u},\mathbf {u}). \end{aligned}$$
Finally, we complete the proofs using Lemmas 3.3 and 2.1.

Remark 3.4

Note that, in \(C_0^2\), the factor \(1+\frac{H^2}{{\hat{\delta }}^2}\) is theoretically optimal for smooth \(H(\mathrm {div})\) problems [33]; the factor \(1+\log ^4(\frac{H}{h})\) comes from the weighted Helmholtz decomposition in Sect. 4 and will degenerate to 1 in a smooth case.

4 The Proof of Decomposition Stability

4.1 The Stability of Coarse Space

Before introducing our coarse space interpolation, we give some useful propositions, projection operators, and lemmas.

Proposition 4.1

If we denote the solution\((\mathbf {u}_h^i,p_h^i)\)related to theitheigenvalue\(\lambda _h^i\)for the eigenvalue problem (3.3)–(3.4), then the following equality holds:
$$\begin{aligned} a_{\omega _i}\left( \mathbf {u}_h^i,\mathbf {u}_h^j\right) =\frac{\lambda _h^i\lambda _h^j}{\delta }\left( \kappa p_h^i,\kappa p_h^j\right) -\lambda _h^i\left( \kappa p_h^i,p_h^j\right) . \end{aligned}$$

Proof

By the equality (3.4), we have the following:
$$\begin{aligned} \mathrm {div}\mathbf {u}_h^i=Q_h(\lambda _h^i\kappa p_h^i-\delta p_h^i), \end{aligned}$$
(4.1)
where \(Q_h\) is the \(L^2\) projection onto \(W_h^{\omega _i}\). We take the special test functions in (3.3) and (3.4) as \(\mathbf {v}_h=\mathbf {u}_h^j\) and \(q_h=\frac{1}{\delta }\mathrm {div}\mathbf {u}_h^j\) for \((\mathbf {u}_h^i,p_h^i)\), and this leads to
$$\begin{aligned} \left( \kappa ^{-1}\mathbf {u}_h^i,\mathbf {u}_h^j\right) -\left( \mathrm {div}\mathbf {u}_h^j, p_h^i\right)= & {} 0,\\ \left( \mathrm {div}\mathbf {u}_h^i, \frac{1}{\delta }\mathrm {div}\mathbf {u}_h^j\right) +\left( p_h^i,\mathrm {div}\mathbf {u}_h^j\right)= & {} \frac{\lambda _h^i}{\delta }\left( \kappa p_h^i, \mathrm {div}\mathbf {u}_h^j\right) . \end{aligned}$$
Adding these two equalities together gets
$$\begin{aligned} \left( \kappa ^{-1}\mathbf {u}_h^i,\mathbf {u}_h^j\right) +\left( \mathrm {div}\mathbf {u}_h^i, \frac{1}{\delta }\mathrm {div}\mathbf {u}_h^j\right) =\frac{\lambda _h^i}{\delta }\left( \kappa p_h^i, \mathrm {div}\mathbf {u}_h^j\right) . \end{aligned}$$
Inserting the equality (4.1) into the right-hand side of the above equality and using the assumption that \(\kappa \) is a constant on each element in \({\mathcal {T}}_h\), the proof is completed.

Remark 4.1

Without loss of generality, we may require that \((\kappa p_h^i,p_h^j)=0\), if \(i\ne j\), then, in this case, the above result may be reduced to \(a_{\omega _i}(\mathbf {u}_h^i,\mathbf {u}_h^j)=\frac{\lambda _h^i\lambda _h^j}{\delta }(\kappa p_h^i,\kappa p_h^j)\) with \(i\ne j\).

As is well known, the kernel of \(\mathrm {div}\) and \(\mathbf {curl}\) operator should be treated carefully in the stable decomposition, and the Helmholtz decomposition is often used. Here, we only state its discrete counterpart.

Lemma 4.1

(Discrete Helmholtz decomposition) If\(\omega _i\)is convex, then we have discrete decompositions for Raviart–Thomas space\(V_h\)and edge element space\(N_h\). Thus,
$$\begin{aligned} N_h(\omega _i)=\mathbf {grad}\ S^h(\omega _i)\oplus N_h^\bot (\omega _i), \end{aligned}$$
and
$$\begin{aligned} V_h(\omega _i)=\, & {} \mathbf {curl}\ N_h(\omega _i)\oplus V_h^\bot (\omega _i)\\=\, & {} \mathbf {curl}\ N_h^\bot (\omega _i)\oplus V_h^\bot (\omega _i), \end{aligned}$$
where\( N_h^\bot (\omega _i)\)and\(V_h^\bot \)are orthogonal complements, respectively.

Definition 4.1

(Projections into complement space) Let \(\Theta _{\mathbf {curl}}^\bot \) and \(\Theta _{\mathrm {div}}^\bot \) be orthogonal projections from \(H(\mathbf {curl};\Omega )\) onto \(H^\bot (\mathbf {curl};\Omega )\) and from \(H(\mathrm {div};\Omega )\) onto \(H^\bot (\mathrm {div};\Omega )\), respectively. Furthermore, we define projections \(P_h^{ND}:H(\mathbf {curl};\Omega )\rightarrow V_{ND}^+ \) and \(P_h^{RT}:H(\mathrm {div};\Omega )\rightarrow V^+_{RT}\), respectively, by the following:
$$\begin{aligned} (\mathbf {curl}\ P_h^{ND}\mathbf {u},\mathbf {curl}\ \mathbf {v})= & {} (\mathbf {curl}\ \mathbf {u},\mathbf {curl}\ \mathbf {v}), \,\mathbf {v}\in V^+_{ND},\\ (\mathrm {div}P_h^{RT}\mathbf {u},\mathrm {div}\mathbf {v})= & {} (\mathrm {div}\mathbf {u},\mathrm {div}\mathbf {v}), \,\mathbf {v}\in V^+_{RT}, \end{aligned}$$
where \( V^+_{ND}:=\Theta _{\mathbf {curl}}^\bot (N_h^\bot )\) and \(V^+_{RT}:=\Theta _{\mathrm {div}}^\bot (V_h^\bot )\).

Remark 4.2

By these definitions, we may easily check that \(P_h^{ND} \mathbf {u}^\bot = \Theta _{\mathbf {curl}}^\bot \mathbf {u}^\bot \)\(=\mathbf {u}^\bot \), \(\forall \ \mathbf {u}^\bot \in N_h^\bot \), and \(P_h^{RT} \mathbf {v}^\bot = \Theta _{\mathrm {div}}^\bot \mathbf {v}^\bot =\mathbf {v}^\bot \), \(\forall \ \mathbf {v}^\bot \in V_h^\bot \).

The following lemma holds for these two projection operators (the proof may be found in [33]).

Lemma 4.2

Let \(\Omega \)be convex. Then, the operators\(P_h^{ND}\)and\(P_h^{RT}\)satisfy the following error estimates:
$$\begin{aligned} \Vert \mathbf {v}_h^\bot -P_h^{ND}\mathbf {v}_h^\bot \Vert\le\, & {} C h\Vert \mathbf {curl}\ \mathbf {v}_h^\bot \Vert ,\,\mathbf {v}_h^\bot \in V_{ND}^+,\\ \Vert \mathbf {u}_h^\bot -P_h^{RT}\mathbf {u}_h^\bot \Vert\le\, & {} C h\Vert \mathrm {div}\ \mathbf {u}_h^\bot \Vert ,\,\mathbf {u}_h^\bot \in V_{RT}^+, \end{aligned}$$
whereCis independent of h, \(\mathbf {v}_h^\bot \), and\(\mathbf {u}_h^\bot \).

Now, for each \(\omega _i\), we are in a position to define the local interpolation function \(I_{L_i}^{\omega _i}\mathbf {u}\) for \(\mathbf {u}\in V_h\).

(i) If it does not contain any inclusion or channel: we have discrete Helmholtz decomposition for \(\mathbf {u}|_{\omega _i}\)
$$\begin{aligned} \mathbf {u}|_{\omega _i}=\mathbf {curl}\ \mathbf {w}^\bot + \mathbf {v}^\bot , \end{aligned}$$
with \(\mathbf {w}^\bot \in N_h^\bot \) and \(\mathbf {v}^\bot \in V_h^\bot .\) Then, we define the following:
$$\begin{aligned} I_{L_i}^{\omega _i}\mathbf {u}=\mathbf {curl}(Q_{ND}^H(P_h^{ND}\mathbf {w}^\bot ))|_{\omega _i}+Q_{RT}^H(P_h^{RT}\mathbf {v}^\bot )|_{\omega _i}, \end{aligned}$$
(4.2)
where \(Q_{ND}^H\) and \(Q_{RT}^H\) are the \(L^2\) projection operators into edge element space and Raviart–Thomas space in \({\mathcal {T}}_H\), respectively.
(ii) If it contains some inclusions or channels, we define \(I_{L_i}^{\omega _i}\mathbf {u}\in V_{\omega _i}\) by the following:
$$\begin{aligned} a_{\omega _i}(I_{L_i}^{\omega _i}\mathbf {u},\mathbf {v})=a_{\omega _i}(\mathbf {u},\mathbf {v}),\ \forall \mathbf {v}\in V_{\omega _i}, \end{aligned}$$
(4.3)
where \(V_{\omega _i}=\text {span}\{\varvec{\Psi }_i, \varvec{\Phi }_{i,l},\ 1\le l \le L_i \}\).
With the help of the interpolation function \(I_{L_i}^{\omega _i}\mathbf {u}\) for each \(\omega _i\), we can define the coarse interpolation \(I_0:V_h \rightarrow V_0\) by the following:
$$\begin{aligned} I_0\mathbf {u}=\sum _{i=1}^{N_0}\pi _{RT}^h( I_{L_i}^{\omega _i}\mathbf {u}). \end{aligned}$$
Next, we shall give the stability estimate of this coarse interpolation operator.

Lemma 4.3

For all\(\mathbf {u}\in V_h\), let\(\mathbf {u}_0:=I_0\mathbf {u}\)be the coarse interpolation function obtained by above procedures. Then,
$$\begin{aligned} a(\mathbf {u}_0,\mathbf {u}_0)\le Ca(\mathbf {u},\mathbf {u}), \end{aligned}$$
whereCis independent of possible jumps of\(\kappa \), the mesh size, and is a uniformly bounded constant only related to\(\lambda _h^{i,L_i+1}\)which is the\((L_i+1)\)th eigenvalue for\(\omega _i\).

Proof

Since \(a(\mathbf {u}_0,\mathbf {u}_0)=\sum _{K\in {\mathcal {T}}_H}a_K(\mathbf {u}_0,\mathbf {u}_0)\), it suffice to estimate \(a_K(\mathbf {u}_0,\mathbf {u}_0).\) We find that
$$\begin{aligned} a_K(\mathbf {u}_0,\mathbf {u}_0)\lesssim\, & {} \frac{1}{\delta }\int _K\left| \mathrm {div}\pi _{RT}^h\left( \sum _{{\mathbf {x}}_i\in K}\left( I_{L_i}^{\omega _i}\mathbf {u}\right) \right) \right| ^2 \mathrm{d}x +\int _K\kappa ^{-1}\left| \pi _{RT}^h\left( \sum _{{\mathbf {x}}_i\in K}\left( I_{L_i}^{\omega _i}\mathbf {u}\right) \right) \right| ^2 \mathrm{d}x\\\lesssim & {} \frac{1}{\delta }\int _K\left| \mathrm {div}\left( \sum _{{\mathbf {x}}_i\in K}(I_{L_i}^{\omega _i}\mathbf {u})\right) \right| ^2 \mathrm{d}x +\int _K\kappa ^{-1}\left| \sum _{{\mathbf {x}}_i\in K}\left( I_{L_i}^{\omega _i}\mathbf {u}\right) \right| ^2 \mathrm{d}x, \end{aligned}$$
where the last inequality uses the commuting diagram property and weighted stability of \(\pi _{RT}^h\).
Then, using each K must be a subset of some \(\omega _i\), we obtain the following:
$$\begin{aligned} a_K(\mathbf {u}_0,\mathbf {u}_0)\lesssim & {} \sum _{{\mathbf {x}}_i\in K} a_{\omega _i}\left( I_{L_i}^{\omega _i}\mathbf {u},I_{L_i}^{\omega _i}\mathbf {u}\right) . \end{aligned}$$
(i) If \(\omega _i\) does not contain any contrast, then we may assume that \(\kappa (x)^{-1}=O(1)\). For the first term, using the definition of (4.2), the property of projections \(Q_{RT}^H\) and \(Q_{ND}^H\), Lemma 4.2, the spaces \(V^+_{ND}\) and \(V^+_{RT}\) are continuously embedded in \(H^1(\Omega )\), Remark 4.2, and we have the following:
$$\begin{aligned} a_{\omega _i}(I_{L_i}^{\omega _i}\mathbf {u},I_{L_i}^{\omega _i}\mathbf {u})\lesssim\, & {} a_{\delta }\left( I_{L_i}^{ \omega _i}\mathbf {u},I_{L_i}^{\omega _i}\mathbf {u}\right) \\\lesssim\, & {} a_{\delta }\left( Q_{RT}^H\left( P_h^{RT}\mathbf {v}^\bot \right) ,Q_{RT}^H\left( P_h^{RT}\mathbf {v}^\bot \right) \right) + \left\| \mathbf {curl}\left( Q_{ND}^H\left( P_h^{ND}\mathbf {w}^\bot \right) \right) \right\| _{0,\omega _i}^2\\\lesssim\, & {} \frac{1}{\delta }\left| P_h^{RT}\mathbf {v}^\bot \right| _{1,\omega _i}^2+\left\| P_h^{RT}\mathbf {v}^\bot \right\| _{0,\omega _i}^2+ \left| P_h^{ND}\mathbf {w}^\bot \right| _{1,\omega _i}^2\\\lesssim\, & {} \frac{1}{\delta }\left\| \mathrm {div}P_h^{RT}\mathbf {v}^\bot \right\| _{0,\omega _i}^2+\left\| \Theta _{\mathrm {div}}^\bot \mathbf {v}^\bot \right\| _{0,\omega _i}^2+ \left\| \mathbf {curl}\ P_h^{ND}\mathbf {w}^\bot \right\| _{0,\omega _i}^2\\\lesssim\, & {} \frac{1}{\delta }\left\| \mathrm {div}\mathbf {v}^\bot \right\| _{0,\omega _i}^2+\left\| \mathbf {v}^\bot \right\| _{0,\omega _i}^2+ \left\| \mathbf {curl}\ \mathbf {w}^\bot \right\| _{0,\omega _i}^2\\\lesssim\, & {} a_{\delta }(\mathbf {u},\mathbf {u})\\\lesssim\, & {} a_{\omega _i}(\mathbf {u},\mathbf {u}),\\ \end{aligned}$$
where \(a_{\delta }(\mathbf {u},\mathbf {u}):=\frac{1}{\delta }\Vert \mathrm {div}\mathbf {u}\Vert _{0,\omega _i}^2 +\Vert \mathbf {u}\Vert _{0,\omega _i}^2\), and the last second inequality uses the orthogonal property of Helmholtz decomposition.
(ii) If \(\omega _i\) does contain some inclusions or channels, then, by (4.3), we may get
$$\begin{aligned} a_{\omega _i}(I_{L_i}^{\omega _i}\mathbf {u},I_{L_i}^{\omega _i}\mathbf {u})\le & {} C a_{\omega _i}(\mathbf {u},\mathbf {u}), \end{aligned}$$
where C is a uniformly bounded constant which is only related to eigenvalues that we do not choose. The proof is completed.

Remark 4.3

Numerical experiments verify that the constant C is very insensitive to the coarse meshsize H and different contrast ratios of \(\kappa (x)\), if the eigenvectors to all near-vanishing eigenvalues are used to form our coarse space. This implies that all “bad” eigenvectors, indeed, need to be contained in our optimal spectral coarse space.

Remark 4.4

Note that we do not consider the local eigenvalue problem of \(H(\mathrm {div})\) formulation (1.2) for the coarse space construction, because the high-contrast coefficient \(\kappa \) is located in the lower order term of (1.2), the direct eigenvalue problems cannot capture the multiscale information, while the coefficient \(\kappa \) is located in the second-order term in our scalar eigenvalue problem. The eigenvalue is dominated by this term and the multiscale information is captured in the corresponding eigenfunctions.

So far, we have obtained the stability estimate of the coarse component; now, we have to deal with local spaces in our DD method.

4.2 The Stability of Local Spaces

In this section, we first derive a weighted Helmholtz decomposition under a mild assumption, then local stability can be derived naturally. In a fixed nonoverlapping subdomain \(\Omega _i\), we derive a weighted Helmholtz decomposition for the edge element space \(N_h(\Omega _i)\), then glue them to get a global weighted decomposition for \(N_h\). First, we recall some classical and useful results for orthogonal component in the classical Helmholtz decomposition.

Lemma 4.4

For\(\mathbf {w}^\bot \in H^\bot (\mathbf {curl};\Omega _i)\cup N_h^\bot (\Omega _i)\)and\(\mathbf {v}\in H^\bot (\mathrm {div};\Omega _i)\cup V_h^\bot (\Omega _i)\), we have the following estimate:
$$\begin{aligned}&\Vert \mathbf {w}^\bot \Vert \le C H_i\Vert \mathbf {curl}\ \mathbf {w}^\bot \Vert, \quad \forall \mathbf {w}^\bot \in H^\bot (\mathbf {curl};\Omega _i)\cup N_h^\bot (\Omega _i),\\&\Vert \mathbf {v}^\bot \Vert \le C H_i\Vert \mathrm {div}\mathbf {v}^\bot \Vert, \quad \forall \mathbf {v}^\bot \in H^\bot (\mathrm {div};\Omega _i)\cup V_h^\bot (\Omega _i), \end{aligned}$$
where\(H_i\)is the diameter of\(\Omega _i\)andCis independent of the mesh size.

Proof

See Lemma 4.4 in [28].

Next, we give an assumption concerning the distribution and value range of \(\kappa ^{-1}\). Under this assumption, we derive the weighted Helmholtz decomposition.

Assumption 4.1

In each subdomain \(\Omega _i\), \(\kappa ^{-1}\) is a constant or mild varying in each inclusion and channel. These inclusion or channel domains contained in \(\Omega _i\) are denoted by \(\{\Omega _r^0\}_{r=1}^{n_1}\), and \(n_1\) is the number of them. The rest domain, i.e., \(\Omega _i\backslash \{\Omega _r^0\}_{r=1}^{n_1}\), can be partitioned into some disjoint domains \(\{\Omega _r^0\}_{r=n_1+1}^{n_2}\), such that, for any fixed \(\Omega _r^0\), \(r=1,\ldots , n_1\), there are at most two domains \(\Omega _{r_1(r)}^0\) and \(\Omega _{r_2(r)}^0\), which both may be the union of some domains in \(\{\Omega _r^0\}_{r=n_1+1}^{n_2}\), such that
$$\begin{aligned} \kappa ^{-1}|_{\Omega ^0_{r_1(r)}\cup \Omega ^0_{r_2(r)}}> \kappa ^{-1}|_{\Omega ^0_{r}} \end{aligned}$$
(4.4)
with \(\overline{\Omega ^0_{r_1(r)}}\cap \overline{\Omega ^0_{r}} \ne \emptyset \), \(\overline{\Omega ^0_{r_2(r)}}\cap \overline{\Omega ^0_{r}} \ne \emptyset \) and \(\overline{\Omega ^0_{r_1(r)}}\cap \overline{\Omega ^0_{r_2(r)}} = \emptyset \).

Remark 4.5

Since we assume that \(\kappa =1\) on the background, while, \(\kappa \gg 1\) in the inclusions and channels, the inequality (4.4) is trivial if at most two domains \(\Omega ^0_{r_1(r)}\) and \(\Omega ^0_{r_2(r)}\) may always be found for any fixed \(r=1,\ldots ,n_1\). In Fig. 2, we use a typical subdomain which consists of inclusions and channels to show how to partition the background and choose \(\Omega ^0_{r_1(r)}\) and \(\Omega ^0_{r_2(r)}\) for any fixed \(r=1,\ldots ,n_1=3\). This example shows that this assumption is reasonable for many general high-contrast cases.

Fig. 2

For \(\Omega ^0_1\): \(\Omega ^0_{r_1(1)}=\Omega ^0_5\), \(\Omega ^0_{r_2(1)}=\Omega ^0_6\); for \(\Omega ^0_3\): \(\Omega ^0_{r_1(3)}=\Omega ^0_7\cup \Omega ^0_8\cup \Omega ^0_9\cup \Omega ^0_{11}\cup \Omega ^0_{12},\), \(\Omega ^0_{r_2(3)}=\Omega ^0_{13}\); for \(\Omega ^0_2\): \(\Omega ^0_{r_1(2)}=\Omega ^0_9\), \(\Omega ^0_{r_2(2)}=\Omega ^0_{12}\)

Under this assumption, a weighted Helmholtz decomposition for \(H(\mathbf {curl};\Omega _i)\) and the stability estimate of this decomposition are derived. We note that a similar result for the case of finite-layered coefficient distribution is given in [23].

Lemma 4.5

If Assumption 4.1holds, then any\( \mathbf {w}\in N_h(\Omega _i)\)admits a discrete weighted decomposition of the form:
$$\begin{aligned} \mathbf {w}=\nabla p_h +\mathbf {w}^\bot \end{aligned}$$
for some\(p_h\in S^h(\Omega _i)\)and\(\mathbf {w}^\bot \in N_h(\Omega _i)\), and\(\mathbf {w}\)satisfies
$$\begin{aligned} (\kappa ^{-1}\mathbf {w}^\bot ,\nabla q_h)=0,\quad \forall q_h\in S^h(\Omega _i). \end{aligned}$$
Moreover, \(\mathbf {w}^\bot \)has the following weighted stability estimate:
$$\begin{aligned} \frac{1}{H_i^2}\Vert \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i}^2\le C \left( 1+\log ^4\left( \frac{H_i}{h_i}\right) \right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\kappa ^{-1},\Omega _i}^2, \end{aligned}$$
(4.5)
where C is independent of the jumps of \(\kappa (x)\) and the mesh size.

Proof

First, in each domain of \(\{\Omega _r^0\}_{r=n_1+1}^{n_2}\), we have the following classical non-weighted discrete Helmholtz decomposition:
$$\begin{aligned} \mathbf {w}|_{\Omega _r^0}=\nabla p_{h,r}+\mathbf {w}^\bot _r, \quad r=n_1+1,\dots ,n_2, \end{aligned}$$
and \(\mathbf {w}^\bot _r\) satisfies
$$\begin{aligned} \frac{1}{H_i^2}\Vert \mathbf {w}^\bot _r\Vert _{0,\Omega _r^0}^2+\Vert \mathbf {curl}\ \mathbf {w}^\bot _r\Vert _{0,\Omega _r^0}^2 \lesssim \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2. \end{aligned}$$
(4.6)
We denote by \(\widetilde{p_{h,r}}\) the zero extension of \(p_{h,r}\) to the whole \(\Omega _i\), and by \(\widetilde{\mathbf {w}_{r}^\bot }\) the \(\mathbf {curl}\mathbf {curl}\)-extension of \(\mathbf {w}_{r}^\bot \) to \(\Omega _i\), where \(\mathbf {curl}\mathbf {curl}\)-extension operator is defined as [23]. Then, for each \(\Omega _r^0\), \(r=1,\ldots ,n_1\), without loss of generality, we assume that \(\Omega ^0_{r_1(r)}\) and \(\Omega ^0_{r_2(r)}\) satisfy Assumption 4.1; moreover, \(\overline{\Omega ^0_{r_1(r)}} \cap \overline{\Omega ^0_{r}}=\mathbf {V}_{r}\) and \(\overline{\Omega ^0_{r_2(r)}}\cap \overline{\Omega ^0_{r}}=E_r\). Then, we obtain the following function in each \(\Omega _r^0(r=1,\dots ,n_1)\):
$$\begin{aligned} \mathbf {w}^*_r:=\mathbf {w}|_{\Omega _r^0}-({\widetilde{{\mathbf {w}_{{r_1}(r)}}}}+ \widetilde{\mathbf {w}_{r_2(r)}})|_{\Omega _r^0}, \end{aligned}$$
(4.7)
where \(\widetilde{\mathbf {w}_{r_1(r)}}:=\nabla \widetilde{p_{h,r_1(r)}}+\widetilde{\mathbf {w}_{r_1(r)}^\bot }\) and \(\widetilde{\mathbf {w}_{r_2(r)}}:=\nabla \widetilde{p_{h,r_2(r)}}+\widetilde{\mathbf {w}_{r_2(r)}^\bot }\). According to Lemma 4.6 in [23], there exist \(p^*_r\in S^h(\Omega _r^0)\) and \(\mathbf {w}^*_r \in N_h(\Omega _r^0)\), such that
$$\begin{aligned} \mathbf {w}^*_r=\nabla p^*_r+\mathbf {w}^{*,\bot }_r \quad \text { on } \quad \Omega _r^0, \end{aligned}$$
and
$$\begin{aligned} p^*_r(\mathbf {V}_{r})=0,\quad \ p^*_r|_{E_r}=0 \text { on }E_r \text { and } \lambda _e(\mathbf {w}^{*,\bot }_r)=0, \quad \forall e \subset E_r, \end{aligned}$$
where \(\lambda _e(\mathbf {w}^{*,\bot }_r)\) is the degree freedom of \(\mathbf {w}^{*,\bot }_r\) in fine mesh edge \(e\in {\mathcal {T}}_h\). Moreover, \(\mathbf {w}^{*,\bot }_r\) has the following estimate:
$$\begin{aligned} \frac{1}{H_i^2}\Vert \mathbf {w}^{*,\bot }_r\Vert _{0,\Omega _r^0}^2\lesssim \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}^*_r\Vert _{0,\Omega _r^0}^2. \end{aligned}$$
(4.8)
So far, we have completed the decomposition as follows:
$$\begin{aligned} \mathbf {w}|_{\Omega _r^0}=\nabla p_{h,r}+\mathbf {w}^\bot _r, \quad r=n_1+1,\dots ,n_2, \end{aligned}$$
and
$$\begin{aligned} \mathbf {w}|_{\Omega _r^0}=\, & {} \nabla p^*_r+\mathbf {w}^{*,\bot }_r+(\widetilde{\mathbf {w}_{r_1(r)}}+\widetilde{\mathbf {w}_{r_2(r)}})|_{\Omega _r^0}\\=\, & {} \nabla (p^*_r+\widetilde{p_{h,r_1(r)}}+\widetilde{p_{h,r_2(r)}}) +\left( \mathbf {w}^{*,\bot }_r+\widetilde{\mathbf {w}_{r_1(r)}^\bot }+\widetilde{\mathbf {w}_{r_2(r)}^\bot }\right) ,\quad r=1,\ldots ,n_1. \end{aligned}$$
Next, we prove the estimate (4.5). For simplicity, we denote \(\beta :=\kappa ^{-1}\), then using estimate (4.6), we have the following:
$$\begin{aligned} \frac{1}{H_i^2}\Vert \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i}^2= & {} \frac{1}{H_i^2}\Vert \mathbf {w}^\bot \Vert _{0,\beta ,\Omega _i}^2\nonumber \\\lesssim\, & {} \frac{1}{H_i^2}\sum _{r=n_1+1}^{n_2}\beta _r\Vert \mathbf {w}_r^\bot \Vert _{0,\Omega _r^0}^2+ \frac{1}{H_i^2}\sum _{r=1}^{n_1}\beta _r\Vert \mathbf {w}_r^\bot \Vert _{0,\Omega _r^0}^2\nonumber \\\lesssim\, & {} \sum _{r=n_1+1}^{n_2}\beta _r\Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2+ \frac{1}{H_i^2}\sum _{r=1}^{n_1}\beta _r\Vert \mathbf {w}_r^\bot \Vert _{0,\Omega _r^0}^2; \end{aligned}$$
(4.9)
then, the estimate of the first term is already completed. We estimate the second sum term (\(r=1,\ldots ,n_1\)) using (4.7), (4.8), \(\mathbf {curl}\mathbf {curl}\)-extension estimate in [23] and Lemma 4.4. Actually, we have
$$\begin{aligned} I_r:= & {} \frac{1}{H_i^2}\beta _r\Vert \mathbf {w}_r^\bot \Vert _{0,\Omega _r^0}^2\\\lesssim\, & {} \frac{1}{H_i^2}\beta _r\left( \Vert \mathbf {w}^{*,\bot }_r\Vert _{0,\Omega _r^0}^2+ \Vert \widetilde{\mathbf {w}_{r_1(r)}^\bot }\Vert _{0,\Omega _r^0}^2 +\Vert \widetilde{\mathbf {w}_{r_2(r)}^\bot }\Vert _{0,\Omega _r^0}^2\right) \\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}^*_r\Vert _{0,\Omega _r^0}^2 +\beta _r \frac{1}{H_i^2}\left( \Vert \widetilde{\mathbf {w}_{r_1(r)}^\bot }\Vert _{0,\Omega _r^0}^2 +\Vert \widetilde{\mathbf {w}_{r_2(r)}^\bot }\Vert _{0,\Omega _r^0}^2\right) \\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2\\&+\beta _r \log ^2\left( \frac{H_i}{h_i}\right) \sum _{j=1}^2\left( \Vert \mathbf {curl}\ \widetilde{\mathbf {w}_{r_j(r)}^\bot }\Vert _{0,\Omega _r^0}^2+ \frac{1}{H_i^2}\Vert \widetilde{\mathbf {w}_{r_j(r)}^\bot }\Vert _{0,\Omega _r^0}^2\right) \\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2 + \beta _r \log ^3\left( \frac{H_i}{h_i}\right) \Vert \widetilde{\mathbf {w}_{r_2(r)}^\bot }\times n\Vert _{0,E_r}^2\\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2 + \beta _r \log ^4\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}_{r_2(r)}^\bot \Vert _{0,\Omega _{r_2(r)}^0}^2\\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2 + \frac{\beta _{r}}{\beta _{r_2(r)}}\beta _{r_2(r)}\log ^4\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _{r_2(r)}^0}^2\\\lesssim\, & {} \beta _r \log ^2\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _r^0}^2 + \beta _{r_2(r)}\log ^4\left( \frac{H_i}{h_i}\right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _{r_2(r)}^0}^2,\\ \end{aligned}$$
where the last inequality uses the fact that \(\frac{\beta _{r}}{\beta _{r_2(r)}}\le 1\). By inserting this estimate into inequality (4.9), we may obtain the following:
$$\begin{aligned} \frac{1}{H_i^2}\Vert \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i}^2\lesssim\, & {} \sum _{r=1}^{n_2}\beta _r\left( 1+\log ^4\left( \frac{H_i}{h_i}\right) \right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\Omega _{r}^0}^2\\\lesssim\, & {} \left( 1+\log ^4\left( \frac{H_i}{h_i}\right) \right) \Vert \mathbf {curl}\ \mathbf {w}\Vert _{0,\kappa ^{-1},\Omega _i}^2. \end{aligned}$$
The proof is completed.

Let \(\mathbf {v}=\mathbf {u}-\mathbf {u}_0\). Using Lemma 4.1, we may find \(\mathbf {w}\in N_h\) and \(\mathbf {v}^\bot \in V_h^\bot \), such that \(\mathbf {v}=\mathbf {curl}\ \mathbf {w}+\mathbf {v}^\bot \). As for \(\mathbf {w}\in N_h\), using Lemma 4.5, we may find \(\mathbf {w}^\bot \in N_h^\bot \) and \(p_h\in S^h\), such that the discrete weighted Helmholtz decomposition holds: \(\mathbf {w}=\nabla p_h+\mathbf {w}^\bot \). Therefore, we have the discrete decomposition of \(\mathbf {v}:\mathbf {v}=\mathbf {curl}\ \mathbf {w}^\bot +\mathbf {v}^\bot \) with \(\mathbf {w}^\bot \in N_h^\bot \) and \(\mathbf {v}^\bot \in V_h^\bot \).

Let \(\{\theta _i\}_{i=1}^N\) be a partition of unity, a set of piecewise linear functions subordinated to the overlapping \(\{\Omega _i^{'}\}_{i=1}^N\); such a set is constructed in [33]. We note that \(\Vert \theta _i\Vert _{\infty }\le \frac{C}{{\hat{\delta }}_i}\), \(0\le \theta _i\le 1\), and \(\begin{aligned}\sum _{i=1}^N\theta _i=1.\end{aligned}\) Here, the distance parameter \({{\hat{\delta }}_i}\) measure the width of the overlapping region \(\Omega _i^{'}\backslash \Omega _i\). We note that the special case where there exists \(C>0\), independent of i, such that \(\hat{\delta }_i\ge CH_i\), is referred to the generous overlap. Next, we first introduce a useful tool concerning about the interpolation stability of the partition of unity multiplied by some finite-element functions. They may be seen as the corollary of interpolation error.

Lemma 4.6

Let\(\mathbf {u}\in V_h\), \(\mathbf {v}\in N_h\), and\(\theta _i\)be defined as above. Then,
$$\begin{aligned}&\Vert \mathrm {div}(\Pi _h^{RT}( \theta _i\mathbf {u})\Vert _{0,\tau }^2\le C\Vert \mathrm {div}(\theta _i\mathbf {u})\Vert _{0,\tau }^2,\\&\Vert \Pi _h^{RT} (\theta _i\mathbf {u})\Vert _{0,\tau }^2\le C\Vert \theta _i\mathbf {u}\Vert _{0,\tau }^2,\\&\Vert \mathbf {curl}\ (\Pi _h^{ND} (\theta _i\mathbf {v}))\Vert _{0,\tau }^2\le C\Vert \mathbf {curl}\ (\theta _i\mathbf {v})\Vert _{0,\tau }^2, \end{aligned}$$
where\(\tau \)is any fine-scale element in\(\Omega _i^{'}\)andCdepends only on the shape regularity of the elements of\({\mathcal {T}}_H\)and\({\mathcal {T}}_h\).

Now, we are in a position to give our local decomposition and prove the stability of this local decomposition.

Lemma 4.7

Let\(\mathbf {v}_i=\pi _h^{RT}(\theta _i \mathbf {v}^\bot )\), \(\mathbf {w}_i=\pi _h^{ND}(\theta _i\mathbf {w}^\bot )\) and \({\hat{a}}_i(\mathbf {u},\mathbf {v}):=a_{\Omega _i^{'}}(\mathbf {u},\mathbf {v})\). Then,
$$\begin{aligned} \sum _{i=1}^N {\hat{a}}_i(\mathbf {v}_i,\mathbf {v}_i)\le C\left( \frac{H}{\hat{\delta }}\right) ^2 a(\mathbf {v}^\bot ,\mathbf {v}^\bot ), \end{aligned}$$
(4.10)
and
$$\begin{aligned} \sum _{i=1}^N {\hat{a}}_i(\mathbf {curl}\ \mathbf {w}_i,\mathbf {curl}\ \mathbf {w}_i)\le C \left( \frac{H}{\hat{\delta }}\right) ^2 \left( 1+\log ^4\left( \frac{H}{h}\right) \right) a\left( \mathbf {curl}\ \mathbf {w}^\bot ,\mathbf {curl}\ \mathbf {w}^\bot \right) , \end{aligned}$$
(4.11)
withCindependent of\(\kappa \), \(H_i\), \(h_i\), and\(\frac{H}{\hat{\delta }}:=\max \limits _{1\le i\le N}(\frac{H_i}{\hat{\delta }}_i)\), \(\frac{H}{h}:=\max \limits _{1\le i\le N}(\frac{H_i}{h_i})\).

Proof

We note that \(\mathrm {div}(\theta _i\mathbf {v}^\bot )=\nabla \theta _i\cdot \mathbf {v}^\bot +\theta _i\mathrm {div}\mathbf {v}^\bot .\) Using Lemmas 4.4 and 4.6, we have the following:
$$\begin{aligned} {{\hat{a}}_i}(\mathbf {v}_i,\mathbf {v}_i)=\, & {} \frac{1}{\delta }\Vert \mathrm {div}\mathbf {v}_i\Vert _{0,\Omega _i^{'}}^2+\Vert \mathbf {v}_i\Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\=\, & {} \frac{1}{\delta }\Vert \mathrm {div}\left( \pi _h^{RT}\left( \theta _i \mathbf {v}^\bot \right) \right) \Vert _{0,\Omega _i^{'}}^2 +\Vert \pi _h^{RT}\left( \theta _i \mathbf {v}^\bot \right) \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \frac{1}{\delta }\left( \Vert \nabla \theta _i \cdot \mathbf {v}^\bot \Vert _{0,\Omega _i^{'}}^2 +\Vert \theta _i\Vert _{\infty }^2\Vert \mathrm {div}\mathbf {v}^\bot \Vert _{0,\Omega _i^{'}}^2\right) +\Vert \pi _h^{RT}\left( \theta _i \mathbf {v}^\bot \right) \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \frac{1}{\delta }\left( \sum _{\Omega _{j}\cap \Omega _{i^{'}}\ne \emptyset } \frac{H_{j^2}}{{{\hat{\delta }}_j}^2}\left( \frac{1}{{H_j}^2}\Vert \mathbf {v}^\bot \Vert _{0,\Omega _j}^2\right) +\Vert \mathrm {div}\mathbf {v}^\bot \Vert _{0,\Omega _i^{'}}^2\right) +\Vert \pi _h^{RT}\left( \theta _i \mathbf {v}^\bot \right) \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, &{} \frac{H^2}{\hat{\delta }^2}\frac{1}{\delta }\sum _{\Omega _j\cap \Omega _i^{'}\ne \emptyset }\Vert \mathrm {div}\mathbf {v}^\bot \Vert _{0,\Omega _j}^2 +\Vert \pi _h^{RT}\left( \theta _i \mathbf {v}^\bot \right) \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2. \end{aligned}$$
For the second term, note that \(\kappa ^{-1}\) is a piecewise constant in \({\mathcal {T}}_h\), then we have
$$\begin{aligned} \Vert \pi _h^{RT}(\theta _i \mathbf {v}^\bot )\Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2= & {} \sum _{\tau \subset \Omega _i^{'}}\kappa ^{-1}_{\tau }\Vert \pi _h^{RT}(\theta _i \mathbf {v}^\bot )\Vert _{0,\tau }^2\\\le & {} \sum _{\tau \subset \Omega _i^{'}}\kappa ^{-1}_{\tau }\Vert (\theta _i \mathbf {v}^\bot )\Vert _{0,\tau }^2 \le \Vert \mathbf {v}^\bot \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2. \end{aligned}$$
Combining these two estimates, summing over the subdomains, and using Assumption 3.2, we obtain (4.10).
We now consider the estimate (4.11). Note that \(\mathbf {curl}(\theta _i\mathbf {w}^\bot ) =\nabla \theta _i\times \mathbf {w}^\bot +\theta _i\mathbf {curl}\ \mathbf {w}^\bot .\) Using Lemmas 4.5, 4.6, we have the following:
$$\begin{aligned} {{\hat{a}}_i}(\mathbf {curl}\ \mathbf {w}_i,\mathbf {curl}\ \mathbf {w}_i)=\, \, {} \Vert \mathbf {curl}\ \mathbf {w}_i\Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\=\, & {} \Vert \mathbf {curl}\ (\pi _h^{ND}(\theta _i\mathbf {w}^\bot ))\Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \Vert \mathbf {curl}\ (\theta _i\mathbf {w}^\bot )\Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \Vert \nabla \theta _i\Vert _{\infty }^2\left( \sum _{j:\Omega _j\cap \Omega _i^{'}\ne \emptyset }\Vert \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _j}^2\right) + \Vert \theta _i\Vert _{\infty }^2\Vert \mathbf {curl}\ \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \left( \sum _{j:\Omega _j\cap \Omega _i^{'}\ne \emptyset }\frac{H_j^2}{ \hat{\delta }_j^2}\frac{1}{H_j^2}\Vert \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _j}^2\right) + \Vert \mathbf {curl}\ \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2\\\lesssim\, & {} \frac{H^2}{\hat{\delta }^2}\left( \sum _{j:\Omega _j\cap \Omega _i^{'}\ne \emptyset }\left( 1+\log ^4\left( \frac{H_j}{h_j}\right) \right) \Vert \mathbf {curl}\ \mathbf {w}^\bot \Vert _{0, \kappa ^{-1},\Omega _j}^2\right) \\&+\Vert \mathbf {curl}\ \mathbf {w}^\bot \Vert _{0,\kappa ^{-1},\Omega _i^{'}}^2. \end{aligned}$$
Summing over the overlapping subdomains and using Assumption 3.2, we may obtain the estimate (4.11). The proof is finished.

5 Numerical Experiments

In this section, we apply our block-diagonal preconditioner \(D_{\delta }^{-1}\) to the equivalent transformed linear system (2.5) discretized by mixed finite elements. We choose the computational domain \(\Omega =(0,1)^2\), and the pair of lowest Raviart–Thomas element and piecewise constant element are used for velocity and pressure approximation, which is a stable pair and satisfies the well-known inf-sup condition. The domain \(\Omega \) is decomposed into nonoverlapping subdomains, then each one is extended with adding neighboring fine mesh to get overlapping subdomains with \(O(H_i)\) diameter size. The initial nonoverlapping subdomains are used to form the coarse mesh which readily satisfies Assumption 3.3.
Fig. 3

\(\kappa (x)\) with all inclusions for Example 5.1

We use the preconditioned MINRES method to solve the transformed linear system. We stop the iteration when the relative residual \(l_2\) norm has been reduced by a factor of \(10^{-6}\). In our DD method, generous overlap is used by choosing \(\hat{\delta }_i=\frac{H_i}{2}\). Software FreeFem++ [18] is used for our numerical simulation.

Example 5.1

In the first example, we use the coefficient depicted in Fig. 3, where \(\kappa (x)\) denotes a binary medium with \(\kappa (x)=\hat{\kappa }\) on a square area inclusion lying in the middle of each nonoverlapping subdomain, and at a distance of 3h both from the horizontal and the vertical edge of each subdomain, and \(\kappa (x) = 1\) in the rest of domain. Here, h denotes the fine mesh size, and we define the coarsening ratio as \(cr:=\frac{H}{h}\), which may measure the size of each local problem.

We first fix the coarse mesh size H and the coarsening ratio cr, and test our method with respect to the contrast with source term \(f=1\), homogeneous Dirichlet boundary condition is adopted, and two parameters \(\delta \) and \(\delta _1\) are chosen to be \(\delta =\delta _1=1\). Table 1 shows the result, and we can conclude that the iteration counts is quite independent of the jumps of coefficient \(\kappa (x)\).

Table 1

Number of preconditioned MINRES iterations (iter) with cr = 32, \(H=\frac{1}{8}\)

\(\hat{\kappa }\)

\(10^5\)

\(10^6\)

\(10^7\)

\(10^8\)

\(10^9\)

Iter

34

34

34

34

34

Next, we test the scalability of our method, fix the coarsening ratio cr and vary the coarse mesh size H, and Table 2 shows the result. We find that the iteration counts almost keep constant with decreasing the coarse mesh size H, and it means that the weak scalability of our method is quite good. In addition, for testing the impact of two parameters \(\delta \) and \(\delta _1\) on the behavior of convergence, we choose two different groups of parameters \(\delta _1\) and \(\delta _1\) to implement our method, respectively. We find that if two parameters \(\delta =\delta _1\) become small, the iteration count also slightly decreases. However, it seems that the decrease is slight, which also suggests that there is no need to choose them too small.
Table 2

Number of preconditioned MINRES iterations (iter) with fixed cr = 8, \(\hat{\kappa }=10^6\)

H

\(\frac{1}{8}\)

\(\frac{1}{12}\)

\(\frac{1}{16}\)

\(\frac{1}{20}\)

Iter (\(\delta =\delta _1=1\))

34

38

38

38

Iter (\(\delta =\delta _1=0.1\))

34

37

37

37

Example 5.2

In our second example, we deal with more complicated coefficients \(\kappa (x)\) which contains channels and inclusions in the computational domain \(\Omega \). We use the coefficients depicted in Fig. 4, where channels or inclusions are distributed in the whole domain without aligning with the subdomains or the coarse grid. As above example, we first fix \(cr=32\), \(H=\frac{1}{4}\), and two parameters \(\delta =\delta _1=1\), and we test our method with respect to the contrast. Table 3 shows the result. We may conclude that the iteration counts are also independent of the contrast, even regular channels or not are included in the coefficient \(\kappa (x)\).

Fig. 4

Left: permeability field with regular inclusions and channels. Right: permeability field with many irregular inclusions and channels

Table 3

Number of preconditioned MINRES iterations (iter) with cr = 32, \(H=\frac{1}{8}\)

\(\hat{\kappa }\)

\(10^5\)

\(10^6\)

\(10^7\)

\(10^8\)

\(10^9\)

Iter (left case)

69

70

70

70

70

Iter (right case)

53

53

53

53

53

Table 4

Number of preconditioned MINRES iterations (iter) with fixed cr = 8

H

\(\frac{1}{8}\)

\(\frac{1}{12}\)

\(\frac{1}{16}\)

\(\frac{1}{20}\)

\(\frac{1}{22}\)

Iter (right case:\(\delta =\delta _1=1\))

70

73

81

85

85

Iter (right case:\(\delta =\delta _1=0.1\))

65

69

78

79

79

Iter (left case:\(\delta =\delta _1=1\))

53

63

69

71

71

Iter (left case:\(\delta =\delta _1=0.1\))

51

61

63

68

68

Next, we test the scalability of our method to this case. We fix \(cr=8\) and vary the coarse mesh size H and two parameters \(\delta , \delta _1\). Table 4 shows the result. We may find that the iteration counts are quite stable with decreasing the coarse mesh size and only increase very slightly. For more complicated case, the right case, the weak scalability is also quite good. Then, we also two different choices of parameters \(\delta \) and \(\delta _1\). When \(\delta =\delta _1\) becomes smaller, note that the iteration count is slightly decreasing. It seems that these two parameters have small effect on the convergence behavior and we also do not need them to approach zero theoretically. This example indicates that, even in the presence of high contrast within the subdomains, our preconditioner is still robust.
Fig. 5

\(\kappa (x)\) with a realization of the random distribution for the Example 5.3 with \(\frac{\kappa _{\max }}{\kappa _{\min }}\sim 10^6\)

Example 5.3

In this example, we consider a random distributed permeability in the model. We choose the permeability \(\kappa (x)\) as a realization of a random field, namely, \(\mathrm{log}(\kappa (x))\) is a realization of a Gaussian random field with a spherical variance function (Fig. 5). The spherical covariance function has a parameter \(\theta \), and a larger one leads to the higher contrast. We test our method with respect to the different parameters \(\theta \), namely, to different contrasts. Table 5 shows the results, which implies that our method is stable with the different contrast even in random cases. Note that, in this case, the partition of unity is chosen to be the multiscale partition of unity(POU), which had been used in [16] to reduce the dimension of the spectral coarse space, since one multiscale POU function can capture the multiscale information of all inclusions in one subdomain.

Table 5

Number of preconditioned MINRES iterations (iter) with cr = 32, \(H=\frac{1}{8}\)

\(O(\frac{\kappa _{\max }}{\kappa _{\min }})\)

\(10^5\)

\(10^6\)

\(10^7\)

\(10^8\)

\(10^9\)

Iter

57

57

57

57

57

6 Conclusions

In this paper, with the help of the equivalent transform, the mixed finite-element discretization is transformed to a new saddle-point problem. For this problem, an abstract preconditioner was discussed. The essence of this abstract preconditioner is about solving a multiscale \(H(\mathrm {div})\) system. We succeeded in using the domain decomposition method to construct a robust preconditioner for this system. The core of our DD method is our newly developed coarse space, which is formed by the selected solutions of mixed formulation of the generalized eigenvalue problem. The stability of coarse interpolation is derived and the stability of local components is obtained by a weighted Helmholtz decomposition in each subdomain. The numerical results show that our DD preconditioners are quite robust with respect to the high contrast and the coarse meshsize with fixed subdomain DOFs.

Notes

Acknowledgements

The authors would like to thank the editor and anonymous referees who made many helpful comments and suggestions which lead to an improved presentation of this paper. The work of the second author was supported by the National Natural Science Foundation of China (No. 11671302).

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Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.LSEC, Institute of Computational Mathematics, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesTongji UniversityShanghaiChina

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