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

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## 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.

*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:

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

*f*satisfies the following compatibility condition:

*V*and

*W*are the spaces \(H_0(\mathrm {div};\Omega )\) and \(L^2(\Omega )/{\mathbb {R}}\), respectively. Here:

*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.

*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*:

*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*

*contains the negative eigenvalues, and*

*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

*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:

### Assumption 3.1

*x*belongs to \(\Omega _{i}^{'}\), then

*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

*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.

### Lemma 3.1

*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*

*then*

### Lemma 3.2

*If Assumption*3.2

*is satisfied and the exact coarse bilinear form and local bilinear forms are used, namely*:

*and*

*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*

*T*

*is 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*

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.

*l*th 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:

### 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

### 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 Lemma*3.1, *our DD preconditioner for*\(M_{\delta }\), *denoted by*\(M_{DD}^{-1}\), *is quasi-optimal with*\(C_0^2=C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))\), *where**C**is 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

## 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 the*

*i*

*th*

*eigenvalue*\(\lambda _h^i\)

*for the eigenvalue problem*(3.3)–(3.4),

*then the following equality holds*:

### Proof

### 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

*If*\(\omega _i\)

*is convex, then we have discrete decompositions for Raviart–Thomas space*\(V_h\)

*and edge element space*\(N_h\).

*Thus,*

*and*

*where*\( N_h^\bot (\omega _i)\)

*and*\(V_h^\bot \)

*are orthogonal complements, respectively.*

### Definition 4.1

### 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*:

*where*

*C*

*is 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 \(\mathbf {u}\in V_h\).

### 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,*

*where*

*C*

*is 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

*K*must be a subset of some \(\omega _i\), we obtain the following:

*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*:

*where*\(H_i\)

*is the diameter of*\(\Omega _i\)

*and*

*C*

*is 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

### 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.

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*4.1

*holds, then any*\( \mathbf {w}\in N_h(\Omega _i)\)

*admits a discrete weighted decomposition of the form*:

*for some*\(p_h\in S^h(\Omega _i)\)

*and*\(\mathbf {w}^\bot \in N_h(\Omega _i)\)

*, and*\(\mathbf {w}\)

*satisfies*

*Moreover,*\(\mathbf {w}^\bot \)

*has the following weighted stability estimate*:

*C*is independent of the jumps of \(\kappa (x)\) and the mesh size.

### Proof

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,*

*where*\(\tau \)

*is any fine-scale element in*\(\Omega _i^{'}\)

*and*

*C*

*depends 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,

*and*

*with*

*C*

*independent 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

## 5 Numerical Experiments

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 3*h* 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 \(\kappa (x)\).

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 |

*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.

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

| \(\frac{1}{8}\) | \(\frac{1}{12}\) | \(\frac{1}{16}\) | \(\frac{1}{20}\) |
---|---|---|---|---|

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 \(\kappa (x)\).

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 |

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

| \(\frac{1}{8}\) | \(\frac{1}{12}\) | \(\frac{1}{16}\) | \(\frac{1}{20}\) | \(\frac{1}{22}\) |
---|---|---|---|---|---|

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 |

*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.

### 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.

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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