# Overlapping Schwarz methods with adaptive coarse spaces for multiscale problems in 3D

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## Abstract

We propose two variants of the overlapping additive Schwarz method for the finite element discretization of scalar elliptic problems in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces. In the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges. The functions that constitute the coarse spaces are chosen adaptively, and they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.

## Mathematics Subject Classification

65N55 65N30 65F08## 1 Introduction

Additive Schwarz methods are considered among the most popular domain decomposition methods for solving partial differential equations because of their simplicity, and inherently parallel algorithms, cf. [8, 25, 33, 36]. We consider in this paper variants of the overlapping additive Schwarz method as preconditioners for solving scalar elliptic equations with highly varying coefficient in the space \(\mathbb {R}^3\). Based on the idea of adaptively constructing the coarse space through solving generalized eigenvalue problems in the lower dimensions, we propose two variants of the algorithm in three dimensions. The resulting system has a condition number which is inversely proportional to \(\lambda ^{*}\), where \(\lambda ^{*}\) is the threshold for coarse basis selection of eigenfunctions corresponding to eigenvalues below \(\lambda ^{*}\). The threshold is used as a parameter for the coarse space in the algorithm. The term “lower dimensions” refers to the fact that the eigenvalue problems are solved either on subdomain faces or edges. The methods are effective, inherently parallel, and simple to construct. Additive Schwarz method with adaptive coarse space was recently considered using a different idea based on solving generalized eigenvalue problem in the overlap, cf. [8, 9, 26].

To see the motivation behind such an approach, we take a brief look into the steps of deriving convergence estimates for two-level overlapping Schwarz methods in their abstract Schwarz framework (cf. [25, 33, 36] for the abstract framework). These methods are analyzed in the abstract Schwarz framework. And the abstract Schwarz framework is based on the assumption that any function *u* in the finite element space can be split into its coarse component \(u_0\) and local components associated with the subdomains, and that this splitting is stable with respect to the energy norm. The condition number bound of the preconditioned system then depends explicitly on the constant \(C_0^2\) which enters into the stability estimate, cf. [25, 33, 36]. Consequently, how robust the method will be with respect to the mesh parameters and the varying coefficients, depends on how robust \(C_0^2\) is concerning those parameters and varying coefficients. Its already known, cf. e.g., [15], that, even with highly varying coefficients as long as the variations are strictly inside the subdomains, this constant is not depend on the variation. The difficulty, however, arises when we have to deal with coefficients which vary along subdomain boundaries. In which case, the crucial term which needs to be bounded in the stability estimate is the boundary term \(\sum _i C/h^2\Vert \sqrt{\alpha }(u-u_0)\Vert ^2_{L^2(\varOmega ^h_i)}\). Here \(\varOmega ^h_i\) denotes the layer of all elements along the subdomain boundary \(\partial \varOmega _i\), \(\alpha \) the varying coefficient, and *C* a constant. Because the \(\alpha \) is varying, estimating the \(L^2\) term say using the Poincaré or a weighted Poincaré inequality introduces the contrast into the estimate, i.e., the ratio between the largest and the smallest values of \(\alpha \), unless strong assumptions on the distribution of the coefficient are made, cf. [31]. A standard multiscale coarse space alone, cf. [15, 29], cannot make the method robust with respect to the contrast unless some form of enrichment of the coarse space is used that can capture the strong variations along the subdomain boundary and improves the approximation. Including selected eigenfunctions of properly defined eigenvalue problems over the subdomain boundary into coarse spaces, enable us to capture those variations and provide estimates with constants independent of the contrast, which is the primary motivation behind the methods presented here.

The idea of adaptively constructing coarse spaces using eigenfunctions of certain eigenvalue problems has already attracted much interest in recent years. Some earlier works in this direction are found in [1, 2] around the Neumann-Neumann type substructuring domain decomposition method, and [6] around the algebraic multigrid method, although they were not addressed to solve multiscale problems. In the context of multiscale problems this idea has only recently started to emerge, with its first appearance in [12, 13], as well as in [26], and later on in [7, 9, 10, 11, 14, 35] in the additive Schwarz framework. This idea of adaptively constructing coarse space, in other words, the primal constraints, for the FETI-DP and BDDC substructuring domain decomposition methods has also been developed and extensively analyzed, cf. [17, 19, 20, 24, 34] in 2D and [5, 16, 18, 27, 30] in 3D.

The methods above are preconditioners to iterative methods, designed to obtain fine-scale solutions to elliptic equations. In this recent development of coarse space enrichment, it is important to take note of the parallel development in methods designed to solve elliptic equations directly by a coarse approximation, such as the Reduced Basis method, see, e.g., [4, 28], cf. also [32], and the Localized Orthogonal Decomposition method, see, e.g., [21, 23]. The constructions of these coarse approximations and the constructions of the coarse spaces in the iterative methods show many similarities.

The rest of this paper is organized as follows: In Sect. 2 we define our problem and its discrete formulation. In Sect. 3 we describe the overlapping Schwarz preconditioner, in Sect. 4 we introduce the two new coarse spaces, and in Sect. 5 we establish the convergence estimate for the preconditioned system. Finally, in Sect. 6, we present some numerical results of our method.

## 2 Differential problem and discrete formulation

*j*. The resulting symmetric system is in general very ill-conditioned; any standard iterative method, like the Conjugate Gradients, e.g. cf [22], may perform poorly due to the ill-conditioning of the system. The aim is to introduce an additive Schwarz preconditioner for the original problem (3) to obtain a well-conditioned system for which the convergence of the conjugate gradient method is independent of any variations in the coefficient, thereby improving the overall performance.

## 3 Additive Schwarz method

The two-level additive Schwarz method is well known and well understood in the literature, and we refer to [36, Chapter 3] for an overview of the method.

### 3.1 Geometric structures

Let \(\varOmega \) be partitioned into a set of *N* nonoverlapping subdomains (or generalized subdomains), \(\{\varOmega _i\}_{i=1}^N\), such that each \(\overline{\varOmega }_i\) (the closure of \({\varOmega }_i\)) is a sum of elements (fine elements) from \(\mathcal {T}_h\), \({\varOmega }_i\cap {\varOmega }_j = \emptyset \) for \(i\ne j\), and \(\overline{\varOmega } = \cup _{i\in I}\overline{\varOmega }_i\). The intersection between two closed subdomains is either an empty set, a closed face (or a closed generalized face which is a sum of closures of fine element faces), a closed edge (or a closed generalized edge which is a sum of closures of fine element edges), or a vertex. Thus, the triangulation \(\mathcal {T}^h(\varOmega )\) is aligned with the subdomains \(\varOmega _i\). Each subdomain \(\varOmega _i\) inherits its own local triangulation \(\mathcal {T}^h(\varOmega _i)=\{\tau \in \mathcal {T}^h(\varOmega ):\tau \subset \overline{\varOmega }_i\}\) such that \(\bigcup _i\mathcal {T}^h(\varOmega _i)=\mathcal {T}^h(\varOmega )\). The corresponding set of overlapping subdomains \(\{\varOmega _i'\}_{i\in I}\) is then defined as follows: extend each subdomain \(\varOmega _i\) to \(\varOmega _i'\), by adding to \(\varOmega _i\) a layer of elements, i.e. sum of \(\overline{\tau }_k\in \mathcal {T}^h(\varOmega )\) such that \(\overline{\tau }_k\cap \partial \varOmega _{i}\ne \emptyset \).

In the same way as each subdomain inherits a 3D triangulation, each face \(\mathcal {F}_{k l}\) inherits a 2D triangulation which we denote by \(\mathcal {T}_h(\mathcal {F}_{k l})\), and each edge \(\mathcal {E}_k\) inherits a 1D triangulation which we denote by \(\mathcal {T}_h(\mathcal {E}_k)\). For each of the structures, \(\varOmega \), \(\overline{\varOmega }\), \(\varOmega _k\), \(\overline{\varOmega }_k\), \(\mathcal {F}\), \(\mathcal {E}\), and \(\mathcal {W}\), we use \(\varOmega _h\), \(\overline{\varOmega }_h\), \(\varOmega _{k,h}\), \(\overline{\varOmega }_{k,h}\), \(\mathcal {F}_h\), \(\mathcal {E}_h\), and \(\mathcal {W}_h\), respectively, to denote the corresponding set of nodal points (vertices of the elements of \(\mathcal {T}_h\)) which are on the structure.

### 3.2 Space decomposition, subproblems, and preconditioned system

*N*local subspaces

## 4 Coarse spaces with enrichment

For our the additive Schwarz method we introduce two alternative coarse spaces. The first one is based on enriching a wire basket coarse space, and the second one is based on enriching a vertex based coarse space.

Discrete harmonic extensions are used to extend our functions from subdomain boundaries into subdomains. We define our discrete harmonic extension operator below.

### Definition 1

*u*restricted to \(\partial \varOmega _k\). We define \(\mathcal {H}_k:V_h(\varOmega _k)\rightarrow V_h(\varOmega _k)\) as the discrete harmonic extension operator in \(\varOmega _k\) as follows,

Functions that are locally discrete harmonic have the minimum energy property locally. This property is well known, but for completeness we restate it in the following lemma, cf. [36] for further details on discrete harmonic extensions.

### Lemma 1

### 4.1 Wire basket based coarse space

The wire basket based coarse space consists of basis functions, one for each node in the wire basket \(\mathcal {W}\), plus eigenfunctions corresponding to the first few eigenvalues of generalized eigenvalue problems associated with the faces, cf. Definition 3.

### Definition 2

For each face \(\mathcal {F}_{k l}\), we define the following generalized eigenvalue problem.

### Definition 3

### 4.2 Vertex based coarse space

The vertex based coarse space consists of basis functions, one for each node in \(\mathcal {V}\), plus eigenfunctions corresponding to the first few eigenvalues of generalized eigenvalue problems associated with the edges, cf. Definition 5, and the faces, cf. Definition 6.

### Definition 4

For each edge \(\mathcal {E}_k\), we define the following generalized eigenvalue problem.

### Definition 5

Note that, by definition, the bilinear forms \(a_{\mathcal {E}_{k}}(\cdot ,\cdot )\) and \(b_{\mathcal {E}_k}(\cdot ,\cdot )\) in (16) are both symmetric and positive definite on \(V^0_h(\mathcal {E}_k)\).

For each face \(\mathcal {F}_{k l}\), we now define the following generalized eigenvalue problem.

### Definition 6

Note that, by definition, the bilinear form \(b_{\mathcal {F}_{k l}}(\cdot ,\cdot )\) is symmetric and positive definite on \(V_h^0(\mathcal {F}_{k l})\), while the bilinear form \(a_{\mathcal {F}_{k l,I}}(\cdot ,\cdot )\) in (18) is symmetric and positive semidefinite on this space. We know its kernel, it is the one dimensional space containing functions that are constant over \(\mathcal {F}_{k l}^I\), consequently, we have \(0=\lambda _{\mathcal {F}_{k l, I}}^1< \lambda _{\mathcal {F}_{k l, I}}^2 \le \cdots \le \lambda _{\mathcal {F}_{k l, I}}^{\hat{M}}\).

### Remark 1

It should be pointed out here that many of the calculations indicated by the use of piecewise discrete harmonic extensions in the definitions above are redundant since, in practice, they are often trivial extensions of zero on subdomain boundaries.

## 5 Convergence estimate for the preconditioned system

In this section, we prove that the condition number of our preconditioned system can be kept low, and independent of the contrast if our coarse space enrichments are appropriately chosen. The main result is stated in Theorem 1.

### Theorem 1

The proof is based on the abstract Schwarz framework, cf. e.g. [25, 33, 36] and is given at the end of the section. The following lemmas are required for the proof.

### Remark 2

In case of constant \(\alpha \) and regular mesh the minimal eigenvalue of the problems are of \(\mathcal {O}((\frac{h}{H})^2)\) and the maximal eigenvalue is of \(\mathcal {O}(1)\).

### Lemma 2

*u*is zero at \(\mathcal {W}_h\), then

*u*is zero at all vertices of \(\partial \varOmega _k\) then

### Proof

*u*on \(\partial \varOmega _k\) and zero at the interior nodes \(\varOmega _{k,h}\). The other case means that \(u=\hat{u}\). Consequently,

*h*boundary layer that is the sum of elements of \(\mathcal {T}_h(\varOmega _k)\) that touch (has a vertex on) the boundary \(\partial \varOmega _k\). We used a local inverse inequality and the discrete equivalence of the \(L^2\) norm on each \(\tau \). Finally, utilizing the fact that \(\hat{u}\) is zero at the interior nodal points and taking the maximum over \(\alpha _\tau \) such that \(x\in \partial \tau \) we get

### Remark 3

The constant in the estimate of Lemma 2 equals \(C_1\,C_2\,C_3\), where \(C_1\) is the constant of the inverse inequality \(|u|_{H^1(\tau )}^2\le C_1 h^{-2} \Vert u\Vert _{L^2(\tau )}^2, \; u\in V^h, \;\tau \in T_h\), \(C_2\) is the squared constant of the inequality stating local equivalence of the \(L^2\) norm to the discrete local nodal \(l_2\) norm, i.e. \(\Vert u\Vert _{L^2(\tau )}^2 \le C_2 h \sum _{x_k \in \partial \tau } |u(x_k)|^2, \; u\in V^h,\; \tau \in T_h\), and \(C_3\) is the maximum over all \(x\in \partial \varOmega _{k,h}\) of the number of \(\tau \in T_h(\varOmega _k)\) such that *x* is a vertex of \(\tau \).

We see from the proof that we could define the coefficient \(\alpha _x\) as equal to the sum of \(\alpha _x\) instead of taking the maximum of them.

We restate [14, Lemma 2.2] which contains important estimates for the eigenfunctions found in (11) and (16).

### Lemma 3

*V*be a finite dimensional real space and consider the generalized eigenvalue problem: Find the eigenpair \((\lambda _k,\xi _k) \in \mathbb {R}\times V\) such that \(b(\xi _k,\xi _k)=1\) and

We have the following lemma estimating the coarse space interpolant.

### Lemma 4

### Proof

*x*,

*y*are vertices of 3D element \(\tau \)) we get

*x*on a subdomain face.

### Lemma 5

### Proof

*x*is a face node, or zero when \(x\in \partial \varOmega _k\cap \mathcal {W}_h\) and \(x\in \varOmega _{k,h}\). By Lemma 2 and (35) we thus get

We have the following lemma.

### Lemma 6

### Proof

*u*linear, \(|u|_{H^1(\tau _e)}^2\) is equivalent to \(h^{-1}|u(x)-u(y)|^2\) (

*x*,

*y*are the ends of the 1D element \(\tau _e\)), we get

*x*,

*y*of an 1D edge element \(\tau _e\). Note that \(h|u(x)-u(y)|^2 \preceq \int _\tau |\nabla u|^2\, dx \) if

*x*,

*y*are vertices of \(\tau \in \mathcal {T}_h\). Thus we get

*u*with \(\hat{u}\) and get

### Lemma 7

### Proof

*w*at interior nodes \(\varOmega _{k,h}\), \(\frac{1}{2} w\) at the nodes \(\mathcal {F}_{k l,h}\) on each face \(\mathcal {F}_{k l}\) of \(\varOmega _k\), \(\frac{1}{n(\mathcal {E}_i)}w\) at the nodes \(\mathcal {E}_{i,h}\) on each edge \(\mathcal {E}_i\) of \(\varOmega _k\), and zero at all remaining nodal points of \(\varOmega _h\). Here \(n(\mathcal {E}_i)\) is the number of subdomains that share the edge \(\mathcal {E}_i\). As in the proof of Lemma 5, we can write

Combining those estimates, we get the proof. \(\square \)

The next and final lemma provides estimates for the stability of decomposition for the two preconditioners presented in this paper, which are required in the proof of Theorem 1.

### Lemma 8

### Proof

For the wire basket based coarse space, let \(u_0^{(1)}=I_0^{\mathcal W}u\) and \(u_i^{(1)}=I_h(\theta _i(u-I_0^{{\mathcal {W}}}u))\). The corresponding statement of the lemma then follows immediately from the Lemmas 4 and 5.

For the vertex based coarse space, let \(u_0^{(2)}=I_0^{{\mathcal {V}}}u\) and \(u_i^{(2)}=I_h(\theta _i(u-I_0^{{\mathcal {V}}}u))\). It’s statement of the lemma then follows immediately from the Lemmas 6 and 7. \(\square \)

### Proof of Theorem 1

## 6 Numerical results

In this section, we present four numerical tests. The test problems are defined on a unit cube domain, with a Dirichlet boundary condition and a constant right-hand side \(f(x) = 100\). For all the test problems, we use coefficient distributions with inclusions across subdomain boundaries (faces and edges), as is illustrated in the Figs. 2 and 3, and channels illustrated in Fig. 4. The coefficient either has a background-value of 1, or a significant value of \(10^6\). For any given distribution the region with high-value coefficient is independent of mesh parameters. We use the PCG method as a solver, with a stopping condition when the residual norm is reduced by a factor \(10^{-6}\).

The algorithms have been implemented in MATLAB, using the functions meshgrid and delaunayTriangulation for discretization, and routines from PDEToolbox for assembling the stiffness and mass matrices. For the iterative solver, we used pcgeig, an extension of MATLAB’s pcg routine, available at *m2matlabdb.ma.tum.de*. The pcgeig solver returns an estimate of the condition number for the preconditioned system. We use the built-in function eigs for solving the eigenvalue problems.

*c*. Another reasonable choice of threshold is the smallest non zero eigenvalue of the eigenvalue problems with the uniform background-value distribution. We refer to [18] for other threshold choices.

Condition number estimates and iteration counts (inside brackets) for Test 1

\(\alpha \) distribution | No enrichment | Face enrichment with \(\lambda _{\mathcal {F}_I} < \lambda ^*\) | ||
---|---|---|---|---|

\(\lambda ^*=0.0375\) | \(\lambda ^*=0.0187\) | \(\lambda ^*=0.0094\) | ||

\(H/h=8\) | \(H/h=8\) | \(H/h=16\) | \(H/h=32\) | |

Vertex based coarse space | ||||

Face 1 | \(3.53\mathrm {e}{5} ~ (41)\) | 20.87 (23) | 39.27 (29) | 75.41 (41) |

Face 2 | \(1.90\mathrm {e}{5} ~ (40)\) | 19.72 (22) | 37.08 (29) | 70.75 (42) |

Face 3 | \(2.22\mathrm {e}{5} ~ (42)\) | 18.48 (28) | 32.60 (30) | 62.00 (41) |

Face 4 | \(2.13\mathrm {e}{5} ~ (114)\) | 21.46 (23) | 38.62 (30) | 72.00 (39) |

Face 1–4 | \(3.42\mathrm {e}{5} ~ (170)\) | 18.49 (29) | 24.52 (30) | 45.29 (39) |

\(\alpha \) distribution | No enrichment | Face enrichment with \(\lambda _{\mathcal {F}} < \lambda ^*\) | ||
---|---|---|---|---|

\(\lambda ^*=0.075\) | \(\lambda ^*=0.035\) | \(\lambda ^*=0.0187\) | ||

\(H/h=8\) | \(H/h=8\) | \(H/h=16\) | \(H/h=32\) | |

Wirebasket based coarse space | ||||

Face 1 | \(1.14\mathrm {e}{4} ~ (34)\) | 12.56 (19) | 19.55 (24) | 34.92 (32) |

Face 2 | \(9.23\mathrm {e}{1} ~ (32)\) | 12.84 (21) | 20.36 (30) | 35.18 (33) |

Face 3 | \(2.50\mathrm {e}{5} ~ (42)\) | 12.93 (22) | 20.37 (27) | 35.55 (33) |

Face 4 | \(1.21\mathrm {e}{5} ~ (79)\) | 12.90 (20) | 19.54 (25) | 34.75 (33) |

Face 1–4 | \(2.50\mathrm {e}{5} ~ (152)\) | 12.41 (24) | 19.52 (32) | 30.65 (34) |

*h*. We observe in the first column that the jump in coefficient is reflected in the condition numbers. From the last three columns, we observe that both our suggested preconditioners improve the performance of the PCG method.

Showing the smallest eigenvalues (column wise) for the eigenvalue problems on the four different faces, with inclusions, for \(H/h=32\)

Face 1 | Face 2 | Face 3 | Face 4 | ||||
---|---|---|---|---|---|---|---|

\(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) | \(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) | \(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) | \(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) |

0.0 | \(6.1\mathrm {e}{-}8\) | 0.0 | \(5.3\mathrm {e}{-8}\) | 0.0 | \(5.0\mathrm {e}{-8}\) | 0.0 | \(6.5\mathrm {e}{-8}\) |

\(6.1\mathrm {e}{-8}\) | \(1.1\mathrm {e}{-7}\) | \(5.6\mathrm {e}{-8}\) | \(1.2\mathrm {e}{-7}\) | \(1.6\mathrm {e}{-7}\) | \(1.7\mathrm {e}{-7}\) | \(7.9\mathrm {e}{-8}\) | \(1.6\mathrm {e}{-7}\) |

\(7.5\mathrm {e}{-8}\) | \(1.2\mathrm {e}{-7}\) | \(1.7\mathrm {e}{-7}\) | \(2.0\mathrm {e}{-7}\) | \(5.4\mathrm {e}{-3}\) | \(5.4\mathrm {e}{-3}\) | \(7.9\mathrm {e}{-8}\) | \(1.6\mathrm {e}{-7}\) |

\(\mathbf {2.5}\mathrm {e}{\mathbf{-2}}\) | \(\mathbf {1.1}\mathrm {e}{\mathbf{-1}}\) | \(\mathbf {1.3}\mathrm {e}{\mathbf{-2}}\) | \(1.3\mathrm {e}{-2}\) | \(5.4\mathrm {e}{-3}\) | \(5.4\mathrm {e}{-3}\) | \(1.6\mathrm {e}{-7}\) | \(2.4\mathrm {e}{-7}\) |

\(1.3\mathrm {e}{-2}\) | \(\mathbf {1.9}\mathrm {e}{\mathbf{-2}}\) | \(\mathbf {1.9}\mathrm {e}{\mathbf{-2}}\) | \(2.3\mathrm {e}{-7}\) | \(2.6\mathrm {e}{-7}\) | |||

\(1.3\mathrm {e}{-2}\) | \(2.3\mathrm {e}{-7}\) | \(2.6\mathrm {e}{-7}\) | |||||

\(\mathbf {5.0}\mathrm {e}{\mathbf{-2}}\) | \(2.8\mathrm {e}{-7}\) | \(3.2\mathrm {e}{-7}\) | |||||

\(2.9\mathrm {e}{-7}\) | \(3.2\mathrm {e}{-7}\) | ||||||

\(3.7\mathrm {e}{-7}\) | \(3.8\mathrm {e}{-7}\) | ||||||

\(\mathbf {2.8}\mathrm {e}{\mathbf{-2}}\) | \(\mathbf {1.4}\mathrm {e}{\mathbf{-1}}\) |

To have an idea of the number of eigenfunctions added to the coarse spaces in Test 1, we have listed the lowest eigenvalues of the generalized eigenvalue problems for the case \(\frac{H}{h} = 32\), in Table 2. Each distribution in the first row of Fig. 2 represents inclusions on one face. The eigenvalues in each column of Table 2 are from eigenvalue problems on each of these faces respectively. Any eigenvalue printed in boldface is an eigenvalue above the threshold. The first few eigenvalues in each column are several magnitudes lower than the threshold. In all observed cases, the number of eigenvalues that are several magnitudes lower than the threshold is equal to the number of separate inclusions on that particular face. The number of eigenfunctions included into the coarse space in each case is low.

Condition number estimates and iteration counts (inside brackets) for Test 2

\(\alpha \) distribution | No enrichment | Face enrichment with \(\lambda _{\mathcal {F}_I} < 0.0094\) | ||
---|---|---|---|---|

Cap \(=\) 1 | Cap \(=\) 2 | Cap \(=\) 3 | No cap | |

\(H/h=32\) | \(H/h=32\) | \(H/h=32\) | \(H/h=32\) | |

Vertex based coarse space | ||||

Face 1 * | \(1.51\mathrm {e}{2} ~ (54)\) | \(1.45\mathrm {e}{2} ~ (46)\) | 72.44 (41) | 72.44 (41) |

Face 2 * | \(1.60\mathrm {e}{2} ~ (55)\) | \(1.21\mathrm {e}{2} ~ (48)\) | 68.38 (42) | 68.38 (42) |

Face 3 * | \(1.23\mathrm {e}{2} ~ (52)\) | 61.99 (48) | 61.99 (45) | 61.99 (40) |

Face 4 * | \(1.37\mathrm {e}{2} ~ (63)\) | \(1.21\mathrm {e}{2} ~ (61)\) | \(1.21\mathrm {e}{2} ~ (61)\) | 63.75 (40) |

\(\alpha \) distribution | No enrichment | Face enrichment with \(\lambda _{\mathcal {F}} < 0.0187\) | ||
---|---|---|---|---|

Cap \(=\,\)0 | Cap \(=\) 1 | Cap \(=\) 2 | No cap | |

\(H/h=32\) | \(H/h=32\) | \(H/h=32\) | \(H/h=32\) | |

Wirebasket based coarse space | ||||

Face 1 * | \(7.32\mathrm {e}{3} ~ (53)\) | 52.07 (41) | 41.94 (35) | 35.42 (33) |

Face 2 * | 80.59 (49) | 45.07 (45) | 38.41 (44) | 35.45 (33) |

Face 3 * | \(1.38\mathrm {e}{2} ~ (49)\) | 62.60 (44) | 62.60 (39) | 35.56 (33) |

Face 4 * | \(4.05\mathrm {e}{4} ~ (70)\) | \(1.90\mathrm {e}{4} ~ (64)\) | \(1.58\mathrm {e}{2} ~ (51)\) | 35.34 (33) |

The numerical results of Test 2 are presented in Table 3. Each row corresponds to a specific distribution. The first column lists the results for the methods with no enrichment. The second and third column lists the results for the methods with capped adaptive coarse spaces. The last column lists the results for the methods with adaptive coarse spaces without a cap beyond the threshold. The condition numbers of the first column in Table 3 are notably lower then the condition numbers in the first column of Table 1. From the last column of Table 3 we see that our method improves the performance of PCG.

In Test 3 the distribution has inclusions on edges. The inclusions are two rectangular slabs horizontally placed, as is illustrated by in Fig. 3. The slabs trigger the solution of generalized eigenvalue problems both on edges and faces. The corresponding eigenvalues from the generalized eigenvalue problems with \(\frac{H}{h} = 32\) are presented in Table 4. Due to the symmetry of the distribution, the eigenvalue problems on the edges are identical. Moreover, there are only two unique eigenvalue problems on the faces.

Showing the smallest eigenvalues (column wise) for the eigenvalue problems on the edge and the two face types, with inclusions, for \(H/h=32\)

Each vertical edge | Each face on the | Each face on the | ||
---|---|---|---|---|

\(\lambda _{\mathcal {E}}\) | \(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) | \(\lambda _{\mathcal {F}_I}\) | \(\lambda _{\mathcal {F}}\) |

\(3.3\mathrm {e}{-8}\) | 0.0 | \(7.2\mathrm {e}{-3}\) | 0.0 | \(\mathbf {2.7}\mathrm {e}{\mathbf{-2}}\) |

\(7.5\mathrm {e}{-8}\) | \(4.5\mathrm {e}{-8}\) | \(\mathbf {4.0}\mathrm {e}{\mathbf{-2}}\) | \(5.8\mathrm {e}{-7}\) | |

\(\mathbf {6.8}\mathrm {e}{\mathbf{-2}}\) | \(8.2\mathrm {e}{-3}\) | \(3.8\mathrm {e}{-3}\) | ||

\(1.8\mathrm {e}{-2}\) | \(1.4\mathrm {e}{-2}\) | |||

\(3.1\mathrm {e}{-2}\) | \(3.2\mathrm {e}{-2}\) | |||

\(3.1\mathrm {e}{-2}\) | \(\mathbf {3.9}\mathrm {e}{\mathbf{-2}}\) | |||

\(\mathbf {4.3}\mathrm {e}{\mathbf{-2}}\) |

Condition number estimates and iteration counts (inside brackets) for the two preconditioners applied in Test 3

\(\alpha \) distribution | No enrichment | Edge-, Face enrichment: \(\lambda _{{\mathcal {E}}} < \lambda ^*\), \(\lambda _{\mathcal {F}_I} < \lambda ^*\) | ||
---|---|---|---|---|

\(\lambda ^*=0.0375\) | \(\lambda ^*=0.0187\) | \(\lambda ^*=0.0094\) | ||

\(H/h=8\) | \(H/h=8\) | \(H/h=16\) | \(H/h=32\) | |

Vertex based coarse space | ||||

Edge | \(8.29\mathrm {e}{5} ~~ (69)\) | 33.84 (27) | 37.53 (35) | 44.98 (46) |

Face 1–4 and Edge | \(3.90\mathrm {e}{5} ~ (167)\) | 25.72 (31) | 34.96 (39) | 46.95 (49) |

\(\alpha \) distribution | No enrichment | Face enrichment: \(\lambda _{\mathcal {F}} < \lambda ^*\) | ||
---|---|---|---|---|

\(\lambda ^*=0.075\) | \(\lambda ^*=0.035\) | \(\lambda ^*=0.0187\) | ||

\(H/h=8\) | \(H/h=8\) | \(H/h=16\) | \(H/h=32\) | |

Wirebasket based coarse space | ||||

Edge | 21.07 (26) | 12.49 (26) | 18.28 (27) | 31.42 (33) |

Face 1–4 and Edge | \(1.26\mathrm {e}{5} ~ (99)\) | 15.63 (25) | 21.09 (31) | 29.26 (32) |

Up to now, we have seen that our method can handle various challenging distributions on a small number of subdomains. In Test 4, we divide the unit cube into 64 subdomains. This decomposition leads to 27 vertex nodes, 108 edges, and 144 faces, excluding all vertices, edges, and faces that are entirely part of the boundary of the domain. The distribution in this test, see Fig. 4, are channels that are parallel to the *y* axis and run through the entire domain touching the boundary at both sides. A third of all the faces have 4 inclusions each. For this test problem, we use a heterogeneous right-hand side \(f(x,y,z)=10^5e^{-5\sqrt{(x-0.25)^2+(y-0.25)^2+(z-0.25)^2}}\).

Condition number estimates and iteration counts (inside brackets) for Test 4

| Adaptive wire coarse space | Adaptive vertex coarse space | ||||||
---|---|---|---|---|---|---|---|---|

\(\lambda ^*\). | \(\#I_\mathcal {W}\) | \(\#I_\mathcal {F}\) | Cond. | \(\lambda ^*\). | \(\#I_\mathcal {V}\) | \(\#I_\mathcal {F}\) | Cond. | |

8 | 0.075 | 783 | 192 | 11.09 (20) | 0.0375 | 27 | 288 | 11.14 (22) |

16 | 0.035 | 1647 | 192 | 16.10 (25) | 0.0187 | 27 | 288 | 22.04 (33) |

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