Fakultät für Mathematik und Informatik

Fakultät für Mathematik und Informatik

Preprint 2016-09

Alexander Heinlein, Axel Klawonn, Jascha Knepper, Oliver Rheinbach

ISSN 1433-9307

Multiscale Coarse Spaces for Overlapping Schwarz Methods Based

on the ACMS Space in 2D

Alexander Heinlein, Axel Klawonn,

Jascha Knepper, and Oliver Rheinbach

Multiscale Coarse Spaces for Overlapping Schwarz Methods Based on the ACMS Space in 2D

TU Bergakademie Freiberg

Fakultät für Mathematik und Informatik

Prüferstraße 9

09599 FREIBERG

http://tu-freiberg.de/fakult1

ISSN 1433 – 9307

Herausgeber: Dekan der Fakultät für Mathematik und Informatik

Herstellung: Medienzentrum der TU Bergakademie Freiberg

MULTISCALE COARSE SPACES FOR OVERLAPPING SCHWARZMETHODS BASED ON THE ACMS SPACE IN 2D

ALEXANDER HEINLEIN∗, AXEL KLAWONN∗, JASCHA KNEPPER∗, AND OLIVER

RHEINBACH†

August 3, 2016

Abstract. Two-level overlapping Schwarz domain decomposition methods for second orderelliptic problems in two dimensions are proposed using coarse spaces constructed from the Ap-proximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarsespaces are based on eigenvalue problems using Schur complements on subdomain edges. It is thenproven that the convergence of the resulting preconditioned Krylov method can be controlled bya user-specified tolerance and thus can be made independent of heterogeneities in the coefficientof the partial differential equation. The relations of this new approach to other known adaptivecoarse space approaches for overlapping Schwarz methods are also discussed. Compared to one ofthe competing adaptive approaches, the new coarse space can be significantly smaller. Comparedto other competing approaches, the eigenvalue problem is significantly cheaper, i.e., the dimensionof the eigenvalue problems is minimal among the competing adaptive approaches under consid-eration. Our local eigenvalue problems can be solved using one or two iterations of LobPCG foressentially the same cost as an LU-decomposition of a Schur complement on a subdomain edge.

Key words. Domain Decomposition, Multiscale, Approximate Component Mode Synthesis(ACMS), Scientific Software, Parallel, Overlapping Schwarz,

AMS subject classifications. 65F08,65F10,65N55,68W10,74S30

1. Introduction. We consider coarse spaces for two-level overlapping Schwarzmethods [38] based on variations of the Approximate Component Mode Synthesis(ACMS) special finite element method, which was introduced in [23] for the dis-cretization of second order elliptic problems with highly varying coefficients. For ournew two-level overlapping Schwarz method, we obtain a condition number boundwhich is independent of large variations in the coefficient; see section 6 for thecondition number bound and section 7 for corresponding numerical results.

The ACMS discretization method builds on the traditional Component ModeSynthesis (CMS) method [7, 25] and uses eigenvectors from local discrete eigenvalueproblems to construct the approximation space. A discretization error estimateand an a-posteriori error indicator are available [22] for ACMS as well as a paral-lel implementation [20]. The improved approximation properties for heterogeneousproblems, compared to standard FEM spaces, relies on the adaptation of the basisfunctions to the heterogeneities, i.e., the basis functions are computed in a first (of-fline) step. They are used in the later (online) simulations, which can be performedon a relatively coarse mesh. However, for large problems, this mesh can still re-quire parallel solution methods, and in [20], it has been shown that parallel domaindecomposition solvers can robustly solve the discretized ACMS (online) problems.

Many multiscale discretization methods have been proposed in the past, amongthem are Multiscale Finite Element Methods (MsFEM) [24, 14], Heterogeneous Mul-tiscale Finite Element Methods (HMM) [12, 13], adaptive multiscale methods [33],and Generalized Finite Element Methods (GFEM) [3, 2]; see also the referencestherein.

Different exotic coarse spaces for overlapping Schwarz methods are also known,

∗Mathematisches Institut, Universitat zu Koln, Weyertal 86-90, 50931 Koln, Germany,alexander.heinlein, axel.klawonn, [email protected], url: http://www.numerik.uni-koeln.de†Institut fur Numerische Mathematik und Optimierung, Fakultat fur Mathematik und

Informatik, Technische Universitat Bergakademie Freiberg, Akademiestr. 6, 09599 Freiberg,[email protected], url: http://www.mathe.tu-freiberg.de/nmo/mitarbeiter/oliver-rheinbach

1

2 A. Heinlein, A. Klawonn, J. Knepper, O. Rheinbach

e.g., energy minimizing coarse spaces [8, 9, 10, 21, 5, 6, 1], where the latter threereferences use coarse spaces built on MsFEM basis functions. Adaptive coarsespaces from eigenvalue problems for overlapping Schwarz methods were alreadyproposed in [15, 16], considering unions of neighboring subdomains. Their idea wasto replace a Poincare inequality by a computable bound from a discrete eigenvalueproblem involving a Neumann matrix and a mass matrix. In [36], Spillane et al.have introduced the GenEO coarse space for overlapping Schwarz methods. Thismethod uses the bilinear form on both sides of the generalized eigenvalue problems.By elimination, the generalized eigenvalue problems can be reduced to the overlapof the subdomains, resulting in smaller eigenvalue problems. This helped to makeadaptive overlapping Schwarz methods practical.

The size of the eigenvalue problems is reduced further in the more recent meth-ods in the focus of this paper, where (generalized) eigenvalue problems on thesubdomain boundary or on edges are solved. We will briefly review the ACMSmethod as a discretization before we will introduce the ACMS-based coarse spacefor overlapping Schwarz, using eigenvalue problems on edges. We will also discusstwo approaches closely related to our approach, i.e., the method introduced in [11],where eigenvalue problems on the boundaries of the overlapping subdomains areused, and the very recent method introduced in [17], also using edges.

The methods [15, 16], [11], and [17] are described in subsection 5.1, subsec-tion 5.2, and subsection 5.3, respectively, in order to illustrate differences andrelations to our approach. Let us note that, compared to [15, 16] and [11], theeigenvalue problems in our approach are more local, i.e., defined on edges. Theeigenvalue problems used in [17] are also defined on edges and are typically compu-tationally slightly cheaper than ours (if the standard, non-economic variant of ourapproach is used), but our resulting ACMS coarse space can be significantly smallerin certain cases; see section 7.

Adaptive coarse space approaches for nonoverlapping domain decompositionmethods are, of course, also related, e.g., [4, 32, 31, 28, 34, 29, 37, 27, 35, 26], butare out of the scope of this paper. For further references to the literature, we referto the references listed in the aforementioned publications.

2. ACMS Special Finite Element Method. We consider problems of theform

(1)−∇ · (A(x)∇u(x)) = f(x) in Ω ⊂ R2,

u = 0 on ∂Ω,

where the coefficient function A is highly heterogeneous, possibly with high jumps.The model problem (1) can be transformed into the variational formulation: findu ∈ H1

0 (Ω), such that

(2) aΩ (u, v) = L(v) ∀v ∈ H10 (Ω)

with the bilinear form and the linear functional

aΩ (u, v) :=

∫Ω

(∇u(x))TA(x)∇v(x) dx and L (v) :=

∫Ω

f(x)v(x) dx,

respectively, where f ∈ L2(Ω). In addition to that, we denote the semi-normcorresponding to the bilinear form aΩ (·, ·) as

|u|2a,Ω := aΩ (u, u) .

We assume that the coefficient function A(x) is a uniformly symmetric positivedefinite matrix and that it satisfies

0 < αminξT ξ ≤ ξTA(x)ξ ≤ αmaxξ

T ξ ∀x ∈ Ω and ξ ∈ R2 \ 0 ,

Overlapping Schwarz with ACMS Coarse Spaces 3

with constants αmin and αmax independent of x.We consider a family (τH)H of conforming triangulations of Ω into triangles or

convex quadrilaterals, where each triangulation introduces an interface

(3) Γ =

( ⋃T∈τH

∂T

)\ ∂Ω.

The decomposition

(4) H10 (Ω) =

(⊕T∈τH

VT

)⊕ VΓ

is orthogonal with respect to aΩ (·, ·) and is standard in domain decompositiontheory. However, here, the decomposition (τH)H of Ω corresponds to a triangulationby ACMS elements with typical diameter H. Only later, when we will use ACMS asa coarse space for overlapping Schwarz methods, this decomposition will correspondto a nonoverlapping domain decomposition with typical diameter H. Note that thisdomain decomposition may then also be irregular.

We have

(5) VT =v ∈ H1

0 (Ω) : v|T ∈ H10 (T ) and v|Ω\T = 0

⊂ H1

0 (Ω)

for all T ∈ τH and

(6) VΓ =HΓ→Ω(τ) ∈ H1

0 (Ω) : τ ∈ H1/2 (Γ)⊂ H1

0 (Ω),

where HΓ→Ω corresponds to the harmonic extension operator, i.e., to the solutionof

(7)

−∇ · (A(x)∇HΓ→Ω (τ)) = 0 in T, ∀T ∈ τH ,HΓ→Ω (τ) = τ on Γ,

HΓ→Ω (τ) = 0 on ∂Ω.

Based on the decomposition (4), problem (2) can equivalently be written as:find uT ∈ VT and uΓ ∈ VΓ, such that

(8)aΩ (uT , vT ) = L (vT ) ∀T ∈ τH ,∀vT ∈ VT ,aΩ (uΓ, vΓ) = L (vΓ) ∀vΓ ∈ VΓ,

where the solution of (2) is then given by

(9) u =∑T∈τH

uT + uΓ ∈ H10 (Ω).

In the CMS method [7, 25], eigenfunctions are used as basis functions to solvethe problems (8); see also Kolmogoroff’s n-width from approximation theory, whichexplains the approximation properties of eigenfunctions. The ACMS finite elementspace, which was introduced by Hetmaniuk and Lehoucq in [23], is designed as anapproximation of the CMS finite element space where only basis functions with localsupport are employed. This applies, in particular, to the functions in VΓ, which haveglobal support, whereas the functions in VT are already local in the CMS method.

We employ, in total, three different types of basis functions in the ACMSmethod, which we refer to as vertex-specific, edge-based, and fixed-interface basisfunctions.


A vertex-specific basis function which corresponds to a vertex P of the triangu-lation τH , is given by the boundary value problem

(10)

−∇ · (A(x)∇ϕP ) = 0 in T, ∀T ∈ τH ,ϕP = 0 on ∂Ω,

ϕP 6= 0 on Γ,

ϕP (P ′) = δP,P ′ on Γ,

where P ′ is also a vertex of the triangulation τH and δP,P ′ is the Kronecker deltafunction. These functions originate from the Multiscale Finite Element Method(MsFEM) [24, 14], and we will refer to them also as MsFEM basis functions.

They are harmonic extensions of trace functions defined on Γ, and thus ϕP ∈ VΓ.The trace values of ϕP on Γ can, e.g., be chosen to be linear in between the verticesor discrete harmonic in one dimension. In the latter case, on each edge e ⊂ Γ, wechoose them to be the solution of

(11)

∂

∂τ〈Ae,max(x)τ,∇ϕP (x)〉 = 0 on e,

ϕP (P ′) = δP,P ′ on Γ,

where τ denotes the tangential vector of the edge with ‖τ‖ = 1 and 〈·, ·〉 is thestandard l2-inner product. Here, we use for an edge eij = Ti ∩ Tj with Ti, Tj ∈ τH ,i 6= j,

Aeij ,max(x) := max

lim

yi∈Ωi→xA(yi), lim

yj∈Ωj→xA(yj)

(12)

instead of A in case of discontinuities across the edge; cf., e.g., the coefficient func-tion in Figure 1.

For any open edge e ⊂ Γ, we define the edge-based basis function by the corre-sponding eigenvalue problem in the space of harmonic extensions: find

(τ∗,e, λ∗,e) ∈ H1/200 (e)× R such that

(13) aΩ

(HΓ→Ω(τ∗,e),HΓ→Ω(θ)

)= λ∗,e (τ∗,e, θ)L2(e) ∀θ ∈ H1/2

00 (e)

with τ∗,e and θ being the trivial extension of τ∗,e and θ, respectively, by zero toΓ \ e. In order to approximate the solution of the local problems in (8), fixed-interface eigenfunctions with

(14) aΩ (z∗,T , v) = λ∗,T (z∗,T , v)L2(Ω) ∀v ∈ VT ,

on each of the T ∈ τH are employed. The eigenvalues λi,T ∞i=1 and λi,e∞i=1are assumed to be ordered non-decreasingly, and the corresponding eigenmodesaccordingly. The eigenmodes z1,T , z2,T , ... and τ1,e, τ2,e, ... form orthonormal basesfor the L2-inner product of VT and of VΓ on the element T and on the edge e,respectively.

The finite element space of the ACMS special finite element method is then


given by

(15)

VACMS =

(⊕T∈τH

span zi,T : 1 ≤ i ≤ IT

)

⊕

⊕P∈Γ

P vertex

span ϕP

⊕

⊕e⊂Γe edge

span HΓ→Ω (τi,e) : 1 ≤ i ≤ Ie

with positive integers IT , ∀T ∈ τH , and Ie, ∀ edges e ⊂ Γ, corresponding to thenumber of eigenmodes used as basis functions in each element and on each edge,respectively.

In practice, the ACMS basis functions are not computed explicitly, but they arereplaced by their approximation on a submesh; cf. [23, 20]. This submesh is alsodenoted as the ACMS fine mesh τh, and its finite elements have a typical diameter ofh. Next, in the overlapping Schwarz method with ACMS coarse space, the standardfinite element mesh will correspond to the ACMS fine mesh, and the ACMS elementsT ∈ τH with a typical diameter of H will correspond to the nonoverlapping domaindecomposition underlying the overlapping Schwarz method.

3. Two-Level Overlapping Schwarz Methods. Let

Ku = f

be the discretization of problem (2) by piecewise linear or bilinear finite elements ona family of triangulations (τh)h, K the stiffness matrix, f the right hand side, and uthe discrete solution vector on the finite element space V := V h (Ω). Furthermore,

let ΩiNi=1 be a nonoverlapping domain decomposition of Ω with the interface

Γ =N⋃i=1

∂Ωi \ ∂Ω.

The typical diameter of the subdomains is H.The set Ω′i

Ni=1 is a corresponding overlapping decomposition of Ω with overlap

δ. In the preconditioning context, we assume that the coefficient function A isconstant on each finite element. We define as Ri : V → Vi := V h (Ω′i), i = 1, ..., N ,the restriction to the local finite element space on the overlapping subdomain Ω′i;RTi is the corresponding prolongation to V h (Ω). In addition, let V0 be a globalcoarse space. We use exact local solvers, and therefore the local bilinear forms onthe subspaces are given by

ai (ui, vi) = aΩ

(RTi ui, R

Ti vi)∀ui, vi ∈ Vi(16)

and the operators Pi : V → Vi by

ai

(Piu, vi

)= aΩ

(u,RTi vi

)∀vi ∈ Vi(17)

for i = 1, ..., N ; cf., e.g., [38]. Then, the two-level overlapping Schwarz precondi-tioner can be written as

M−1OS−2 = RT0 K

−10 R0︸︷︷︸

coarse level

+N∑i=1

RTi K−1i Ri︸︷︷︸

first level


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 1: Discontinuous coefficient function A with inclusion at the vertices of thedecomposition. The light blue color corresponds to a coefficient of 1.0 and the darkblue color to a coefficient of 108; 1/H = 4.

V0 it. κVQ1 98 8.87 · 105

VMsFEM 21 5.56

Table 1: Iteration counts and condition numbers for PCG with a two-level Schwarzpreconditioner using Q1 (piecewise bilinear) or MsFEM coarse basis functions;1/H = 4, H/h = 16, and δ = 2h; relative stopping criterion

∥∥r(k)∥∥

2/∥∥r(0)

∥∥2<

10−8; for the coefficient distribution, see Figure 1.

with local and coarse stiffness matrices

Ki = RiKRTi

for i = 0, ..., N . Here, the coarse operator K0 is formed by a Galerkin productinstead of using a discretization on a coarse grid.

The condition number of the standard two-level Schwarz preconditioner forproblem (2), where standard Lagrange basis functions are used for the discretizationof the coarse level, depends on the contrast of the coefficient function A, i.e.,

κ(M−1

OS−2K)≤ C max

x,y∈ωK

(A (x)

A (y)

)(1 +

H

δ

).

Here, ωK corresponds to the union of all coarse mesh elements which touch a coarsemesh element K. This bound can be made sharper but a dependence on the coef-ficient contrast remains; see [18].

3.1. MsFEM Coarse Space. As proposed by Aarnes and Hou [1] as wellas by Buck, Iliev, and Andra [5, 6], MsFEM basis functions can be used in thecoarse space of an overlapping Schwarz preconditioner to enhance the performancein presence of rough coefficients.

Considering the overlapping Schwarz method and the underlying domain de-composition into nonverlapping subdomains Ω1, ...,ΩN of typical diameter H, weidentify each Ωi with an element T ∈ τH from section 2. The MsFEM basis func-tions ϕP , as defined in (10), are then discretized in the spaces V h (Ωi), i.e., on afine triangulation of Ωi into finite elements of typical diameter h; we also refer tothe fine triangulation as the fine mesh.


In particular, the discretized basis functions, which we will also denote as ϕP ,are employed to build the coarse MsFEM space

VMsFEM :=

⊕P∈Γ

P vertex

span ϕP

.(18)

Since these functions are thus defined as discrete harmonic extensions, they can becomputed also for unstructured domain decompositions, as in the GDSW precondi-tioner [8, 9, 21], without the need for an additional coarse triangulation. This is anadvantage over the use of standard Lagrange coarse basis functions. The GDSWpreconditioner is a two-level overlapping Schwarz preconditioner with energy mini-mizing coarse space functions, i.e., it uses discrete harmonic coarse space functionsas well. MsFEM basis functions can also improve significantly the robustness of theoverlapping Schwarz preconditioner for coefficient functions of a certain (simple)type; cf., e.g., Figure 1 and the results in Table 1.

Note that, for a constant coefficient function A and a structured domain de-composition into triangles or quadrilaterals, the MsFEM coarse space correspondsto standard piecewise linear or bilinear coarse basis functions, respectively.

4. Coarse Spaces Based on the ACMS Space. In adaptive coarse spaces,generalized eigenvalue problems are typically used in order to obtain condition num-ber estimates that are independent of the contrast of the coefficient function. Thus,it seems natural to use the ACMS edge-based basis functions as coarse basis func-tions in a two-level Schwarz method for heterogeneous problems; these ACMS edgefunctions are computed from local generalized eigenvalue problems (13) and con-tribute to the excellent approximation properties of the ACMS discretization forsecond order elliptic PDEs with highly heterogeneous coefficients. As we will ob-serve, after applying some modifications to the eigenvalue problems, that we will beable to construct a coarse space for overlapping Schwarz methods which is robustwith respect to the contrast of the coefficient function; cf. section 7. This theoreticalresult will be proven in section 6.

As in the two-level Schwarz preconditioner with MsFEM coarse space (cf. sub-section 3.1), let us consider our overlapping Schwarz method using the underlyingdomain decomposition into nonverlapping subdomains Ω1, ...,ΩN of typical diam-eter H. In order to define our ACMS coarse space for the overlapping Schwarzmethod, we again identify each Ωi with an ACMS element T ; cf. section 2. Thetriangulation of Ωi into finite elements of typical diameter h then represents theACMS fine mesh τh; for simplicity, we assume quasi-uniformity of the fine mesh.

4.1. Coarse Space with Dirichlet Boundary Conditions. We propose acoarse space for overlapping Schwarz methods which is based on the construction ofthe ACMS finite element space. In particular, we use the MsFEM basis functions,i.e., the vertex-specific basis function (see (10)), and a modified version of the edge-based coupling eigenmodes (see (13)), where we replace the L2-inner product onthe right hand side of (13) by a scaled version. Therefore, we define for an edgeeij = Ωi ∩ Ωj the bilinear form

beij (u, v) :=1

h

(Aeij ,max u, v

)L2(eij)

(19)

and the corresponding norm

‖u‖2b,eij := beij (u, u) ,

where Aeij ,max is defined as in (12). Using this bilinear form, we obtain a discreteeigenvalue problem, which is a modification of (13): find


eij

ηkhijΩi Ωj

kh kh

Fig. 2: Graphical representation of the slab ηkhij corresponding to the edge eij .We denote the methods which construct the eigenvalue problems using functionssupported on a slab as “economic”.

τ∗,eij ∈ V h0 (eij) :=v|eij : v ∈ V, v = 0 on ∂eij

such that

aΩ

(HΓ→Ω(τ∗,eij ),HΓ→Ω(θ)

)= λ∗,eij beij

(τ∗,eij , θ

)∀θ ∈ V h0 (eij) .(20)

Again, the tilde corresponds to the extensions of τ∗,eij and θ by zero on Γ \ eij .Let the eigenvalues be sorted in non-descending order, i.e., λ1,eij ≤ λ2,eij ≤ ... ≤λm, and the eigenmodes accordingly, where m = dim

(V h0 (eij)

). We select all

eigenmodes τ∗,eij where the eigenvalues are below a certain tolerance, i.e., λ∗,eij ≤tol. Then, the coarse basis functions corresponding to the edge eij are the discreteharmonic extensions of the selected τ∗,eij to the interior degrees of freedom, i.e., thesolutions v∗,eij of

(21)

aΩl

(v∗,eij , v

)= 0 ∀v ∈ V h0 (Ωl) , l = 1, ..., N,

v∗,eij = τ∗,eij on eij ,

v∗,eij = 0 on Γ \ eij ,

where V h0 (Ωi) := v : v ∈ V, v = 0 in Ω \ Ωi.The coarse space based on the ACMS finite element method is then given by

V tolACMS−D = VMsFEM ⊕

⊕e⊂Γe edge

span vk,e : λk,e ≤ tol

,

where VMsFEM is the MsFEM coarse space, which is the identical to the spacespanned by the discretized vertex-based ACMS basis functions; cf (15).

As can be observed in section 7, this coarse space is robust for many coefficientdistributions with jumping coefficients, while the eigenvalue problems are definedon the open edges and are thus of comparably modest size.

Here, we enforce Dirichlet boundary conditions on the whole interface for thediscrete harmonic extensions on the left hand side of the generalized eigenvalueproblem (20). However, for our proof, a different choice of the boundary conditionson the left hand side of (20) will be beneficial; see the subsequent subsection 4.2.

4.2. Coarse Space with Neumann Boundary Conditions. In order toimprove the coarse space and, in particular, to be able to prove an estimate for thecondition number, we replace the left hand side of the eigenvalue problem.

We first define ηkhij to be a slab of several layers of fine elements around theedge eij ; cf. Figure 2. Then, we replace the Dirichlet boundary condition in the left


hand side of the eigenvalue problem by a Neumann boundary condition:

aηkhij

(Heij→ηkh

ij(τ∗,eij ),Heij→ηkh

ij(θ))

= λ∗,eij beij(τ∗,eij , θ

)∀θ ∈ V h0 (eij) .(22)

In this version of the eigenvalue problem, no Dirichlet boundary conditions are pre-scribed on ∂ηkhij \eij . The coarse basis functions v∗,eij are again obtained by selectingeigenmodes with eigenvalues up to a certain tolerance and by extending these valueson the edge discrete harmonically to the subdomains, i.e., by solving (21).

The resulting coarse space depends on the width of ηkhij and the threshold for

the selection of the eigenvalues tol. If ηkhij = Ωi ∪Ωj , i.e., if the slab coincides withthe complete subdomain, the resulting coarse space is

V tolACMS−N = VMsFEM ⊕

⊕e⊂Γe edge


,

and if the width of ηkhij consists of k layers of fine elements in each of the subdomains

Ωi and Ωj the coarse space is denoted as V tolACMS−N,k; see section 6 for the proof ofthe condition number of the corresponding Schwarz operator.

4.3. Using a Lumped Mass Matrix. The solution of generalized eigenvalueproblems can be expensive, and therefore typical criticism concerns the construc-tion of the adaptive coarse spaces (in the offline stage). In order to reduce thecomputational cost for the solution of the generalized eigenvalue problem, one canlump the mass matrix on the right hand side.

In particular, for the edge eij , both eigenvalue problems (20) and (22) can bewritten in discrete form as

SeijV = λBeijV(23)

using the Schur complement Seij where all degrees of freedom except for thoseon the interior edge have been eliminated. Depending on which version of theACMS coarse space is used, zero Dirichlet boundary conditions have been appliedonly to endpoints of the edge (Neumann version) or to all degrees of freedom on(∂Ωi ∪ ∂Ωj) \ eij (Dirichlet version). The matrix Beij corresponds to the scaledmass matrix on the edge, i.e., to the discretization of the bilinear form (19).

Now, since the mass matrix is spectrally equivalent to its diagonal, we canreplace Beij by diag

(Beij

)in (23), and obtain

diag(Beij

)−1SeijV = λV

or, in a symmetric version,

diag(Beij

)−1/2Seij diag

(Beij

)−1/2W = λW

with diag(Beij

)1/2V = W in a computationally inexpensive step.

Thus, we are able to solve a standard eigenvalue problem instead of a generalizedeigenvalue problem. We denote the corresponding coarse spaces with a bar, i.e., asV tol

ACMS−D, V tol

ACMS−N. Accordingly, we write V tol

ACMS−N,kfor the slab version.

As we will observe in section 7, lumping the mass matrix does not affect theperformance of the preconditioner.

4.4. Properties of the Spectral Projection. We consider the projectiononto the space spanned by the discrete harmonic extensions of the selected eigen-


functions, i.e., by the vk,e e⊂Γe edge

,

Πv :=∑e⊂Γe edge

∑λk,e<tol

be (v, vk,e) vk,e.(24)

This projection, which will be part of the construction of a stable decompositionof the two-level Schwarz preconditioner in section 6, has typical properties, whichare summarized in the following lemma for the Neumann version of the eigenvalueproblem; cf. (22). However, the properties in case of the Dirichlet version couldbe proven analogously. The proof of the next lemma is based on arguments fromclassical spectral theory and follows arguments which are standard in the theoryof adaptive coarse spaces; see, e.g., [28, Lemma 5.3], [36, Lemma 2.11], and [29,Lemma 4.6].

Lemma 1. Consider for an edge e the eigenvalue problem (22) and the corre-

sponding eigenpairs (τk,e, λk,e)dim(V h

0 (e))k=1 , and let vk,e be the discrete harmonic

extension of τk,e; cf. (21). Then, the projection

Πev :=∑

λk,e≤tol

be (v, vk,e) vk,e.

is orthogonal with respect to the bilinear form

a∗,e,η (u, v) := aη (He→η(u),He→η(v))

and therefore

|v|2a,∗,e,η = |Πev|2a,∗,e,η + |v −Πev|2a,∗,e,η .

The semi-norm is defined as

|u|2a,∗,e,η := a∗,e,η (u, u) .(25)

In addition, the estimate

‖v −Πev‖2b,e ≤1

tol|v −Πev|2a,∗,e,η

holds.

Proof. Since both bilinear forms are positive definite, the eigenvectors span thespace V h0 (e), and therefore they can be chosen to be an orthogonal basis of V h0 (e)with

a∗,e,η (vk,e, vl,e) = a∗,e,η (τk,e, τl,e) = be (τk,e, τl,e) = be (vk,e, vl,e) = 0 ∀k 6= l

and

|vk,e|2a,∗,e,η = |τk,e|2a,∗,e,η = λk,e,

‖vk,e‖2b,e = |τk,e|2b,e = 1.

Therefore, since

v =

dim(V h0 (e))∑

k=1

be (v, vk,e) vk,e ∀v ∈ V h0 (e) ,


we obtain

|v|2a,∗,e,η=

∣∣∣∣∣∣∑

λk,e≤tol

be (v, vk,e) vk,e

∣∣∣∣∣∣2

a,∗,e,η

+

∣∣∣∣∣∣∑

λk,e>tol

be (v, vk,e) vk,e

∣∣∣∣∣∣2

a,∗,e,η

= |Πev|2a,∗,e,η + |v −Πev|2a,∗,e,η .

Moreover,

‖v −Πev‖2b,e =

∥∥∥∥∥∥∑

λk,e>tol

be (v, vk,e) vk,e

∥∥∥∥∥∥2

b,e

=∑

λk,e>tol

be (v, vk,e) ‖vk,e‖2b,e

=∑

λk,e>tol

be (v, vk,e)1

λk,e|vk,e|2a,∗,e,η

≤ 1

tol

∑λk,e>tol

be (v, vk,e) |vk,e|2a,∗,e,η

=1

tol

∣∣∣∣∣∣∑

λk,e>tol

be (v, vk,e) vk,e

∣∣∣∣∣∣2

a,∗,e,η

=1

tol|v −Πev|2a,∗,e,η .

Since

Πv =∑e⊂Γ

Πev

the projection Π inherits the bounds for the local projections Πe. In particular, thiswill be essential in the proofs of section 6.

4.5. Heuristic Construction of Edge Functions. In order to reduce thecost for the computation of the edge-based coupling basis functions even further, onecan construct the trace values by inspecting the values of the coefficient function.In particular, we construct one basis function for each heterogeneity with a highcoefficient intersecting an edge and extend these values discrete harmonically to theinterior of the subdomain. Precisely, we set the functions to one for those degreesof freedom which belong to an element inside a heterogeneity with a high coefficientand to zero elsewhere; cf. Figure 3. Therefore, we define a tolerance tol whichcharacterizes the threshold for the identification of a coefficient jump: we identify

a coefficient jump if the contrast of the coefficient,At1

At2or

At2

At1, of two neighboring

elements t1 and t2 is larger than tol.We denote the resulting coarse space, which consists of the constructed edge-

based basis functions and the ACMS vertex-specific basis functions, as V tolACMS−R.Let us note that the vectors constructed from this approach could also be usedin other ACMS approaches as initial values in iterative methods for eigenvalueproblems. The first results for the use of the V tolACMS−R coarse space can be foundin [19] and in the master thesis of the third author.

5. Related Adaptive Coarse Spaces for Overlapping Schwarz. Whilethere are several different approaches for adaptive coarse spaces for overlapping


eij

Ωi

Ωj

eij

Fig. 3: Heuristic construction of the edge-based coarse basis functions in ACMS-Rin order to avoid eigenvalue problems; see subsection 4.5: coefficient function on thetwo subdomains Ωi and Ωj (left), coefficient function on the edge eij (top,right),and the resulting two coarse basis functions (bottom,right).

Schwarz preconditioners, we would like to mention two of them (in subsection 5.2and subsection 5.3) which are closely related to our approach. But first, we willdescribe an early approach by Galvis and Efendiev [15] in the subsequent subsec-tion 5.1.

5.1. Local Spectral Multiscale Coarse Spaces. In [15], Galvis and Efen-diev proposed to circumvent the need for a weighted Poincare inequality∫

Ω

A(x)v2 dx ≤ C∫

Ω

A(x) (∇v)2dx,(26)

with a constant C which depends on the contrast of the coefficientAΩ,max

AΩ,min, by using

eigenfunctions as coarse space functions.Therefore, the eigenvalue problem

div (A(x)∇ψωP

k ) = λiA(x)ψωP

k(27)

on the union of the neighboring subdomains (“coarse grid blocks”)

ωP :=⋃

T∈τH ,P∈T

T

of the vertex P of the coarse mesh is considered. The discrete form of the eigenvalueproblem is: find ψωP

i ∈ V h (ωP ) :=v ∈ V h (ωP ) : v|∂ωP∩∂Ω = 0

such that

AωPψωP

k = λkMωPψωP

k ,(28)

where AωP is the discretized matrix of the bilinear form aωP(·, ·) and MωP is the

discretized matrix of the bilinear form∫ωP

A(x)uv dx.

Now let χP P∈ΓP vertex

be a partition of unity, which is subordinated to the

covering ωP P∈ΓP vertex

of Ω, with χP ∈ V h (Ω) and |∇χP | ≤ 1H . Then, the coarse


basis functions are given as

ΦP,k = Ih (χPψωP

k ) for P ∈ Γ vertex and 1 ≤ i ≤ LP ,(29)

where LP is the number of eigenfunctions selected from the eigenvalue problem onthe patch ωP . The local spectral multiscale coarse space is then defined as

VLSM := ΦP,k : P ∈ Γ, 1 ≤ k ≤ LP .

For a two-level Schwarz preconditioner using this coarse space, the authorsprove the condition number bound

κ(M−1LSMK

)≤ C

(1 +

H2

δ2

),

where the constant C depends on the number of eigenfunctions selected for thecoarse space but not on the contrast of the coefficient function and the mesh size.However, the solution of the eigenvalue problems (28) is quite expensive: eacheigenvalue problem is defined on the (coarse) neighborhood ωP of a coarse node,i.e., it involves all neighboring subdomains of a vertex in the domain decomposition.Moreover, in this approach, unnecessary coarse space functions can be added to thecoarse space resulting from the interior part of the subdomain.

As a result, improved approaches were later proposed. For example, in [16],the authors have presented an improved version, where the dimension of the coarsespace is reduced by introducing a suitable second partition of unity which modifiesthe scaling of the mass matrix. Thereby, basis functions which correspond to interiorinclusions are eliminated.

We will now describe the two approaches more closely related to our new algo-rithm which, like us, use smaller eigenvalue problems, i.e., eigenvalue problems ontraces instead of volumes in order to further reduce computational cost, i.e., [11]and [17].

5.2. Coarse Spaces Based on Local Dirichlet-to-Neumann Maps. In[11], the authors Dolean, Nataf, Scheichl, and Spillane have introduced a coarsespace which is related to the ACMS approach. However, the corresponding eigen-value problem is defined on the complete boundary of the overlapping subdomains,i.e., it is defined using the Schur complements on the subdomain boundaries ∂Ω′i;see (30).

In particular, on each overlapping subdomain Ω′i, the eigenvalue problem isgiven by

div (A(x)∇vk) = 0 in Ω′i,

A(x)∂vk∂ni

= λkA(x)vk on ∂Ω′i,

where ∂∂ni

is the derivative in normal direction. The corresponding variationalformulation is∫

Ω′i

A(x)∇vk · ∇w dx = λk

∫∂Ω′i

tri (A(x)) vkw ds ∀w ∈ H1 (Ω′i) ,

where triA (x) := limy∈Ω′i→x

A (x) for a.e. x ∈ ∂Ω′i. In discrete form the eigenvalue

problem can be written as

(30) S(i)Γ′ v

(i)k,Γ′ = λkM

(i)Γ′ v

(i)k,Γ′ .


Here, M(i)Γ′ corresponds to the 1D mass matrix on ∂Ω′i, and S

(i)Γ′ corresponds to

the Schur complement where the interior degrees of freedom of the overlapping

subdomains have been eliminated. Then, v(i)k,Γ is extended discrete harmonically to

Ω′i and denoted by v(i)k . The corresponding coarse space is given by

VDtN := span

ΦHi,k : 1 ≤ i ≤ N and 1 ≤ k ≤ mi

where mi is the number of eigenfunctions selected in subdomain Ω′i. The basisfunctions are defined by

ΦHi,k := Ih(χiv

(i)k

)and the functions χiNi=1 form a partition of unity corresponding to the overlapping

decomposition Ω′iNi=1 with

χi(xj) :=dist (xj , ∂Ω′i)∑

xj∈Ω′l

dist (xj , ∂Ω′l).

In order to prove the condition number estimate

κ(M−1

DtNK)≤ C +

Nmaxi=1

1

δλ(i)mi+1

,

the authors assume a quasi-monotone coefficient function A. Here, δ denotes thewidth of the overlap.

Note that the eigenvalue problems are still rather large since each corresponds tothe degrees of freedom on the complete boundary of a single overlapping subdomain.

5.3. Spectral Harmonically Enriched Multiscale Coarse Space. In [17],the authors Gander, Loneland, and Rahman have introduced a coarse space whichis very closely related to our approach of using the ACMS space as a coarse space:first, MsFEM basis functions corresponding to the vertices of the decomposition areused in the coarse space, second, eigenvalue problems on the degrees of freedom ofthe edges of the nonoverlapping decomposition are employed.

In particular, the eigenvalue problem

aeij

(ψkeij , v

)= λkeij beij

(ψkeij , v

)∀v ∈ V h0 (eij)

is considered, where

aeij (u, v) :=(Aeij ,max(x)Dxtu,Dxtv

),(31)

beij (u, v) := h−1∑xk∈eij

βku (xk) v (xk) , and

βk :=∑

t∈τh,k∈dof(t)

At.

Here, Aeij ,max is defined as in (12), At is the constant coefficient on the elementt ∈ τh, Dxt denotes the tangent derivative with respect to the edge eij , and the xkcorrespond to the finite element nodes on the edge. Eigenfunctions correspondingto the first meij eigenvalues are selected and extended discrete harmonically bysolving

aΩl

(pkeij , v

)= 0 ∀v ∈ V h0 (Ωl) , l = 1, ..., N,

pkeij = ψkeij on eij ,

pkeij = 0 on Γ \ eij .


The Spectral Harmonically Enriched Multiscale (SHEM) coarse space is thengiven by

VSHEM :=⊕eij⊂Γ

spanpkeij : k ≤ meij

⊕ VMsFEM.

Using this choice, the authors are able to prove an estimate for the conditionnumber of the preconditioned system

κ(M−1

SHEMA)≤ C

(1 +

1

λm+1

),

where λm+1 := maxeij⊂Γ

λmeijeij

.

The authors thus use the bilinear form on the edge (rather than a Schur comple-ment). The eigenvalue problems are small and local, and therefore inexpensive toset up and solve. However, a drawback of this coarse space is that, due to the reduc-tion of the eigenvalue problem to the edges, the coarse space can become large forcertain coefficient functions where a much smaller coarse space would be sufficient;cf. section 7, where we observe that this smaller coarse space will be found by thenon-economic versions of the new methods described in this paper, i.e., using theeigenvalue problems from subsection 4.1 or subsection 4.2. Of course, if in the eco-nomic versions the slab is wide enough then the resulting adaptive method will alsobe successful. Note that among our methods, only the methods from subsection 4.2are covered by our theory in section 6.

6. Convergence Analysis for the Overlapping Schwarz method withACMS Coarse Space. Since the first level of our Schwarz preconditioner is stan-dard, strengthened Cauchy-Schwarz inequalities (Assumption 1) and local stability(Assumption 2) can be assumed; cf. [38].

Assumption 1 (Strengthened Cauchy-Schwarz Inequality). There exist con-stants 0 ≤ εij ≤ 1, 1 ≤ i, j ≤ N such that∣∣aΩ

(RTi vi, R

Tj vj)∣∣ ≤ εij ∣∣aΩ

(RTi vi, R

Ti vi)∣∣1/2 ∣∣aΩ

(RTj vj , R

Tj vj)∣∣1/2 ,

for vi ∈ V h (Ω′i) and vj ∈ V h(Ω′j). We denote the spectral radius of the matrix

ε = (εij)i,j as ρ (ε).

Then, the maximum eigenvalue of the Schwarz operator depends on ρ (ε), whichcan be bounded by the coloring constant NC . One way to obtain the coloringconstant NC is to determine, for each point x ∈ Ω, the number of overlappingsubdomains it belongs to. The maximum of these numbers can be taken as NC ;see, e.g., [38, Section 3.6].

Assumption 2 (Local Stability). There exists an ω > 0 such that

aΩ

(RTi vi, R

Ti vi)≤ ωai (vi, vi)

where vi ∈ range(Pi

)⊂ V h (Ω′i) , 1 ≤ i ≤ N .

Since we use exact local solvers, i.e., ai (vi, vi) = aΩ

(RTi vi, R

Ti vi), we have

ω = 1 in our case; cf. (16). For the definition of Pi, see (17).To prove an estimate for the convergence of the OS-ACMS preconditioner, we

have to prove the existence of a stable decomposition; cf., e.g., [38]. Therefore, wehave to provide a suitable coarse interpolation I0 into the coarse space

V0 =

⊕P∈Γ

P vertex

span ϕP

⊕ ⊕

e⊂Γe edge


,(32)


in particular, into the space V tolACMS−N or V tolACMS−N,k. We construct I0 from apoint-wise interpolation

IMsFEMu :=∑P∈Γ

P vertex

u (P )ϕP(33)

to the coarse multiscale space VMsFEM (see also (10)) and the projection Π onto thespace spanned by the edge-based coupling eigenfunctions; cf. (24).

We start with the proof of a lemma which states the a-stability of the MsFEMinterpolation operator in 1D. The semi-norms

|u|2a,ηkhij

:= aηkhij

(u, u) :=

∫ηkhij

(∇u(x))TA(x)∇u(x) dx

|u|2a,eij := aeij (u, u)

will be used in the following proofs to shorten the notation, where aeij (u, v) is givenin (31).

Lemma 2. The MsFEM interpolation operator, which is defined in (33), is sta-

ble with respect to the |·|2a,eij semi-norm, i.e.,

|IMsFEMv|2a,eij ≤ |v|2a,eij

∀v ∈ V h0 (eij) .

Proof. The MsFEM interpolation is exact in the vertices of the nonoverlappingdecomposition, i.e.,

IMsFEMv (P ) = v (P )

for all vertices P ∈ Γ. Furthermore, the MsFEM basis functions are defined asdiscrete harmonic, i.e., energy minimizing, functions on the edges of the decompo-sition; cf. (11). Therefore, IMsFEMv is the discrete harmonic extension of the valuesof v in the vertices to the edges, and thus

|IMsFEMv|2a,eij ≤ |v|2a,eij

∀v ∈ V h0 (eij) .

We define the coarse component of the stable decomposition as

u0 := I0u := IMsFEMu+ Π (u− IMsFEMu) ∈ V0.

Using this coarse component, we prove the existence of a stable decomposition;cf. Figure 4 for a graphical representation of the overlapping region Ωi,δi and thenonoverlapping region Ωi , which are used for the proofs in this section. In thefollowing proofs, we assume that the edges of the subdomains are straight.

In order to prove the existence of a stable decomposition, the following lemmawill be useful.

Lemma 3. Let u ∈ V h (Ω), u0 := I0u := IMsFEMu + Π (u− IMsFEMu), andweij ∈ V h (Ωi ∪ Ωj) with

weij =

u− u0 on eij ,0 on ∂Ωk \ eij ,piecewise linear/bilinear in Ωk \ Ωk,0 in Ωk,

for an edge eij = Ωi ∩ Ωj and k = i, j. Then,∣∣weij ∣∣2a,Ωi≤ C

tol

(|u|2a,Ωi

+ |u|2a,Ωj

),

where C > 0 is a constant independent of H, h, and the contrast of the coefficientfunction.


Ωi,δi

Ωi

2h

Fig. 4: In our proof of Theorem 4, we consider a partition of unity correspondingto an overlapping decomposition with overlap h. The overlapping region is calledΩi,δi and the nonoverlapping region Ωi .

Proof. From the construction of weij and a standard inverse inequality, weobtain ∣∣weij ∣∣2a,Ωi

≤ C ‖u− u0‖2b,eij .

Using the properties of the projection Π, cf. Lemma 1, we obtain

‖u− u0‖2b,eij = ‖u− IMsFEMu−Π (u− IMsFEMu)‖2b,eij

≤ 1

tol|u− IMsFEMu−Π (u− IMsFEMu)|2a,∗,eij ,ηkh

ij

≤ 1

tol|u− IMsFEMu|2a,∗,eij ,ηkh

ij.

We can decompose the semi-norm into parts belonging to Ωi and Ωj , i.e., ηkhii :=ηkhij ∩ Ωi and ηkhj := ηkhij ∩ Ωj , respectively, and obtain

|u− IMsFEMu|2a,∗,eij ,ηkhij

= |u− IMsFEMu|2a,∗,eij ,ηkhi

+ |u− IMsFEMu|2a,∗,eij ,ηkhj.

Next, we estimate each part separately and obtain

|u− IMsFEMu|2a,∗,eij ,ηkhi

=∣∣∣Heij→ηkh

i(u− IMsFEMu)

∣∣∣2a,ηkh

i

≤ Ch |u− IMsFEMu|2a,eij≤ Ch |u|2a,eij≤ C

(|u|2a,ηhi + |u|2a,ηhj

)≤ C

(|u|2a,Ωi

+ |u|2a,Ωj

).

In particular, the energy minimality of the discrete harmonic extension is usedin the second step, and the a-stability of the 1D MsFEM interpolation operator,cf. Lemma 2, is used in the third step.

Summing over all terms, we finally obtain

∣∣weij ∣∣2a,Ωi≤ C

tol

(|u|2a,Ωi

+ |u|2a,Ωj

).

Now, we are able to prove the existence of a stable decomposition


Theorem 4 (Stable Decomposition). For each v ∈ V = V h (Ω), there exists

a decomposition u =N∑i=0

RTi ui with ui ∈ Vi = V h (Ω′i) such that

N∑i=0

|ui|2a,Ω′i ≤ C20 |u|

2a,Ω

where C20 = C

(1 + 1

tol

).

Proof. We first consider the estimate of the coarse component and proceedsubdomain by subdomain. Using

|u0|2a,Ω =N∑i=1

|u0|2a,Ωi,

Lemma 3, and the fact that u0 is discrete harmonic on each subdomain Ωi, weobtain

|u0|2a,Ωi≤ 2

(|H∂Ωi→Ωi

(u)|2a,Ωi+ |H∂Ωi→Ωi

(u− u0)|2a,Ωi

)≤ 2

|u|2a,Ωi+ C

∑eij⊂∂Ωi

∣∣weij ∣∣2a,Ωi

≤ 2 |u|2a,Ωi

+C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)(34)

Now, we consider the local components ui := Ih (θi (u− u0)) based on the

partition of unity θiNi=1; see, e.g., [38, Lemma 3.4], where

θi(xj) :=dist

(xj , ∂Ωi

)∑

xj∈Ωk

dist(xj , ∂Ωk

)

and

Ωi

Ni=1

is an overlapping decomposition with overlap h corresponding to the

nonoverlapping decomposition ΩiNi=1. Then, u =N∑i=0

RTi ui and

|ui|2a,Ω′i = |ui|2a,Ω′i\Ωi+ |ui|2a,Ωi\Ωi

+ |ui|2a,Ωi .

Now, analogously to (34), we have

|ui|2a,Ωi ≤ |u− u0|2a,Ωi≤ 2 |u|2a,Ωi

+ 2 |u0|2a,Ωi

≤ 2 |u|2a,Ωi+ 2

2 |u|2a,Ωi+

C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)≤ 6 |u|2a,Ωi

+C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

).


Furthermore, we have

|ui|2a,Ωi\Ωi≤ 2 |ui − (u− u0)|2a,Ωi\Ωi

+ 2 |u− u0|2a,Ωi\Ωi

≤ C∑

eij⊂∂Ωi

∣∣weij ∣∣2a,Ωi+ 4 |u|2a,Ωi

+ 4 |u0|2a,Ωi

≤ C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)

+ 4 |u|2a,Ωi+ 4

2 |u|2a,Ωi+

C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)≤ 12 |u|2a,Ωi

+C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)and

|ui|2a,Ω′i\Ωi≤ C

∑eij⊂∂Ωi

∣∣weij ∣∣2a,Ωj

≤ C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

).

Therefore, we obtain

N∑i=0

|ui|2a,Ω =N∑i=1

(|u0|2a,Ωi

+ |ui|2a,Ω′i\Ωi+ |ui|2a,Ωi\Ωi

+ |ui|2a,Ωi)

≤N∑i=1

20 |u|2a,Ωi+

C

tol

∑eij⊂∂Ωi

(|u|2a,Ωi

+ |u|2a,Ωj

)≤ C

(1 +

1

tol

)|u|2a,Ω .

From Theorem 4, we directly obtain a condition number estimate for the pre-conditioned system.

Corollary 5. The condition number of the ACMS two level Schwarz operatoris bounded by

κ(M−1

ACMSA)≤ C

(1 +

1

tol

).

The constant C is independent of H, h, and the contrast of the coefficient functionA.

Proof. Using Assumption 1 and Assumption 2, we obtain, using standard ar-guments,

κ(M−1

ACMSA)≤C2

0ω (ρ (ε) + 1) ,

where C20 is the constant of the stable decomposition; cf. [38]. Since we use exact

solvers on the subdomains, we have ω = 1. From ρ (ε) ≤ NC and Theorem 4, weobtain the final estimate.

Let us note that in the numerical experiments considered in section 7, we haveNC = 4.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 5: Discontinuous coefficient function A with different types of channels andinclusions intersecting through the interface. The light blue color corresponds toa coefficient of Amin = 1.0 and the dark blue color to a high coefficient of Amax;1/H = 4.

Amax = 104 Amax = 106 Amax = 108

V0 tol it. κ dim V0 tol it. κ dim V0 tol it. κ dim V0

VMsFEM - 133 7.83 · 103 9 - 199 7.83 · 105 9 - 276 7.82 · 107 9

V tolACMS−D 10−1 21 4.74 69 10−1 22 4.75 69 10−1 22 4.75 69

V tolACMS−D

10−1 21 4.74 69 10−1 22 4.75 69 10−1 22 4.75 69

VtolACMS−N 10−2 20 4.42 69 10−2 22 4.43 69 10−2 22 4.43 69

V tolACMS−N

10−2 20 4.42 69 10−2 22 4.43 69 10−2 22 4.43 69

VtolACMS−N,1 10−2 20 4.31 69 10−2 21 4.33 69 10−2 21 4.33 69

V tolACMS−N,1

10−2 20 4.31 69 10−2 21 4.33 69 10−2 21 4.33 69

VSHEM 10−2 19 4.32 69 10−3 20 4.33 69 10−3 20 4.33 69

V tolACMS−R 2.0 24 6.50 69 2.0 24 6.50 69 2.0 24 6.50 69

Table 2: Numerical results for the coefficient function in Figure 5 with varyingcontrast Amax/Amin: tolerance for the selection of the eigenfunctions, iterationcounts, condition numbers, and resulting coarse space dimension for different coarsespace variants; 1/H = 4, H/h = 30, and δ = 2h; relative stopping criterion∥∥r(k)

∥∥2/∥∥r(0)

∥∥2< 10−8. The theory new presented in this paper covers the two

coarse spaces marked in bold face.

7. Numerical Results. We present numerical results which illustrate the ro-bustness of our overlapping Schwarz method with ACMS coarse space and supportour theory. Our numerical experiments also show that the dimension of our adap-tive coarse space is usually small, i.e., our iterative method does not degenerate toa direct solver. This is important since, as in other domain decomposition methodswith adaptive coarse spaces, the condition number is determined by the user butthe dimension of the resulting coarse space is determined by the algorithm and, ingeneral, only known ex post.

We consider the different variants of ACMS coarse spaces in numerical exper-iments. By V tolACMS−D, we denote the method from subsection 4.1, where Dirichlet

boundary conditions are used. By V tolACMS−N, we denote the method describedin subsection 4.2, where Neumann boundary conditions are used. In both casestol is the user-defined tolerance. If mass lumping is used, we denote the space byV tol

ACMS−Dor V tol

ACMS−N, respectively.

For the Neumann versions, we also consider variants using harmonic extensionsonly on slabs of width h; see subsection 4.2. These are denoted as V tolACMS−N,1 or,


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 6: Discontinuous coefficient functions A: circles and a channel intersectingseveral edges (left); comb-shaped inclusions intersecting the horizontal edges (right).The light blue color corresponds to a coefficient of Amin = 1.0 and the dark bluecolor to a high coefficient of Amax; 1/H = 4.

Coeff. function A from Figure 6 (left) Coeff. function A from Figure 6 (right)V0 tol it. κ dimV0 tol it. κ dimV0

VMsFEM - 318 1.36 · 108 9 - 282 3.77 · 107 9

V tolACMS−D 10−1 33 30.00 33 10−2 52 44.74 33

V tolACMS−D

2 · 10−1 33 20.00 33 10−1 52 44.74 33

VtolACMS−N 10−2 36 31.31 33 10−2 52 45.40 33

V tolACMS−N

10−2 33 31.31 33 10−2 52 45.40 33

VtolACMS−N,1 10−2 30 6.94 45 10−2 29 6.40 93

V tolACMS−N,1

10−2 30 6.94 45 10−2 29 6.40 93

VSHEM 10−3 28 6.92 45 10−3 29 6.39 93

V tolACMS−R 2.0 33 8.33 45 2.0 30 6.58 93

Table 3: Results for the coefficient functions in Figure 6: tolerance for the selectionof the eigenfunctions, iterations counts, condition numbers, and resulting coarsespace dimension for different coarse space variants; 1/H = 4, H/h = 30, and δ = 2h;maximum coefficient Amax = 108; relative stopping criterion

∥∥r(k)∥∥

2/∥∥r(0)

∥∥2<

10−8.

when lumping the mass matrix, V tolACMS−N,1

.

In the method V tolACMS−N,1 (cf. subsection 4.2), the left hand side of the eigen-value problem is assembled using discrete harmonic functions supported only onnarrow slabs of width h, cf. Figure 2, and in the methods V tolSHEM and V tolACMS−R

the work for the assembly and construction of the left hand sides of the eigenvalueproblems corresponds only to the degrees of freedom on the open edge. Therefore,we denote these four approaches as “economic”. Let us note that the right handside in all eigenvalue problems is computed by a one-dimensional integral over theedge and is therefore not affected.

In the non-economic approaches (i.e., V tolACMS−N, V tolACMS−N

, V tolACMS−D, and

V tolACMS−D

), harmonic extensions to the complete subdomains are used, leading toa higer computational cost. However, the use of the coefficient information on thecomplete subdomain can be beneficial: the resulting coarse space can be signifi-cantly smaller; see Table 3.

In all tables, we highlight in bold face the methods VtolACMS−N and Vtol

ACMS−N,1,which are supported by our theory; see section 6.


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 7: Discontinuous coefficient function A with many connected channels inter-secting the interface. The light blue color corresponds to a coefficient of Amin = 1.0and the dark blue color to a high coefficient of Amax; 1/H = 4.

Coeff. function A from Figure 7 (left)V0 tol it. κ dimV0

VMsFEM 460 1.94 · 108 9

V tolACMS−D 10−2 45 23.02 57

V tolACMS−D

10−2 45 23.02 57

VtolACMS−N 10−2 47 22.95 57

V tolACMS−N

10−2 47 22.95 57

VtolACMS−N,1 10−2 21 5.14 213

V tolACMS−N,1

10−2 21 5.14 213

VSHEM 10−3 23 5.03 213

V tolACMS−R 2.0 26 6.22 213

Table 4: Results for the coefficient functions in Figure 7: tolerance for the selectionof the eigenfunctions, iterations counts, condition numbers, and resulting coarsespace dimension for different coarse space variants; 1/H = 4, H/h = 40, and δ = 2h;maximum coefficient Amax = 108; relative stopping criterion

∥∥r(k)∥∥

2/∥∥r(0)

∥∥2<

10−8.

As described in subsection 4.3, all methods which are denoted by a bar, i.e.,V tol

ACMS−D, V tol

ACMS−N, and V tol

ACMS−N,1, use a lumped mass matrix. Here, the eigen-

value problems can be transformed into standard eigenvalue problems with smallcomputational effort, which leads to another reduction of computational work; thisholds true also for the SHEM coarse space. However, we observe that lumping themass matrix does not significantly affect the performance of the method; cf. Table 2,Table 3, Table 4, and Table 5.

In Table 2, for the coefficient function illustrated in Figure 5, we compare thedifferent approaches of adaptive coarse spaces. We see that for this problem theMsFEM coarse space VMsFEM is not sufficient: the condition number and the num-ber of iterations is large since the MsFEM coarse space cannot cope with severalheterogeneities intersecting through an edge. For all other methods, a small con-dition number κ (below 10) of the preconditioned operator can be obtained (if thetolerance tol is chosen appropriately) resulting in a number of approximately 20conjugate gradient iterations. We also observe that all methods are robust withrespect to variations in the contrast of the coefficient function. Note that the sizeof the coarse space is identical in all cases of adaptive coarse spaces.

At first sight, the results are similar for the coefficient functions depicted in Fig-ure 6 (left and right); see Table 3. However, now the dimensions of the coarse


1 LobPCG it. 2 LobPCG it. Direct eigensolverV0 it. κ dimV0 it. κ dimV0 it. κ dimV0

V 10−2

ACMS−D 23 4.75 69 22 4.75 69 22 4.75 69

V 10−1

ACMS−D23 4.75 69 22 4.75 69 22 4.75 69

V10−2

ACMS−N 22 4.43 69 22 4.43 69 22 4.43 69

V 10−2

ACMS−N22 4.43 69 22 4.43 69 22 4.43 69

V10−2

ACMS−N,1 21 4.33 69 21 4.33 69 21 4.33 69

V 10−2

ACMS−N,122 4.33 69 21 4.33 69 21 4.33 69

VSHEM 20 4.33 69 20 4.33 69 20 4.33 69

Table 5: Results for the coefficient function in Figure 5: iterations counts, conditionnumbers, and resulting coarse space dimension for different coarse space variants;1/H = 4, H/h = 30, and δ = 2h; maximum coefficient Amax = 108; relativestopping criterion

∥∥r(k)∥∥

2/∥∥r(0)

∥∥2< 10−8. Relative tolerance of 10−5 for LobPCG;

the tolerance for the selection in the SHEM coarse space is set to 10−3.

Coeff. function A from Figure 5 Coeff. function A from Figure 6 (left)H/h it. κ dimV0 it. κ dimV0

10 20 5.03 69 31 19.34 3320 19 4.30 69 31 25.65 3330 22 4.43 69 36 31.31 3340 21 4.82 69 36 37.04 3350 22 5.28 69 40 42.82 3360 23 5.76 69 45 48.60 3370 24 6.25 69 51 54.38 3380 25 6.75 69 56 60.16 33

Table 6: Results for the V 10−2

ACMS−N coarse space with varying H/h: iterations counts,condition numbers, and resulting coarse space dimension for different coarse spacevariants; 1/H = 4 and δ = 2h; maximum coefficient Amax = 108; relative stoppingcriterion

∥∥r(k)∥∥

2/∥∥r(0)

∥∥2< 10−8.

spaces differ. In the methods, where the construction of the eigenvalue problem iseconomic, the coarse space is larger. The reason is that, e.g., for the ring-like struc-ture in Figure 6 (left), these methods cannot detect that the ring is a connectedstructure and thus a single eigenvector is sufficient. This is even more pronouncedfor the comb-like structures in Figure 6 (right).

To investigate this effect further, we have considered the coefficient function inFigure 7, where on each edge a comb-like structure is placed in addition to a smallinclusion. Here, for each horizontal edge, the non-economic coarse spaces adds twoeigenvalues to the coarse space resulting in a dimension of 9 + 24 = 33. On theother hand, for the economic versions (which includes VSHEM), for each horizontaledge 7 eigenvectors are used. As a result, the coarse space is larger by a factor ofalmost 4 for the economic versions; see Table 4. It is clear that for an (artificial)example of a very fine comb on a very fine grid the coarse space constructed fromthe economic versions can be larger by an arbitrary factor compared to our coarsespaces V tolACMS−D and V tolACMS−N (or the corresponding lumped versions V tol

ACMS−D

and V tolACMS−N

).It is clear that the solution of the eigenvalue problems can be costly. First,

note that we need to solve only standard eigenvalue problems instead of generalizedeigenvalue problems. In particular, for the versions with a lumped mass matrix thisis computationally inexpensive; see subsection 4.3.

To show how the computational cost can be reduced further, in Table 5, we


show numerical experiments, where the eigenvalue problems are solved approxi-mately by using only one or two iterations of LobPCG [30] using the Choleskydecomposition of the Schur complement matrix as a preconditioner: instead of theeigenfunctions corresponding to the smallest eigenvalues of (23), we solve for thesmallest eigenvalues of

−BeijV = λSeijV

and use the inverse of Seij as the preconditioner in the LobPCG iteration. Theresults show that a single iteration of LobPCG is sufficient and that the use of twoiterations gives the same results as a direct eigensolver (LAPACK); see Table 5.The cost for approximating the eigenvectors thus is comparable to a Cholesky de-composition of the Schur complement Seij on the edge.

In Table 6, the results for the V tolACMS−N method are depicted for a varyingnumber of elements per subdomain, i.e., for a growing H/h. We observe that thedimension of the coarse space is robust with respect to different H/h, and for thecoefficient function from Figure 5, we even observe that the number of iterationsand the condition number increase only slightly.

In some methods, the choice of the tolerance tol is more difficult than in others.In the methods using Neumann boundary conditions (i.e., V tolACMS−N, V tol

ACMS−N,

V tolACMS−N,1, V tolACMS−N,1

, VSHEM) there is a clear spectral gap, and we observe that

the tolerance tol can be chosen relatively easy, for different values of H/h; in ourexperiments, we can indeed use tol = 10−2 for all our computations. However, forthe methods using Dirichlet conditions the choice is more difficult since the gap inthe spectrum is smaller.

Let us note that, currently, in the construction of the adaptive coarse space, wehave not taken advantage of the fact that the MsFEM space also removes certainbad eigenvalues. Therefore, in the best case, the size of our coarse could be reducedby one, for each edge with a heterogeneity.

The method described here for overlapping domain decompositions is related(as well as the other approaches [15, 16, 11, 17]) to the approach in [28] for FETI-DPand BDDC methods, where also a Poincare inequality is replaced by an eigenvalueproblem. However, for nonoverlapping domain decomposition, to obtain robustnessfor certain cases, an additional eigenvalue problem is needed to replace an extensiontheorem.

8. Conclusion. We have introduced an overlapping Schwarz method using anACMS-based coarse space. This coarse space uses eigenvalue problems on edges.We then have shown that the condition number of the method can be controlledindependently of the heterogeneities in the problem, and, finally, we have providedsupporting numerical experiments. We have observed that our coarse space canbe smaller by a large factor compared to other competing coarse spaces for specialcoefficient functions while still remaining computationally inexpensive. Indeed, wehave also shown numerically that the approximation of the local eigenvectors by asingle iteration of preconditioned LobPCG can be sufficient. Among the competingapproaches which generate small coarse spaces, to the best of our knowledge, ourapproach uses eigenvalue problems with the smallest dimension.

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