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Technical Report-IRIT: RT-APO-15-1

The anisotropic Steklov-Poincar´e matrix

Mario Arioli 1Daniel Loghin 2

June 12, 2015

1Universit´e de Toulouse-IRIT, 2 rue C. Camichel, Toulouse, 31000, France,

mario.arioli@gmail.com

2School of Mathematics, University of Birmingham, Edgbaston,Birmingham B15 2TT,

United Kingdom

Technical Report-IRIT: RT-APO-15-1

Abstract

In this work we analyse the Steklov-Poincar´e (or interface Schur complement) matrix

arising in a domain decomposition method in the presence of anisotropy. Our problem

is formulated such that three types of anisotropy are being considered: reﬁnements

with high aspect ratios, uniform reﬁnements of a domain with high aspect ratio and

anisotropic diﬀusion problems discretized on uniform meshes. Our analysis indicates

a condition number of the interface Schur complement with an order ranging from

O(1) to O(h−2). By relating this behaviour to an underlying scale of fractional

Sobolev spaces, we propose optimal preconditioners which are spectrally equivalent

to fractional matrix powers of a discrete interface Laplacian. Numerical experiments

to validate the analysis are included; extensions to general domains and non-uniform

meshes are also considered.

RT-APO-15-1

Contents

1 Background and motivation 2

2 The anisotropic Steklov-Poincar´e matrix 2

3 Eigenvalue analysis 4

4 Discrete fractional Sobolev norms for preconditioning 9

5 Numerical experiments 11

5.1 Thetwo-domaincase........................... 11

5.2 Otherexperiments ............................ 12

6 Conclusion 13

1 Background and motivation

Various analyses going back to the 1980s considered the discrete Steklov-Poincar´e op-

erator arising from two-domain decomposition methods for linear systems resulting

from ﬁnite element discretizations of Poisson’s problem on uniform meshes for simple

polygonal shapes [5], [6], [7], [3], [4]. In all cases, the aim was to provide an eigen-

value analysis of the Schur complement (or capacitance) matrix arising in the case of a

tractable decomposition. This eﬀort resulted in the design of several interface precon-

ditioners which are loosely speaking spectrally equivalent to a square-root Laplacian

matrix associated with the interface. In turn, this fact is related to the boundedness

of the Steklov-Poincar´e operator when acting on fractional Sobolev spaces of index

θ= 1/2. Due to the complexity of the matrix square-root computation, the im-

plementation of the preconditioners was expected to require fast Fourier transforms,

under the assumption of mesh uniformity. Generalisations to the non-uniform case

led to expensive implementations.

We consider in this work a generalisation of the contexts considered previously

which allows for various types of anisotropy. In particular, we work with a scaled

discrete Laplacian which can be associated with an anisotropic mesh or with an

anisotropic diﬀusion operator. Both cases arise naturally in practice, not least in

contexts that require fast, parallel algorithms for generating a solution. We pro-

vide an eigenvalue analysis in terms of both geometric and anisotropy parameters,

which allows us to describe the dependence on anisotropy of the conditioning of the

interface Schur complement. In particular, we show that the interface Schur comple-

ment is spectrally equivalent to more general fractional powers θ∈[0,1] of a discrete

Laplacian associated with the interface. We validate this conclusion with a range of

numerical results. We also consider more general experiments inspired by the existing

analysis.

2 The anisotropic Steklov-Poincar´e matrix

Let a∈R+and let Ω = (−1,1) ×(−a, a). Consider the ﬁnite element solution of

−∆u=fin Ω,

u= 0 on ∂Ω,(1)

using a partition of Ω into two equal domains separated by a horizontal boundary

Γ = {(x, 0) : −1≤x≤1}and subdivided uniformly into equal triangles with sides

hx= 2/(n+ 1), hy= 2a/(n+ 1) with n= 2m+ 1, m ∈N. Let

Tk:= tridiag[−1,2,−1] ∈Rk×k

denote a scaled FEM discretisation of −d2/dx2on a mesh with kinterior points and

let Ik∈Rk×kdenote the identity matrix. With this notation, the discrete Laplacian

matrix L∈Rn2×n2is given by

L=hx

hy

Tn⊗In+hy

hx

In⊗Tn=1

aTn⊗In+aIn⊗Tn:= La.

2

The above expression for Lacorresponds also to the following alternative formulations:

•the discretisation of a standard Laplacian on an anisotropic mesh with aspect

ratio a=hy/hx;

•a uniform discretisation of the operator arising in the following anisotropic dif-

fusion problem

−a∂xx +1

a∂yy u=gin Ω,

u= 0 on ∂Ω,

(2)

where Ω is any square.

We will refer to Laabove as the discrete anisotropic Laplacian. We also deﬁne

L0:= 1

aTn⊗In, L∞:= aIn⊗Tn

as the dominant parts of Lafor asmall and large, respectively. Using a lexicographic

ordering, the corresponding matrices are

L0=1

a

2In−In

−In2In−In

.........

2In−In

−In2In

, L∞=a

Tn

Tn...

Tn

Tn

.

Using the natural domain decomposition ordering (corresponding to a permutation

containing the nodes from domains 1, 2 and ﬁnally boundary), the above matrices

have the following structure:

L0=1

a

Tm⊗In−em⊗In

Tm⊗In−e1⊗In

−eT

m⊗In−eT

1⊗In2In

, L∞=a

Im⊗Tn

Im⊗Tn

Tn

,

where eidenotes the ith column of Im. The corresponding Schur complements are

S0=1

a

2In−X

i∈{1,m}

(eT

i⊗In)(T−1

m⊗In)(ei⊗In)

=1

a

2In−X

i∈{1,m}

(eT

iT−1

mei)In

=1

a2−(T−1

m)11 −(T−1

m)mmIn

=2

a1−(T−1

m)11In

and

S∞=aTn.

3

Remark 2.1 If we let dk= det Tk>0then dk> dk−1and since

(T−1

m)11 =dm−1

dm

<1,

the Schur complement S0is always a positive scaling of the identity.

We see therefore that the Schur complement of the two-domain decomposition has

two very diﬀerent limiting cases, as we vary the aspect ratio of a rectangular domain

while keeping the number of nodes ﬁxed in each direction. This change in character

will have an impact on preconditioning.

3 Eigenvalue analysis

Let us consider the Schur complement of Lafor a general value of a > 0. Let Lm,n

denote the Laplacian corresponding to a discretisation with minterior nodes in the

x-direction, respectively, nin the y-direction. Dropping the subscript afor now, the

Laplacian for the original problem is L=Ln,n where

Ln,n =1

a

aLm,n −em⊗In

aLm,n −e1⊗In

−eT

m⊗In−eT

1⊗InaT

where

Lm,n =1

aTm⊗In+aIm⊗Tn

and

T=2

aIn+aTn.

Then the Schur complement is (using the block Toeplitz character of Lm,n)

S=T−1

a2X

i∈{1,m}

(eT

i⊗In)L−1

m,n(ei⊗In) = T−2

a2(eT

m⊗In)L−1

m,n(em⊗In).

Let now Tk∈Rk×khave the eigenvalue decomposition

Tk=VkDkVT

k

with

(Dk)ii = 2 1−cos iπ

k+ 1=: µ(k)

i,

and

Vk= [v1, . . . , vk],(vj)i=r2

k+ 1 sin ijπ

k+ 1, i, j = 1, . . . , k.

4

Let Tm=VmDmVT

m, Tn=VnDnVT

n; with this notation, the inverse of Lm,n is given

by

L−1

m,n = (Vm⊗Vn)aIm⊗Dn+1

aDm⊗In−1

(Vm⊗Vn)T

:= (Vm⊗Vn)D−1

m,n (Vm⊗Vn)T

Now,

(Vm⊗Vn) (em⊗In) = (Vmem)⊗Vn=vT

m⊗Vn

and similarly

(eT

m⊗In) (Vm⊗Vn)T=vm⊗VT

n.

Let (Dm,n)i∈Rn×ndenote the ith diagonal block of Dm,n ∈Rmn×mn with i=

1, . . . , m. It follows that

(Dm,n)i=aDn+1

a(Dm)iiIn.

Hence,

S=T−2

a2(vT

m⊗Vn)D−1

m,n vm⊗VT

n

=Vn2

aIn+aDnVT

n−2

a2

m

X

i=1

(vm)2

iVn(D−1

m,n)iVT

n

=Vn"2

aIn+aDn−2

a2

m

X

i=1

(vm)2

i(D−1

m,n)i#VT

n

=Vn"2

aIn+aDn−2

a2

m

X

i=1

(vm)2

iaDn+1

a(Dm)iiIn−1#VT

n.

Thus, the eigenvalues of Sare

λj(S) = aµ(n)

j+2

a"1−

m

X

i=1

(vm)2

i

a2µ(n)

j+µ(m)

i#,(j= 1, . . . , n).

Note that labeling the eigenvalues of Tnin increasing order, i.e., µ(n)

1≤ · · · ≤ µ(n)

n,

yields a labeling with a similar ordering for the eigenvalues of S:λ1(S)≤ ·· · ≤ λn(S).

Now for large k

µ(k)

1= 2 1−cos π

k+ 1≈π

k+ 12

, µ(k)

k= 2 1−cos kπ

k+ 1≈4−π

k+ 12

.

In order to derive asymptotics for the smallest eigenvalue of Swe let h= 1/(m+ 1) =

hxand note that

(vm)2

i=2

m+ 1 sin2imπh = 2hsin2iπh

5

and

µ(m)

i= 4 sin2iπh

2

so that (vm)2

i

µ(m)

i

=h

2

sin2(iπh)

sin2(iπh/2) = 2hcos2iπh

2=h(1 + cos iπh).

Hence,

λ1(S)≥aµ(n)

1+2

a"1−h

m

X

i=1

(1 + cos iπh)#=aµ(n)

1+2h

a.

Since

µ(n)

1≈πh

22

,

for large mwe ﬁnd

λ1(S)≈aπh

22

+2h

a.

We can similarly derive an upper bound on the spectrum of S. For a≥1 we ﬁnd

λn(S)≤aµ(n)

n+2

a≤4a+2

a.

On the other hand, for 0 < a 1,

λn(S)≤4a+2

a"1−h

m

X

i=1

(1 + cos iπh)#= 4a+2h

a.

Setting a=O(hθ), θ∈R, we get the following asymptotic behaviour for the extreme

eigenvalues of S:

λ1(S) = O(h2+θ), θ ≤ −1/2

O(h1−θ), θ > −1/2., λn(S) = O(hθ), θ ≤1/2

O(h1−θ), θ > 1/2.

Hence, the condition number of Shas the following behaviour for a=O(hθ):

κ2(S)∼

O(h−2), θ < −1/2,

O(h2θ−1), θ ∈[−1/2,1/2],

O(1), θ > 1/2.

It turns out that the bounds are pessimistic for λ1(S) if θ∈(−1,0) and for λn(S) if

θ∈(0,1). This reﬁnement is included below.

Lemma 3.1 The following equations hold:

Zsin2(x)

%2+ sin2(x

2)dx= 2h2%2x+x+ sin(x)−4%p%2+ 1 tan−1p%2+ 1 tan(x

2)

%i; (3)

6

Zπ

0

sin2(x)

%2+ sin2(x

2)dx= 2 2π%2+π−2π%jp%2+ 1; (4)

M= max

x∈[0,π]

sin2(x)

%2+ sin2(x

2)≤4; (5)

Zπ

0

sin2(x)

%2+ sin2(x

2)dx=πh

m

X

i=1

sin2(πih)

%2

j+ sin2(π

2ih)+E(6)

with |E| ≤ 2Mh

Proof: Equations (3) and (4) can be easily veriﬁed. The bound (5) follows from

sin2(x)

%2+ sin2(x

2)≤sin2(x)

sin2(x

2)= 4 cos2((x

2)≤4.

Formula (6) and the error bound follow from the usual Riemann summation formula

for the quadrature. Let subdivide (0, π) in mequal intervals, of size h=π

m+1 ,

(xi, xi+1) and denote

f(x) = sin2(x)

%2+ sin2(x

2).

Owing to f(x0) = f(0) = f(xm+1) = f(π) = 0 and f(x)≥0, the errors

E1=Zπ

0

f(x)dx−h

m

X

i=0

f(xi) and E2=Zπ

0

f(x)dx−h

m+1

X

i=1

f(xi)

are both reduced to

E=Zπ

0

f(x)dx−h

m

X

i=1

f(xi).

Thus, assuming that M∈(xˆ

k, xˆ

k+1), we have

|E| ≤ h

ˆ

k

X

k=0f(xk)−f(xk+1)+h

m

X

k=ˆ

kf(xk)−f(xk−1)+

hM−min(f(xˆ

k), f (xˆ

k+1)=

hmax(f(xˆ

k), f (xˆ

k+1)+hM ≤2hM.

7

Let now %2

j:= 1

4a2µ(n)

j. We have

λj(S) = 1

a"4%2

j+ 2 −2

m

X

i=1

2hsin2(iπh)

a2µ(n)

j+ 4 sin2(π

2ih)#

=1

a"4%2

j+ 2 −πh

π

m

X

i=1

sin2(πih)

%2

j+ sin2(π

2ih)#

=1

a"4%2

j+ 2 −1

πZπ

0

sin2(x)

%2

j+ sin2(x

2)dx+E#

=1

a4%2

j+ 2 −2

π2π%2

j+π−2π%jq%2

j+ 1+E

=1

a4%jq%2

j+1+1

aE.

Thus, if a=O(hθ) with −1≤θ≤0, we have %2

1≈a2πh

22<1 and hence

λ1(S)≈πh. (7)

Similarly, if a=O(hθ) with 0 ≤θ≤1, we have %2

n=1

4a2µ(n)

n=O(h2θ)<1 and hence

λn(S)≈µ(n)

n=O(1).(8)

Finally, if a=O(hθ) with θ > 1, the error Ewill dominate and all eigenvalues will be

O(h1−θ) and greater than 1. The resulting asymptotics for the extreme eigenvalues

of Sare included below:

λ1(S) =

O(h2+θ), θ ≤ −1

O(h), θ ∈(−1,0]

O(h1−θ), θ > 0.

, λn(S) =

O(hθ), θ ≤0

O(1), θ ∈(0,1]

O(h1−θ), θ > 1

.

Hence,

κ2(S)∼

O(h−2), θ < −1,

O(hθ−1), θ ∈[−1,1],

O(1), θ > 1.

(9)

A numerical validation of the above asymptotics is given in Fig. 1, where we compare

the condition number κ2(Sθ) = λn(Sθ)/λ1(Sθ) with the function

zθ=

h−2, θ < −1

hθ−1, θ ∈[−1,1]

1, θ > 1

8

Figure 1: κ2(Sθ) v.s. zθ.

for θ∈[−2,2]. The choice of h= 1/(m+ 1) corresponds to m= 104; however, other

choices yield similar close approximations.

The spectral dependence described by (9) has a direct consequence with regard to

the preconditioning approach required in order to achieve optimal performance with

respect to the geometric parameters of the problem. We discuss this next.

4 Discrete fractional Sobolev norms for precondi-

tioning

It is known that the Schur complement associated with a domain decomposition of

problem (1) is spectrally equivalent to a square-root Laplacian. In general, assuming

an elliptic operator discretized on a quasi-uniform partition of size h, a domain decom-

position approach will yield an interface Schur complement with condition number

κ2(S) = O(h−1). A matrix with the same spectral property is a discrete square-

root Laplacian acting on the interface, which can be seen as a discrete norm for

the fractional Sobolev space of index 1/2. It is clear that for our modiﬁed problem

the asymptotics (9) point to a diﬀerent choice of preconditioner than the standard

square-root Laplacian.

The function spaces relevant in the context of our problem are the fractional

Sobolev spaces Hα

00(Γ) of functions acting on the interface. These are interpolation

spaces (see [8]) parameterized by α∈[0,1]:

Hα

00(Γ) := [H1

0(Γ),L2(Γ)]1−α.

Matrix norms Hαfor continuous piecewise polynomial subspaces of Hα

00(Γ) were in-

troduced in [1]. For our problem, a choice represented in the ﬁnite element basis is

9

given below:

Hα=hxMn1

h2

x

M−1

nTn1−α

,(10)

where

hxMn=hx

6tridiag[1,4,1]

is the mass matrix assembled on the interface Γ. A simpliﬁed version of this precon-

ditioner corresponds to using a lumped version of the mass matrix (i.e., a diagonal

matrix with entries equal to the rowsums of the original mass matrix). This choice

will also be considered in our numerical experiments.

One can show that the orthogonal matrix Vndiagonalises both Mnand Tnand

VnMnVn=DMis such that 1

3I≤DM≤I.

Assuming an increasing ordering of the diagonal elements in DM, the eigenvalues

{φj}n

j=1 of Hαare:

φj=hdj µ(n)

j

djh2!1−α

.

In particular, we have

φ1=hd1 4 sin2πh

2

d1h2!1−α

and φn=hdn 4 sin2(m+ 1)πh

2

dnh2!1−α

.

Taking into account that d1≈1/3 and dn≈1, we have

φ1≈h

63

2π21−α

and φn≈h

2h

22α−2

,

so that we essentially recover the result of Lemma 2.5 in [1]:

κ2(Hα) = O(h2α−2).

If we compare the values of λ1and λn, respectively, to the values of φ1and φnfor

θ∈[−1,1] then, choosing

α∗:= α∗(θ) =

0, θ < −1,

1+θ

2, θ ∈[−1,1],

1, θ > 1.

(11)

we get

κ2(SH −1

α∗(θ)) = O(1).(12)

Thus, an optimal preconditioner for the Schur complement arising in a two-domain

formulation of problems (1), (2) parameterized by ais Hα∗(θ), where θ= log a/ log h.

We verify numerically this conclusion in the next section.

10

Remark 4.1 The proposed preconditioner requires the computation of fractional pow-

ers of a certain matrix followed by the application of its inverse to a given vector. This

is an expensive procedure. In practice, a Krylov method, involving a sparse generalized

Lanczos procedure can be used successfully – for more details, see [1], [2].

5 Numerical experiments

We solved the anisotropic diﬀusion problem

−a∂xx +1

a∂yy u= 1 in Ω,

u= 0 on ∂Ω,

(13)

on the unit square using a continuous piecewise linear ﬁnite element Galerkin method

on a uniform subdivision of Ω; the corresponding linear system has the standard block

structure arising under a non-overlapping domain decomposition permutation:

Lu=LII LIΓ

LΓILΓΓ uI

uΓ=f.

We employed a standard full GMRES method [10] with tolerance 10−6and right-

preconditioner

P=LII LIΓ

Hα∗

with α∗as deﬁned in (11) which ensures that

sp LP −1= sp SH −1

α∗∪ {1},

where sp(·) denotes the spectrum set.

Remark 5.1 As a→ ∞, the solution exhibits directionality (it becomes elongated

and constant in the y−direction. Sharp layers are also typical near the top/bottom

boundaries. Our subdivisions will therefore have interfaces parallel to the x−axis.

5.1 The two-domain case

Given the optimality result (12), we expect performance independent of aand h. This

is indeed the case, as can be seen in Fig. 2where we employed both the mass matrix

and its lumped version. We employed three levels of reﬁnement, corresponding to

sizes n2∈ {16,641,66,049,263,169}. The number of GMRES iterations is displayed

as a function of afor preconditioned runs with both Hα∗and with the standard choice

H1/2. Evidently, the latter choice exhibits both aand h(or level) dependence. Our

analysis yielded preconditioner Hα∗(θ)which eliminates any dependence on the pa-

rameters a, h. As a→ ∞ we ﬁnd α∗= 0 and the performance approaches the optimal

limit of 2 iterations as described in [9] – this is due to the fact that Lapproaches

L∞for which the Schur complement is Tn– a scaling of Hα∗. Finally, we note that

11

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

a

GMRES its

levs 1,2,3

lev1

lev2

lev3

0 200 400 600 800 1000

0

10

20

30

40

50

60

70

a

GMRES its

levs 1,2,3

lev1

lev2

lev3

Figure 2: GMRES iterations for two-domain case using preconditioners Hα∗(blue)

and H1/2(red). The lumped mass matrix was employed for the plot on the right.

the lumped mass matrix results represent an improvement when preconditioning with

H1/2, although the same mesh and domain dependence are observed.

The remaining experiments are performed using a lumped mass matrix in the ex-

pression (10). We do not present the results corresponding to H1/2- in all cases the

behaviour is similar to that displayed in Fig. 2.

5.2 Other experiments

The above preconditioning technique can be extended naturally to the case of de-

compositions into 1 ×Nsubdomains with interfaces parallel to the x-axis. For this

case, the interface preconditioner (2) will now have the matrices Mn, Tnreplaced with

block-diagonal matrices ⊕iMni,⊕iTni, with nithe number of nodes on each of the

N−1 interfaces. The results corresponding to level 3 are included in Table 1for

a range of aand using lumped mass matrices on both uniform and quasi-uniform

meshes, as exempliﬁed in Fig. 3. It is evident that there is no domain dependence;

we also found the performance to be independent of the mesh and the parameter a,

in very much the same way as for the case N= 2.

We also experimented with domains other than the unit square. We chose to

log2a= 0 2 4 6 8 10

N= 2 6/5 8/7 9/8 8/7 5/5 4/4

4 9/8 9/8 9/9 8/8 5/5 4/4

8 9/13 9/9 9/10 8/8 5/5 4/4

Table 1: GMRES iterations with preconditioner Hα∗(lumped mass matrix) for sub-

divisions into Nhorizontal strips using uniform/quasi-uniform meshes.

12

Figure 3: Regular subdivisions into 4 strips: uniform and quasi-uniform meshes.

work with a hexagon and ellipse as illustrated in Fig. 4. In both cases the results

are very similar to the case of the square domain, with independence of geometric

parameters as well as a-independence. Table 2contains the results for the case of the

decomposition into several horizontal strips and quasi-uniform meshes.

log2a= 0 2 4 6 8 10

N= 2 8/6 8/7 7/6 5/4 4/4 4/3

4 10/9 9/10 8/8 6/5 5/5 4/5

8 14/11 10/10 9/9 7/6 5/6 4/7

Table 2: GMRES iterations with preconditioner Hα∗(lumped mass matrix) for sub-

divisions into Nhorizontal strips for the hexagonal/elliptical domains of Fig. 4.

Figure 4: Regular subdivisions into 4 strips using quasi-uniform meshes.

6 Conclusion

We investigated the eﬀect of anisotropy on a standard two-dimensional domain de-

composition problem with two subdomains and for which the interface is parallel to

13

one of the axes. In particular, our test problem is a diﬀusion problem parameterized

so that the resulting linear system corresponds to either

•a discretized Laplacian on an anisotropic mesh reﬁnement of a ﬁxed domain;

•a discretized Laplacian on a uniform reﬁnement of a domain extending in one

direction;

•an anisotropic diﬀusion problem on a ﬁxed domain.

Therefore, our analysis will have diﬀerent implications in each case.

1. Laplacian discretized on an anisotropic mesh using stretched elements. This is

a non-standard approach in practice – however, there may be situations where

due to the shape of the domain the mesh anisotropy is inevitable (e.g., crystal

domain in [2]). For such problems, the optimal α∗is likely to be diﬀerent from

the standard value α∗= 1/2. Our analysis does not extend to complex domains;

however, the optimal value can often be identiﬁed experimentally.

2. Laplacian discretized uniformly on a collapsing (extending or compressing) two-

dimensional strip. Our analysis indicates that discrete Steklov-Poincar´e ma-

trix has an eigenvalue distribution commensurate with an underlying fractional

Sobolev space of index other than 1/2. In particular, for the case where the

strip is compressed (a→0), the limit space is L2(Γ); given that for a= 1,

the index is 1/2, this indicates a loss of regularity of the underlying Sobolev

space akin to applying the trace operator. Indeed, the procedure of collapsing

a dimension is standard in the analysis of trace operators.

3. Anisotropic diﬀusion problem. For the case of diﬀusion dominating in a certain

direction, generating a domain decomposition compliant with the directional-

ity of the solution allows for the deﬁnition of an interface preconditioner with

performance uniform in all the problem parameters.

Finally, we wish to point out once again that working with fractional powers of dis-

crete operators need not be computationally prohibitive in a preconditioning context.

Fast Krylov methods are available for general discretizations [1], [2]; in particular,

working with generalised Lanczos decompositions allows for the fast and sparse im-

plementation of the proposed discrete fractional Sobolev norm matrices.

14

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