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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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