ETNA
Electronic Transactions on Numerical Analysis.
Volume 36, pp. 168-194, 2010.
Copyright  2010, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
CONDITION NUMBER ANALYSIS FOR VARIOUS FORMS OF
BLOCK MATRIX PRECONDITIONERS
OWE AXELSSON AND JÁNOS KARÁTSON Dedicated to Richard S. Varga on the occasion of his 80th birthday
Abstract. Various forms of preconditioners for elliptic finite element matrices are studied, based on suitable
block matrix partitionings. Bounds for the resulting condition numbers are given, including a study of sensitivity to
jumps in the coefficients and to the constant in the strengthened Cauchy-Schwarz-Bunyakowski inequality.
Key words. preconditioning, Schur complement, domain decomposition, Poincaré–Steklov operator, approximate block factorization, strengthened Cauchy-Schwarz-Bunyakowski inequality
AMS subject classifications. 65F10, 65N22
1. Introduction. Preconditioning is an essential part of an efficient iterative solution
method when solving large-scale linear and nonlinear systems of equations. This paper deals
with systems arising from the finite element discretization of elliptic partial differential equations.
The efficiency of a preconditioner is mostly judged by the condition number of the resulting preconditioned operator, and in applications it is important to know whether the condition
number depends critically on certain problem parameters such as jumps in the material coefficients. The most efficient preconditioners are based on some block partitioning of the
matrix. Common structures are block tridiagonal and two-by-two partitionings. Elementwise
constructed preconditioners can be efficient as they can be constructed locally and relatively
cheaply but still can provide a significant reduction of the condition number of the unpreconditioned operator.
Block tridiagonal matrices arise in many applications. For instance, such a structure
arises when decomposing the domain of definition of an elliptic operator using unidirectional
stripes, or more generally, for a decomposition such that (in addition to a corresponding
portion of the original boundary) each subdomain has a common boundary only with its
previous and next neighbours in the sequence of subdomains. This subdivision can often be
done according to different values of the coefficients in the differential operator, i.e., different
materials in the underlying physical domain. Each diagonal block in the matrix corresponds
to the restriction of the operator to one of the subdomains, and ordering the nodes in each
domain in groups and then the domains consecutively, results in a block tridiagonal matrix.
An interesting example of matrices of two-by-two block structure arises by ordering the
interior domain nodes separately from the interface nodes and ordering all interface nodes
last. This in turn results in a block diagonal submatrix with uncoupled blocks, which are only
coupled to the interface nodes ordered last. The part of the system which corresponds to the
different interior node sets can then be solved in parallel.
In both cases there arise Schur complement matrices when solving systems with these
matrices. For the block tridiagonal case, they arise at each step of the consecutive elimination
of the pivot blocks, and in the latter case by elimination of the interior nodes. Schur complement matrices are in general full matrices and must be approximated by some sparse matrix
Received March 29, 2009. Accepted for publication August 12, 2009. Published online on September 7, 2010.
Recommended
by J. Li.
Department of Information Technology, Uppsala University, Sweden, and Institute of Geonics AS CR, Ostrava,
Czech Republic (owea@it.uu.se).
Department of Applied Analysis, ELTE University, H-1117 Budapest, Hungary (karatson@cs.elte.hu).
168
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169
BLOCK MATRIX PRECONDITIONERS
in the construction of the preconditioner. The construction of such approximations and the
analysis of condition numbers of the Schur complements, both on continuous and discrete
level, are the main topic of this paper.
We give here first a general framework for the analysis of approximations
of Schur com
, and let ,
plement matrices. Consider a symmetric positive definite bilinear form be two subspaces of a linear space , where the intersection of , contains only the
trivial element. Here the spaces can be more general function spaces as well, but in our applications is a finite element space, i.e., spanned by a set of finite element basis functions. As
has been shown in early publications [3, 4, 11, 13, 14], the strengthened Cauchy-SchwarzBunyakowski inequality plays a fundamental role in the analysis of matrices partitioned in
two-by-two block form. The inequality takes the form
!" #$ %'&
)
*&
(
,+.-
where
is the smallest such constant and is referred to as the CBS constant. In fact,
is the cosine of the angle between the two subspaces, measured by the inner product .
/
/
For matrices in the form
/1032
/
the CBS inequality
can be written
as
/
/
8 9
8
8 9
$
8
4
5
76
/
: 8 9
/
$
0FE
8
$
/HG
"
#$ /
/
% 8
/IG
/
G
#$
&<;>=@?
8
&<;=BADC
#$ #$
E
C Alternatively, we can define by
5
, where
denotes
$
the spectral radius. Hence measures the size of the off-diagonal blocks in relation to the
/
/
diagonal blocks. It is readily
seen that
0R/
-KJL 8 9
/
/
J
5
where M QP
number satisfies
(1.1)
G $ / 8 8 9
M 8
8 9
8
$
% 8
&N;
=OA is the Schur complement matrix. Hence the condition
4
S
/
G
$
M -
-JL
C
The remainder of the paper is organized as follows. In Section 2 we discuss briefly the
factorization of block tridiagonal matrices. We are in particular interested in approximating
the arising Schur complement matrices in such a way that their quality is insensitive to jumps
in the coefficients in the differential operator. This will be discussed in Section 3. Section 4
is devoted to an algebraic derivation of condition numbers in the approximations of matrices
partitioned in two-by-two block form, where the pivot block is block diagonal, such that the
condition number depends only on the CBS constant. A continuous analogue of the method
of Section 3 is presented on some model problems in Section 5. In Section 6 we analyze the
case of using elementwise approximations of Schur complements, and how to define them so
that they also become insensitive to coefficient jumps.
/
/
Except when it is otherwise stated,
the inequalities
UTV
E
+UT
/
/
TWJ
between/ two symmetric matrices (of the same order)
mean
that
is positive / semidefinite
or positive
definite,
respectively.
The
notation
for
a
symmetric
positive semidefinite
/
01X YZ5[ /
X Y]
/
matrix
stands
for
its
maximal
eigenvalue.
The
spectral
condition
number
of is defined
$\
=
by S
.
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170
O. AXELSSON AND J. KARÁTSON
/
2. Recursive approximation of Schur complements. Let us consider a symmetric,
block structure
positive definite matrix with/ tridiagonal
/
/
/10_^`
/
(2.1)
/
0n/
]ml
c
C7C7C
$
C7CeC
b
b
l]
9
Here
for all o
qp
/
`a
b
C7C7C
d
$
C7CeC
/
Yg
C7f C7Y C
C7CeC
b YKhji
/ C7Y
C7C
k i
G
MNrts(u
Y
7CeC7C€ ]
M
/
u
and su is the strictly lower block triangular part of
are0ndetermined
recursively as
/
gP
M
0n/
$
CƒC„CƒC„C
/
J
P
$
M ‚
CƒC„Cƒ] C„C 01/]„]
M
5
9
MNrs
takes the form
o~@
where M
M Here the Schur complements M
(2.2)
M
C
/
. The/1
exact
block factorization
of
0
G
0wv5xy{ze|}
C7C7C
b
J
P
/
G
5 M
/]f ]
G
.
]G
4
/g]
G
M
f]
G
,
for o
.
The application
]
0 of this factorization to solve a linear system involves the solution of the
block triangular factors
using a forward and a backward sweap. ] At each of them, systems
-BeC7C7C7
, appear that must ] G be solved. In addition, matrix-vector multiwith matrices M , o
plications with s u and s 9u , respectively,
appear. In general,
the M are full matrices and their
]
]
construction
and
the
computation
of
actions
of
can
be
expensive.
M
]
0‡v5xy{ze|}
Y
]G
†
whichG is sparse and the computation
Our goal is to approximate M by some matrix
0
eCƒC„C„
of † and †
applied to vectors are cheap. We
define †
o~@ †' †ˆ
†
,
9
and let the preconditioner ‰ be defined by ‰
.
At
the
same
time,
†
†
†
rLs u
rLs u
/
]
]
]G
G
the approximation must be sufficiently
it hasbeen
shown
in [1] that
/g]„] accurate. For instance,
/‚]ƒ]
,
Š

‹
„
Œ
Ž
the following lower bound
.
‰
S
†
M
] holds for the condition number: S
are generally inexpensive to solve, we could try
Since systems with the matrices
as an approximation of M . At each step of the method we then deal with a two-by-two block
/]„]
/g]f ]„
matrix in the form
2
/]„
f ]‘/]„
f ]ƒ
where
/
/
]„]
^`
/
`
`a
b
0”/]„
f ]ƒ
/
$
5
$
..
.
]ƒ
56
/
/
0
..
..
.
b
..
.
CeC7C
f]
/]„]
.
b
G
C7C7C
b
C7C7C
$d
C7CeC
/g]ƒ
C7CeC
b
/]f ]
0
b
hji
/]„] i
k i
G
..
.
-Bc’eCƒC„Cƒ…wJ,-@C
o
/]f ]„
/“]„
/
]ƒG 
f ]„
]„
0
]„
-{\4-JL f ]ƒ
As pointed
out in J the introduction,
the accuracy of the approximation
]„
of M HP
is given by S
.
M
Here
depends on the stage of the elimination. For a model elliptic problem with constant
coefficients on a unit square and constant mesh size • , it can be seen (see, e.g., [1]) that the
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171
BLOCK MATRIX PRECONDITIONERS
PSfrag replacements
™Ÿž
œ7— 
š —
š — ˜€™O¡
›
F IG . 2.1. The functions
]„
›
and £© ¤jª~§ for which the CBS constant is taken.
£© ¤¦¥¨§
]0
š —$¢
– — ˜€™
–5—
]
ž
œe—  basis
functions
which
at stage
o are as]„shown in Figure 2.1, where
¨give
­ ¯° rise ° to the -constant
0
]„
+
+«J 0.
8
8
8
,
b
• .
]
!
-gJ1 \{²
Y <0±
Since , one finds
•
. In the limit as oK³
0
„
]

0
Y
]„ ²
-IJ
and
, one finds
³
• . For more general problems, such as with variable
-J<´ \O²
-JN´ coefficients, one gets
and
•
• . It follows that the quality
of this approximation deteriorates with increasing stage numbers o .
As discussed in several publications (see, e.g., [1, 9, 18]), the approximation method can
be improved in various ways. A simple method is to use a diagonal compensation in some
form, where
] 0Q/g]ƒ]
]
(2.3)
†
]
µ
where
J¶µ
P
is a diagonal matrix, such
that
] ]
0n/]f ]
µ
(2.4)
]G
G
]
for some
positive vector .
] given
First, let be the eigenvector to
value · of this matrix. Then
/
]f ]
]G
G
G
/¹]f ]
/]
]G
f]
G
corresponding to the smallest eigen
]
·
]
]
]
f]
G
]0
]q¸€]
µ
/
†
]
µ
](0
/]
G
5†
]
]
f]
]
/g]„]
/g]f ]
01v5x¨y{z7|}
where
µ
o~B
u
/
Hence ‰
Š
‰ /
/
‰
/ J
J
]ƒ]>$0 † /gY
Y su
r
7C7CeC€
]
†
/]f ]
G
5†
E
‰
f]
G
u
u
/]
۠
f]
G
is G the smallest
/]„]
J
Š
b
C
/
G
J»µ
9
s
, and
]G
G
r
, which yields
†
/g]
]G
G
G
G
i.e.,
for the o th block. J Since G ·
/]„]
])0 ·
] is a multiple of the identity matrix
Šºµ
and then
eigenvalue,
it follows that
5†
J»µ
. Here
†
/«0
G
«-@C
/
G
G
/
In this method we must estimate the smallest eigenvalue of ‰
, which we will not do here
as the choice of † should rather
be¾!]
such
that
the
smallest
eigenvalue
of ‰
is bounded
] 01
0
!
¾
]
½below by
constant
¼
.
] unity or some positive
-B7CeC7Ce7-¿
Consider now the choice P
, i.e., has all components equal to unity.
Then † is obtained from
] 0n/]„]
]
(2.5)
†
P
J»µ
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172
O. AXELSSON AND J. KARÁTSON
where
]¨¾!])01/]f ]
µ
(2.6)
/
]G
/]
G
5†
Assume here for simplicity/ that
]„G ]
]G
G
f ]¨¾!]
G
C
/] wef ] have componentwise
À ]f ]-matrix. Then
is an /g
G
Š
G
b
G
b
C
b /¹]„]
/g]f ]
Š
J
G
]G
/
]„]
G
/]
]
G
f]
¾ ]
0
It/ follows
induction
is]
b componentwise. Hence
۠
/ µ ]„] / ]by
/ ] that
]ƒ]
f]
f ] ¾ †]
]G
] 0Â
J1
G i.e., G all itsG off-diagonal components are non-positive. Since
a Á -matrix,
J
J1µ
, it holds that if this vector is / nonzero
then †
/ ]ƒ] †
] ¾ ] 0 ]ƒ]
is positive
definite,
and also an À -matrix. Should the matrix lose positive definiteness (by
J1µ
b ), we must perturb the matrices
with some (small) positive
having
number. This has been discussed, e.g., in [1]; see also [5].
]0Q/g]ƒ]
]
/]„] we
/have
]f ]
/]
f]
Assuming that no perturbation
is required,
]G
J¶µ
†
G
J
G
G
5†
/
]„]0 semidefinite
]
/]f ]
/
]
f]
/]„]
with the inequality in the positive
sense.
Therefore
]G
/
that is, ‰
J
‰
G
†
r
and
G
G
۠
X ]
b
/
G
J
Š«-BC
‰
Hence we have a lower bound.
The upper bound follows from a theorem in [18], there stated
/
in a somewhat more0 general form.G
ÄÃW
T HEOREM 2.1. Let be
a symmetric
positive
definite matrix partitioned in /
9
†
†
†
block
Let ‰
, where
is symmetric
positive definite block
rs 0Q† /
r,s
X ]
0
] 0nform.
]
G
diagonal and s is strictly lower block
triangular,
both
with
consistent
partitioning
to . Let
J
J
‹ˆÈ{É Å
+U’
Y
ۮ
Æ
Å
9
P
†
L
Ç
P
Ç
, where
,
let
and
assume
that
. Then
s
s
/
G
S
,
‰
0
In particular, if s
s
u
Å
]
, then
µ
E®/
]ƒ]
Ç
µ
†
E
X YZ5[
Ç
/¹]f ]
/
]G
†
]G
/g]
ÏJ,-®eÐC
E
and
if
u
Š”4-J
†
¼
G
]
Hence
J
Å
+½-®\B’
for some
then by (2.5),
¾!]
Í ]„Î
’¹J
oËÊ(Ì
])0«XD]
(2.7)
(2.8)
]
-
/
]„]
C
]„]
f]
-J
E
+ђC
G
G
R EMARK
2.2. The
above
two G choices
have
somewhat opposite properties. In particular,
/g]f ]
/g]
f]
]G
G eigenvector
G
if is also an
of
for the smallest eigenvalue, then it can be
۠
G
seen that
is a multiple of the identity matrix. In the following we assume
۠
that this does not hold.
In the following section it is analyzed how the Schur complements depend on jumps in
the coefficients in the differential operator.
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BLOCK MATRIX PRECONDITIONERS
3. Schur complements for elliptic problems with jumps in their coefficients. Let us
consider a domain decomposition (DD) method for an elliptic problem discretized with FEM,
such that (in addition to a corresponding portion of the outer boundary) each subdomain has a
common boundary only with its previous and next neighbours in the sequence of subdomains.
Elliptic operators with different constant diffusion coefficient in each subdomain often arise
in the context of various (DD) procedures [2, 15, 16, 17, 19]. Our goal in this section is to
study the sensitivity of Schur complements to coefficient jumps.
In the classical DD approach, the interior domain nodes are ordered separately from
the interface nodes and all interface nodes are ordered last. Like in multigrid methods, to
avoid large condition numbers of the corresponding Schur complements, an efficient method
has proved to be to introduce one or more proper auxiliary coarse spaces that have a global
balancing effect; see, e.g., the BDD method [19] and the approach of so-called exotic coarse
spaces [16] in a Schwarz method framework.
An alternative to the above approach is to take the interface nodes into account together
with the previous subdomain in the mentioned sequence of subdomains. This approach, considered in the present paper, leads to a tridiagonal block structure as in (2.1). It will be verified
for a model problem that the condition numbers of the Schur complements are sensitive to
the jump in the first approach (namely, proportional to the magnitude of the jump) but are not
in the second approach. That is, one can have independence of jumps without introducing
auxiliary problems.
0
For simplicity, 0 the detailed study is given for a decomposition of the domain Ò in three
subdomains Ò , Ò and Òd . According to the above, we have common boundaries Ӂ<P
ÒKÔ
Ò and ÓUP
ғÔ
Òd , but Ò and Òd have no common boundary. We will first
formulate the block forms
of
the stiffness matrix under the two mentioned approaches for an
]
isotropic Poisson equation. Then we rewrite the stiffness matrices under different diffusion
coefficient in each Ò , and study the variation of the corresponding condition numbers.
3.1. Basic block forms for the isotropic Poisson equation. Let us consider the Poisson
]
equation with homogeneous Dirichlet
boundary conditions. The FEM subspace is chosen
]
with piecewise linear basis functions, assumed either to have node points on one of Ó or to
have its support entirely in one of Ò .
/
In the classical DD approach,
the stiffness matrix/ isf Õ written in the block form
/10
/
^`
`
`a /KÕ b
(3.1)
b
?
Here
(3.2)
lf Õ ¯
9
0n/
/Õ
gP
Õ
9
/Ko Õ
for
all
2
0
f
?
A
P
/Õ A
. Then/Kone
lets
Õ
0Q/
/
0Â2
ÕOÕ
(3.3)
Õ
?
/KÕ
f
/KÕ
f
? A
fÕ
?
P
/
?
Õ
A 6
/
and thus obtains the more concise form
/10
(3.4)
^`
`a
/Kb Õ
b
/
Õ
/
b
/ b
/Õ $
b
/Kb Õ
/
Õ
/
Õ
Õ jh i
/KÕO
k i
d
d$d
d
i
k i
A
01/
dIP
€6
C
C
i
/KÕ
fÕ
A
h i
A
b
b
b
fÕ
A
P
/
2
0
9
/
Õ d
b
Õ
“P
b×6
?
d
A
b
b
fÕ
b fÕ
/
?
fÕ
Õ dd f
/
/
?
b
? b
qp
fÕ
/ b
f
/KÕ $
b
/ Õ
f
f
b
/g]f ÕOÖ0n/
/
$
`
0
d
9
Õ
P
2
/Õ
f
b
A
d€6
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174
/
]„]
O. AXELSSON AND J. KARÁTSON
0
]
0
The solution
of the corresponding linear system can be reduced to solving systems with Ø
-Bc’Ù
,o
, and01
an/additional
with
the
ÕOÕ
/Õ system
/ G
/
Õ
/K
Õ
/Schur
/ complement
Õ
/KÕ
/ G matrix
/
Õ
G
(3.5)
J
ØnP
J
$
J
d
d$d
P
C
d
In the other approach, the interface nodes are taken into account together with the previous subdomain. Under this reordering,
the
stiffness matrix in (3.1) can be rewritten as
/
/
fÕ
/Õ
^`
f
`
`
fÕ
/Õ ?
`a
(3.6)
/Õ
/½0
?
(3.7)
b
b
b
/
(3.8)
/KÕ
fÕ
$
Ú $¹P
?
2
0
/
f
$
Ú $“P
A
A
A
2
to obtain the concise form
/
/«0
d$d
/
b
?
/
/
f
b
? b
bÛ6
/Õ
f
A
b
Ú dIP
A 6
d
2
0
b
/Õ
2
0
Ú cgP
b®6
fÕ
Ú $d‚P
hji
i
i
A k i
/ b
d
fÕ
2
b fÕ
/
d
b
0
A 6
(3.9)
/
/
fÕ
A
Ú ‚P
fÕ
/KÕ
0
b
/KÕ b f Õ
A
/KÕ
f
A
A
b
? 6
/
/KÕ
f
b fÕ
/
?
/
?
?
?
A
b
/ the notation
/
fÕ
where we/ introduce
2
0
f
fÕ
/ b
/Õ
? b
Ú
/KÕ
/
?
f
d56
/
a /
/
^
Ú Ú
/
/
Ú c
b
Ú 4
b
/
Ú Ú $d kh
Ú d
d$d
C
0
/
In the Schur complement approach, here only the first block remains unchanged: M gP
Ú ,
and the solution of the original system can now be reduced to solving two additional systems
corresponding to Schur
recursively
as
0
/ complements,
/
/
0Q/
/
/
G determined
G
(3.10)
M P
J
Ú $
Ú 5 M
Using notation (3.7)-(3.8) and Õ letting
0Q/Õ
(3.11)
we obtain
(3.12)
The similar formula for M
?
M
/
/
$
M /Õ
?
f
A
?
$
Ú d M ?
Ú $d
C
fÕ
?
/
fÕ
/
G
/
ÕG
J
dd
/
?
/KÕ
f
? M
J
f
J
?
Õ
P
M d
fÕ
P
0Â2
Ú fÕ
/Õ
fÕ
A
C
A
A 6
will not be needed here.
d
3.2. Conditioning properties for problems with jumps in their coefficients. Now we
]
can turn to the case of our interest. Instead of the above Poisson equation, we consider the
FEM solution of an elliptic problem
with a different
constant diffusion coefficient in each Ò .
Ü&
²
ß Ò
That is, in weak form, one seeks
, such that
ÝVÞ
­‚á
(3.13)
à
°
°
V±
0
à
­Iâ
ã
%
ˆ&
Ý
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175
BLOCK MATRIX PRECONDITIONERS
á
¯å
]
is a weight function on Ò ,­ such
that
where
á“ä
0
á
In our model problem we assume
-Bc’ÙDC
o
á
á
(3.14)
and are interested in the case
á
Š
Š
á
d
á
(3.15)
gæ“æ
C
When varying these coefficients, in order to avoid the loss of ellipticity in the limit, we also
assume that there exists a constant ¼tæUb such that
á
á
(3.16)
Š
d
¼
á
C
á
\
unboundedly, then the condition numbers
Below, we will find that if we vary the ratio also grow to infinity for the Schur complement in (3.5) but remain bounded for the Schur
complements in (3.10).
]
Let us first consider the
] classical DD approach again. The stiffness matrix (3.1) is] then
á
modified as follows. The
entries corresponding to basis functions with support in Ò are
multiplied by the weight . For simplicity, assume that for the node points on one of Ó , the
support of the basis function is symmetric w.r.t the node point, and thus its parts intersecting
]
l
á
á
with the two domains have equal measure. (An opposite case will be mentioned
in Remark
\O’
3.3.) Then the entries corresponding to such basis functions are multiplied by
.
r
/ matrix has the form
/
fÕ
Therefore, the stiffness
á
á
^`
/10
`
`
`a
(3.17)
/ b Õ
b
á
f
?
/
á
$
b
/
á
b Õ
/
á
?
b
á
Õ f
A
f
d /KÕ d$d f
b
ç
?
A
d
fÕ

b
á
/
á
b/
d
/ Õ
?
/

?
?
b
A
ç
fÕ
/ b
á
fÕ
b
A
ç
?
á
fÕ
A
d
d Õ
/K
A
hji
i
k i
b
A
ç è
C
i
fÕ
A
0«é”/Kone
ÕBÕ
/Õ sees
/ G that
/
Õ the Schur
/Õ complement
/ G
/
Õ
/KÕ becomes
/ G
/
Õ
With these modifications,
readily
(3.5)
á
(3.18) é
á
Ø
J
P
á
á
J
z
$
¸5Õ d
?
z
J


A
d
d$d
¸5d Õ
A
á
where
is the two-by-two block diagonal matrix, blockdiag ç ç
ç
çãè
P ROPOSITION 3.1. There exist constants z á ‚æUb z independent of , such that
á
(3.19)
S
Ø
á
KŠ
.
C
r
"
Proof.0 Using (3.2)-(3.3), a simple calculation yields
á
ØÚ
(3.20)
P
á
á
-
0Û2 Ø
0
?
ç (
A Ø
ç
/KÕ
/KÕ
r
fÕ
b
?
/KÕ
J
fÕ
f
?
/Õ
/
4
G /
b
rf Õ
ç
çãAè
/
A
fÕ
J
A 6
Š
/
Õ
/
G
á
/
Õ
J
á
d
/
Õ
d
/
G
dd
d
Õ
?
?
? ? . Here Ø
where ØK“P b (i.e., it is positive semidefinite) and
$
is not a zero matrix since it is a Schur complement, corresponding to the positive definite
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O. AXELSSON AND J. KARÁTSON
/
0
matrix
-Bc’
o
Ú $ modified by setting a zero diffusion coefficient outside Ò
½ 4çãAè
, and , owing to (3.14).
the matrix
/KÕ Hence
fÕ
r
ç
X YZ5[
0Â2
á
ê
ç A
P
ê
“Š
ç A
ç Ø
ç
X YZ5[
á
satisfies
ê
the condition number of
?
?
?
Ø
and
that
S
á
KŠ
fÕ
çãX Aè Y]
r
A /KÕ A 6 f Õ
ç
‚
=
A
, which yields for
A
XDY]
/ Õ
=
Ø
fÕ
A
C
A
ç
á
S
Ø
´<ë á
ì
á
as
³Rí
0
á
á
The
condition
numbers of the other two terms in (3.20) are bounded. Since
S
ØÚ
, we obtain (3.19).
?
C OROLLARY 3.2. If we vary ç 0 A unboundedly,
then á
á
,
X YZ5[
á
á
ê
ê
æ«b
b
á
=
. Further,
/KÕ
?
X Y]
b
á
r
Õ ¯ fÕ ¯
/
Ø
C
³Rí
S
?
R EMARK 3.3. The above sensitivity to ç A may be reduced if the supports of the basis
ç
functions on Ó> are not assumed to be symmetric
with respect to the node point, but their parts
intersecting with Ò have small measure. However, this would in turn lead to inpractically
small element widths and very large gradients of the basis functions near Ӂ .
á
Let us now consider the second approach.
We study the Schur complements (3.10) mod/
ified with respect to the diffusion coefficient . The corresponding modification of the matrix
Ú in (3.6) comes by first replacing the considered blocks of (3.1) by the corresponding blocks
of (3.17), and then using the same reassembling as for (3.6). Then the Schur complement M / as
/ Õ
/
/
f
Õ
fÕ
fÕ
G
in (3.12) becomes modified
á
á
á
0î2 á
á
(3.21)
where M
M Õ
?
P
J
/
á
? Õ
f
A
(3.22)
?
M
Introducing the notation
(3.23)
P
we have
0Â2 á
M P
á
?
M
0
á
á
-
ï ’
á
á
á
á
Since, by assumption,
?
M
)ï
ë
-
Š
/Õ
f
r
á
?
, we obtain
/KÕ
A
fÕ
C
A
f
?
?
?
/KÕ
á
J
?
/
fÕ
d
á
?
J
r
fÕ
f
C
?
A 6
.
?
M
d
r
fÕ
á
fÕ
/
/
/
$
G
?
/
á
G
á
á
ì
/ Õ
fÕ
Õ
/
?
? M
Õ
f
/Õ
-
?
’
á
?
/KÕ
ë á
/KÕ
J
á
/Õ f $
á
M A á
á
Š
Õ
L EMMA 3.4. There holds
Proof. We haveÕ
0
J
$
á
?
á
?
á
P
á
(3.24)
’
0n/
á
M r
fÕ
á
in (3.11) has
been 0 replaced
á
á by /KÕ
Õ
?
M
/
G
$
/
fÕ
?
ì
r
’
/
G
fÕ
?$ð
/KÕ
?
fÕ
?ð
C
A
Õ
A
fÕ
A 6
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177
BLOCK MATRIX PRECONDITIONERS
Õ
M
á
?
á
Š
ï
/KÕ
-
ë
fÕ
/KÕ
?
’
f
J
?
/
?
/
G
fÕ
$
/Õ
-
?
?
’
r
ì
fÕ
0
Õ
á
? ð
C
?
M
/
fÕ
Õ G of
Similarly to (3.23), let us denote the top left block
(3.12)
by
0n/
(3.25)
M $
and then let
P
MÚ /KÕ
M P
f
J
032
(3.26)
/KÕ
? M
?
/
fÕ
f
A
4-
¼
r
?
/KÕ
A
fÕ
A
A 6
á
with ¼ from (3.16). Now we can prove the following required
boundedness.
X
/
Y
5
Z
[
P ROPOSITION 3.5. The condition number of M satisfies
á
S
=
á
Hence it is bounded independently
of .
Proof. Clearly M $
$ , and /
2
á
(3.27)
M To find a lower bound for
yield
á
/KÕ
A
á
M /Õ
fÕ
$
á
d fÕ
/
r
f
C
MÚ á
Ú $
á
/
á
X Y]
M á
A
A
owing
to (3.14). Hence
/
0
C
Ú $
A 6
, note that Lemma 3.4 and the definitions (3.23) and (3.25)
á
(3.28)
M KŠ
C
$
M /
fÕ
Substituting (3.28) into (3.24),2 and
using (3.16)á and
(3.26),
respectively,
we then obtain
á
0
á
M á
0
Here
á
/Õ f
$
M A á
Š
á
4-
¼
r
á
/Õ
A
á
fÕ
A
A 6
MÚ C
á
, since by the above, M Ú is the Schur complement M in the case
¼
d
. Together with (3.27), we obtain the required statement.
Finally,/ we / consider/ the second Schur complement M d from (3.10). When replacing its
á
considered blocks from (3.1) by the corresponding
blocks of (3.17), we observe that each of
0
/
/ Hence theG matrix
/
the blocks d$d , Ú d$ and Ú d á is multiplied
by
d
.
M d becomes modified as
á
á
á
(3.29)
MÚ æñb
M d
P
d
J
d$d
d
Ú d$
M C
Ú $d
á
We can easily prove again the following required boundedness.
X YZ5[ /
P ROPOSITION 3.6. The condition number of M d
satisfies
á
S
M d
/
X Y]
=
á
d$d
M d
C
á
á
Hence it is bounded independently
of .
ò
á
á
Proof. Obviously
d
d
d$d . Further, in (3.24) we can estimate each
M
/
fÕ
by d and M $
below by M $ using (3.28),
such
that
we
obtain
2
0
á
M á
/KÕ
KŠ
M d
A
á
and substitution into (3.29) yields M
bounds imply the desired estimate.
d
óŠ
f
/KÕ
fÕ
A
á
/
d
dd
A 6
J
below
á
A
á
á
/
d
d
Ú d$
M M /
G
0
Ú $d
á
d
M d
. The two
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O. AXELSSON AND J. KARÁTSON
3.3. On the growth of condition number with the number of subdomains. Whereas
we have obtained jump independence in the previous subsection, these estimates are inherently unable to compensate for the number of subdomains. This follows if we relate the new]
á
estimates to those on the original Schur complements (whose condition number is
] known to
á
increase with the number of subdomains).
Namely, the appearance of the new constants
]
makes each inequality worse (or unchanged if the constants/ coincide), therefore M
cannot
á
be better conditioned
than
the
original
M .
/
á
Ú $ , and (3.26) and the ordering
In á fact, for M ,Š definition (3.10) implies M Ú d$d . Hence the bounds in
d
implies M M Ú . Similarly, (3.10) implies M d
X YZ5[ / 3.5 and 3.6,
X YZ5respectively,
X YZ5[ /
X YZ5[
[
Propositions
satisfy
XDY]
=
Ú $
MÚ XDY]
Š
=
M M 0
S
X Y]
M and
=
dd
X Y]
Š
M d
=
M d
M d
0
S
M d
5C
One can see that Proposition 3.5 can be extended to the case of more than the three
subdomains considered in our example, if similar conditions are assumed. In particular, we
assume a stripe-type decomposition] (i.e., each subdomain has a common boundary only with
its previous and next neighboursá in the sequence of subdomains), and the subdomains are
numbered such that the weights
are ordered monotonically. Then the proof of Proposition 3.5
can
be
repeated
such
that
the
role of the 1st, 2nd and
3rd subdomains are played by
]
]
JV-®
-®
á
the o
th, o th and o r th subdomains,
respectively.
Using
the
above arguments, however,
]
the bounds obtained for S M
cannot be less than S M . deteriorate even in the preAs shown in the introduction, the condition numbers S M
conditioned form (1.1). This shows an important motivation for the efficient preconditioning
of the Schur complements. A possible improvement was given in Section 2, and in the sequel
we will study other block orderings to avoid the recursive growth
of the condition numbers.
The next section yields estimates in terms of the constant in the strengthened CauchySchwarz-Bunyakowski inequality.
4. Odd-even partitioning of subdomains. We assume now that we have ordered the
/
/
subdomains in an odd-even fashion so that
the finite element
matrix takes the form
/«0
/
/ b
a
^
/
(4.1)
b
/g]
0
/
d
/ C
$d kh
/
d
d5
-Bc’
/
d$
]
Here , o
, correspond to interior node points and d to edge and vertex node points.
Clearly the matrices
themselves are block diagonal. This matrix can be factored into the
/
¸
/ G
/
form
a
^
(4.2)
/
/
/ b
b
a
d$
(4.3)
M d
d$d
J
]G
b
dc
/
b
b
M
01/
/
where M is the Schur complement
matrix
/
b kh
dc
¸
^
b
/
b
/
4d
d
$d kh
d
/
G
G /
¸
J
/
d$
/
G
$d
and some simpler matrix is used in a corresponding approximate block matrix factorization.
Although the actions of the matrices
can be computed readily separately for each subdomain, the major problem remains how to/ precondition the matrix M d .
As indicated in Section 2, and shown in papers on domain decomposition methods (see,
e.g., [23, 22] and also [1]), if we just use dd as preconditioner then the condition number
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/
179
BLOCK MATRIX PRECONDITIONERS
G
´
\O²Ü
²
•
grows as
(as •t³ôb ), where •
are the characteristic mesh sizes for
the fine mesh and for the subdomains, respectively. Furthermore, as mentioned in Section 3,
the condition number of M d itself deteriorates/ as the magnitude of coefficient jumps increase,
which makes the construction of an efficient preconditioner to M d additionally difficult.
Instead, we will use a preconditioner of that takes contributions from both interior and
boundary points into account.
This is similar to the second approach in Section 3.
T
The preconditioner will be in additive
form
0
S
d$d
M d
T
(4.4)
The matrices
T
$T
T
T
C
r
are formed from the inverse matrices
/
¸
T
^`a
$
T
b
b
T
and
4d
b
d5
^
/
khji
ö Md õ
a
G
b
G
/
¸
0
b
b
dc
¸
d
/ k
b
h
d
b
G
¸
^`a
b
T
b
T
b
]
]ml
01/
/
ö
M dõ
]¨/
J
d
^
/ b
b
b
d$
/]
]G
$d kh
d
0
d
/
/ b
khji
$ö M dõ
/
a
G
d
d
where
b
T
0
d
-B$’÷
o
/
]
T
]Ÿl
T
T
and the matrices
need
not be given as we only aim at a bound on S
in which
T
will not appear. To form , the sub-block identity matrices are deleted from the above, i.e.,
0
T
(4.5)
T
^`a
T
b
b
¹P
T
0
d
b
b
G
dcøb
ö M dõ
T
kh i
^`a
b
b
T
b
“P
T
b
b
T
d
M dõ
d
$
$ö G
kh i
C
ÙDÙù
We will show that by use of perturbations of the subblocks in the position
of
0øE
/ G
/
/ G
/
/
/ G
/
/ G
S of the preconditioned matrix which dethe inverses, we can derive a condition
number
#$ pends only
on the CBS constant
dc
d r
d$ / $d
d
d
½-®\-JÜ #$
since S
.
T
Let first the preconditioner be/«
defined
by/ ú (4.4)-(4.5). The matrix is split as
0ô
/ ú
where
/
/ ú
0
/
/ ú
a
^
J»ûˆ
r
/
b
b
d
/ k
b
h
d
b
d5üb
Then
0
/
a
^
b
/
/ b
b
b
d$
0
/ b
a
^
Ûû
d kh
d
býb
/ b
býb
b
býb
¸
T
/ ú
0
/ ú
a
^
¸b
b
b
b
b
b
üb
k
d
h
T
0
¸
a
^ b
b
¸b
b
b
b
C
k
b
d
h
C
k
d
h
#$ #$
,
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180
O. AXELSSON AND J. KARÁTSON
A/ computation
shows
that
/101/
/ ú
T
0
n
/
T
¿
v5xy{ze |} ¸
^ býb
J»ûH4T
T
/ b
/ b
b
b
r
/
¸b
b
b
b
b
b
b
/
4d
b
G
dcøb
^`a
d
k
b
h
/
b
d$
b
býb
b
G
dc
b
bþb
b
h
d
dc
b
b
b
k
b
h
/ d5üb
b
0
b
d$
$d kh
b
/
kh i
$d
b
b
h
d
/
hjk i
/
k
b
d$
/
/ b
^ b
G
ö M dõ
b
C
P
rtÿ
d
/
/
/10
T
(4.6)
4d
b
d
b
d
Hence
/ b
/
/
Md õ
b
b
/
k
b
a
bþb
bþb
/
/ 4ö $d
/
khji
G
b
b
^`a
^
b
^
¸b
J»ûH
b
$ö Md õ
/
d5
b
r
/
a
b
b
a
býb
býb
ö M dõ
/
r
^`a
/ ú
d
$d kh
b
r
^ b
d kh
b
/
¸
a
b
b
J»ûH "
/
/
b
/ b
/
/ b
^ b
$d kh
d
/
d$
T
b
J»ûH4T
J»ûH4"
/ ú
/
h
a
r
r
k
J¶ûò
/ ú d
r
/ d
d5
/
r
^ b
b
b
J»ûH
a
^
/ ú
T
a
^
/ a
^
r / ú
r
/
0n/
0n/
¸ d
d
/ 4
d kh
r
d / ú
/ ú býb
b
T
a
býb
r
b
/ ú
J»ûH
¸ /
a
r
/ ú
r
/ o~@
r
0n/
T
J
ÿ
and
/
/
/
/
0n¸
0½/
#$ T
/
#$
/
ö
/
G
] #$
/
r
$ö
G
# C
0
ÿ
-@$’
ö
Let d be split as d
, where d õ (o
) arises from contributions to
r
dõ
dõ
/
edge nodes from odd and / even numbered
subdomains, respectively. Then the matrices
a
^
/
b
b
b
d5øb
/
/
and
h
ö
dõ
]
/
^ b
k
b
/
a
d
/
b
/
b
b
]q/
/]
]G
b
/
d$
dõ
$d k
h
ö
0
are the full contributions from odd and even numbered subdomains, respectively, so they are
-B$’
ö J
positive semidefinite.
Hence
) are / also positive
semidefinite,
thus
d
d (o
dõ
G
G
0n/
/
/
/
/
/
G
ö
M dõ
and
similarly,
G
/
/
d
Md õ
$ö ö
r
dõ G
/
dõ
d5
ö d
dõ
$ö G
Md õ
/
d5
G
Š”
/
J
ö
dõ
Š
d
dõ
. Hence
/
/
ö d5
T
/
J
/
$d
Md õ
4ö /
b
/
/ b
dc
d
Š”
/
d$
C
G
/
d
4ö M dõ
/
0n/
d
/ 4
d kh
d
ö /
/
a
^
dõ
G
/
/
or
/
and by (4.6),
ö
dõ
$ö /
/
d
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181
BLOCK MATRIX PRECONDITIONERS
Therefore
/
/
¸
# T
(4.7)
X YZ€[
ђ
#$
/
T
and
Q’C
/
T
To derive a lower T bound, we will use perturbations. Let then
let the preconditioner Ú to be defined as
0
T
(4.8)
Ú
0
P
/
T
where
r
/
/½0Q/
a
P
^ býb
b
/
býb
G
b
býb
/
/
/
be defined as above, and
d
for some
k
h
Š
b
C
/
The intention
isJ to keep/ T sufficiently
small so as not to increase the upper bound too much.
/ J
/
T
We have Ú
, and we wish to find a positive number · sufficiently
r
-¿
T
Š
large to make · r
Ú
.
/
/
/10
/
/
/
/
/
/
Here
-¿
·
T
J
Ú
r
·
-®€
·
r
T
J
r
-®
·
r
C
r
Further,
/
/10
a
(4.9)
^
býb
/ b
býb
b
k
h
býb
and, using (4.6), we have
/
·
/
/
-®€
T
r
/
0
J
Ú
^
b
·
r
0n/
and ·
¼
/
·
r
/
d
/
dc
d
-¿
r
/
Md õ
/
b
/ ö G
/
d
d$
/
M dõ
d
$d
4ö
r,M d õ
khji
$ö a
JW’
·
-¿4"
r
^
býb
/ b
býb
b
d
h
d
0
M dõ
4ö
r,M d õ
k
býb
/
G
d
r
d kh
µ
d
¼
b
/
4d
¼
d5
·
r
G
d5
¼
/
’
$ö
J¶µ
d
d
0
are positive numbers
satisfying
·O¼
(4.11)
M dõ
-¿
/
d
4d
d
/
$ö r
µ
d5ü
¼
G
r
@¨’
where
¼
-¿
/
/b
/
·
^`a
(4.10)
/
/
a
/
d
·
r
-¿
r
-¿
·
·O¼
’÷
0
r
·
0
r
’
-¿
·
-
¨’
·
-¿
=
Ú
r
r
Now we can prove
T HEOREM 4.1. Assume that
Then
X Y]
/
T
(equating the off-diagonal matrices)
r
-®
r
(equating the
/
-terms)
d
d
-terms)
C
/
in (4.1) is positive definite, and let
X YZ5[
-
Š
·
·
µ
(equating the
r
-
and
/
T
Ú
ђ
r
-JÜ
T
be defined in (4.8).
Ú
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182
0
where ·
·
O. AXELSSON AND J. KARÁTSON
satisfies the equation (4.11) and
0
E
/
/
#$ d
As
/
G
/
G
d5
d
/
T
S
Ú
-
’
’
/
‹óŒ„Ž
Proof. We have/
/
a
b
¼
¼
d5
G
d
Md õ
4d
¼
d kh
µ
d
T
S
ß
G
# #$ C
d
G
#$
d
Ú
#$
-
C
-JL
/
¸
a
^
/
/
d5
b /
¼
¸
/
a
/
G
b
¼
"
-JÜ
r
^
¸b
G
d$
a
b
b
b
d
b
b
b
b
b
/
G
/
G /
¸
¼
b kh
h
¸
^
b
k
b
/
b
¼
/
d
/
$ö /
0
¼
¼
d
and
/
^`a
/
¸
/
/b
/
/
G
d
r
’
and
^
/
, ·I³Rí , the condition number is asymptotically bounded by
³îb
/
/
/
d5
b
d
Md õ
d5
G
b
/ ö /
d
d$
/
d
Md õ
kh i
$d
4ö
r,Md õ
$ö /
0
a
^ býb
^`a
4d
k
býb
b
býb
M dõ
býb
b
^ bþb
^`a
b
bþb
$d
bþb
Md õ
býb
/
a
r
h
$ö
b
k
4ö
a
býb
h
^
G
Md õ
býb
b
b
b
b
^ b
G
4ö M dõ
b
b
M dõ
k
$ö
h
a
b
býb
b
d5üb
býb
khji
$ö /
/
b
hk i
b
b
b
b
b
d
Md õ
C
k
ö
h
It follows that the first two terms in (4.10) are positive semidefinite. Since by (4.11) the last
term in (4.10) is zero, it follows that
/
/
/
X Y]
whence
/
=
T
X YZ5[
/
T
Ú
0
T
Š
Ú
r
-
Š
Ú
-¿
·
. Further, using (4.7) and (4.9),
-
·
r
/
[ Î
/
T/
8 9
ß
/
8
Ú
Q’
r
8 9
8
’
0
/
[ Î
r
[ Î
8
d
d
8 9
d
8 9
ß
8 9
M d
8
d
d
0
8
’
r
’
r
-JL
/
[ Î
8
0
8 9
ß
/
/
/
8 9
ß
d
8 9
8
d
d
8
d5
d$ kh
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183
BLOCK MATRIX PRECONDITIONERS
where M d is from (4.3).
A computation from (4.11) shows that
0
·
¼
r
·
and
0
·
0
where
·
·
-
r
. Hence
r
r
+½-
T
T
r
’
Ú
S
ß
’
r
’
·
-B
r
’
, and hence
C
’
-
³
\÷4-JL ’
. / Hence
Ú
’
·
/
T
‹ŒƒŽ
U’
0
·
r
-
Š
Ú
’÷4-JL -
/
For the condition number we have
r
, then ·I³Rí ,
³Âb
X Y]
S
·
’
gr
-
Kr
. If
0
r
-
=
which is minimized for
·
-
-B
-JL
C
0
-J´
\O²Ü
•
As follows from Section 2, for partitioning in subdomains,
, where
•
are the characteristic mesh sizes for the finest elements and for the macroelements,
respectively. Hence it follows from Theorem
4.1 for the
condition number that
/
0
G
$²
S
T
0
Ú
´
with
ù
•
\O²Ü
#$ "C
/
0
0
Therefore, the number of conjugate gradient iterations
to solve a system with matrix
’
using the preconditioner
G
´
ù0 \{²Ü
" 0
Ú
T*
Ú
-JL
´ \O²Ü
0
G •
!
\{²Ü
,
, increases at most as
’
#
, which is fairly minor. For instance, •
, respectively 4, if
’
.
• or
•
/
We remark that for convenience of the derivation of the condition number, weT have formulated the matrix based on an odd-even ordering. However, since the matrix is given
in additive form, we
implement
the
0 can
v5xactually
y{z7|÷}
/ G
/ G actions of the local element inverses in any
order, or even in parallel. The method
can
be
further
improved by use of a perturbation ma o~@ b
b
trix in the form
instead
of (4.8). It turns out that in this
d
M
d
d
case the condition number does not depend on but only on , which can be chosen independently of the coefficient in the differential operator. This shows independence of both the
mesh parameter and coefficient jumps, but will not be discussed further in this paper.
The above method can be further extended by use of a splitting of node points in a coarse
mesh set and a remaining fine mesh set. For the corresponding two-by-two block matrix the
above method can be applied for the pivot block matrix, and the arising Schur complement
matrix in the block preconditioner can likewise be preconditioned elementwise. This will be
discussed in Section 6; see further analysis in [10, 12], and for related results [21]. In the
following section elementwise preconditioners are also analyzed by more analytical means.
²
•
-
#
T
²
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184
O. AXELSSON AND J. KARÁTSON
5. Some model analysis on the continuous level. A continuous analogue of the method
of Section 3 is presented now on some model problems, including the introduction of a certain
modified Poincaré–Steklov operator for the interfaces. This study on the continuous level can
help the understanding of the properties of the studied factorization approach.
5.1. Preliminaries: the Poincaré–Steklov operator. As pointed out in Section 3, the
analysis of standard domain decomposition methods relies strongly on the Poincaré–Steklov
operator; see, e.g., [23, 22]. In this subsection we give a brief description, following [22].
Let us consider a boundary value problem
0
J
"
(5.1)
%$
ä
#
â
­
0
in Ò
b
â
'$
&
where Ò is a bounded domain
with Lipschitz
. The domain Ò is
Ò
s
0
0 boundary and
decomposed in two nonoverlapping domains Òg and Ò , whose common boundary is denoted
by Ó , further, we let Ó>gP
ÒK Ó and ӂP
Ò Ó .
The Poincaré–Steklov operator is then defined in the following way. Let us choose an
½&Q²
²
#$ #$ ß$ß
Ó
arbitrary function
. (For
the definition of ] ßß Ó 0 and other related Sobolev
²
²
spaces, see also [23].) Let and denote the harmonic
extensions
of in Ò and Ò ,
²
-Bc’
respectively, with zero boundary condition on Ò , i.e.,
,o
, is the solution of the
0
]
]
problem
&
&
$
*
(5.2)
J
Õ ¯
] ²
()
)+
²
] ²
ä
0
0
Õ
ä
b
b
C
>
in Ò
G
,
²
#$ ß$ß
Ó
²
# ß$ß
Then the Poincaré–Steklov operator is
, which assigns to
P
³
Ó
jump of the normal derivatives of its harmonic extensions on Ó , i.e.,
$
0
,
(5.3)
P
$
²
Ê
$
$
²
r
Ê
the
C
on Ó
The plus sign represents the jump with the convention that the outward normal vector Ê of
Ò is opposite to Ê of Ò on Ó , which will be understood throughout this paper. That is, for
a smooth function on Ò , the two normal derivatives are the opposite of each other and hence
the jump on Ó equals zero.
R EMARK 5.1. Problem (5.1) can then be 0reduced to equation
(5.4)
-
with
]
.
defined as follows. Let
(5.5)
P
0
]
]
]
0
-Bc’
,o
]
J
and let
,
0
â
'-
" .
$
$
J
Ê
.#
]
âóä
â
, respectively,
0
]denote the solutions of the problems
â
­D¯
$
â
.
:
0
J
0
$
b
Ê
in Ò
.
â
)
on Ó
0
â
â
which
represents
the negative
jump ofJ the corresponding
normal derivatives. Then
P
²
-@$’
on Ò (o
) satisfies
on both Ò and Ò and is continuous on
r
.
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185
BLOCK MATRIX PRECONDITIONERS
. Hence solves (5.1) if and only if its normal derivative has zero jump on Ó , which is
equivalent to (5.4).
R EMARK 5.2. Green’s formula implies that the bilinear form of the Poincaré–Steklov
operator is
0
°
°
°
°
Ò
,
0/ , 21
­
Å
(5.6)
whence
,
­
²
à
*±
?
²
Å
²
à
r
A
±
²
Å
%ã Å
&<²
#$ ß$ß
Ó
5
is a symmetric and strictly positive operator.
On the discrete level, let us now consider a FEM discretization of problem (5.1) and let
/
Õ
us decompose the stiffness matrix as /
/«0
a
^
(5.7)
/KÕ $
b
/
/
/Kb Õ
Õ
/ÕO
k
h
corresponding to the node points in Ò
0Q/ ÕOÕ
be reduced to the Schur complement
(5.8)
, in
/
J
ØQP
Ò
and on Ó , respectively. The linear system can
/
Õ
Õ
/
G
$
/
Õ
/
Õ
J
/
G
$
Õ
i.e., Ø /is
ÕOÕ the Schur complement for Ó with respect to both Ò¹ and Ò . Then, as pointed out
in [22], Ø is the discrete analogue of the Poincaré–Steklov operator (5.3). Essentially, the
term
is responsible for the boundary values of the considered function and the two
other
|
terms represent the procedures involving the two harmonic extensions.
R EMARK 5.3. The generalization of the above notions to the case of more (say, ) subdomains
is straightforward.
Then the Poincaré–Steklov operator involves harmonic extensions
|
|ˆ0
from the union of interfaces
to all subdomains, and its bilinear formulation will contain a sum
Ù
of terms, e.g., for0
the
form° (5.6) is replaced
by °
°
°
°
°
3/ , 21
­
(5.9)
> Å
à
|
­
²
?
±
²
Å
r
­
à
²
A
*±
/g]„]
²
Å
²
à
r
d
±
²
d
Å
C
è
|0
Similarly, the stiffness matrix (5.7) and the corresponding Schur
complement (5.8) will inÙ
0n/ÕOÕ , e.g.,
/KÕ for
/ the
/ above
Õ
/Õ
/ G
/
Õ
/ G
/
Õ
clude interior blocks
example
, /
weÕ have
G
(5.10)
as in (3.5).
ØnP
J
$
J
$
J
$
d
dd
d
Y
5.2. The modified Poincaré–Steklov operator. Let us consider again
a FEM discretizaeC7CeC€] Ò
Ò
Ò
tion of problem (5.1). We decompose
the
domain
in
subdomains
, such that,
]
]ƒ
G of the original boundary Ò , each Ò
in addition
to
a
corresponding
portion
has
a common
]
f]
]f ]„
G
Ò
Ò
boundary
only with its neighbours
and
.
Denoting
here
these
common
boundaries
Y
]f ]„
Ó ]ƒ] by /g
and Ó
we decompose the stiffness matrix as in (2.1), correspond , respectively,
7CeC7] C€
Ò
ing to the subdomains Ò , such that the node points on Ó
are taken into account
in
(i.e., together with Ò ). Our goal is to study the factorization (2.2). Since,
in con]
trast ] to the idea of (5.7), the boundary node points are not considered here separately, the
SchurG complements in (2.2) are understood recursively as complements for Ò with respect
to Ò
. This is an important difference as compared to (5.8), and therefore the continuous analogues of the Schur
complements in (2.2) will also be appropriate modifications of
]
G
the Poincaré–Steklov operator
(5.3). In fact, the proper operator takes only into account the
previous subdomain Ò
.
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186
O. AXELSSON AND J. KARÁTSON
4$
First, for simplicity, let us consider the case of two subdomains Ò and0 Ò , where one
can follow
more clearly how the operator in subsection 5.1 is modified. Similarly as therein,
0
the common
boundary
ÒK ‚Ó
and
0n/
/
/ G
/ of Òg and Ò is denoted by Ó , further, we let ÓÜP
Ó)
P
Ò óÓ . We wish to define the continuous
analogue
of
the
Schur
complement
­
0
Õ
J
ä
ä
c
.
M ‚P
$
/
J
â
A to
Let us take a Õ function on Ò , such that A
b . Applying the operator
ä
(which corrresponds to the term
$ in M ), we want it to equal . Let us further consider
²
the restriction , and calculate its harmonic extension to Òg , i.e., let be the solution
0
of the problem
6$
5&
7&
*
(5.11)
J
²
()
Õ
0
ä Õ 0
ä ?
²
²
)+
b
b
in ÒK
which solves the analogue
of (5.2) only on Ò . Accordingly, the modified Poincaré–Steklov
operator assigns to the jump of the normal derivative of its harmonic extension and of
itself, i.e.,
0
8
8
(5.12)
$
$
²
‚P
Ê
$
$
r
Ê
C
on Ó
R EMARK 5.4. Similarly as in Remark 5.1, problem (5.1) can now be reduced to the
0
equation
9
(5.13)
0
##
0
where
J
â
;.
8
â
.
:9
0
0
0 ²
â
.
â
P
=
with defined
in (5.5). Letting
on Ò and
P
r
on Ò , it is readily seen that solves (5.1) if and only if s in Ò and (5.13)
holds on Ó .
­
R EMARKJ 5.5.ä The analogue of Remark 5.2 holds if, according to our setting, we handle
A and
the operators
together. Using Green’s formula, the pair Ú of these operators
satisfies
å
0
Õ
Õ
Õ
Õ
<
8
8
3= >= ?
Ú
ä
÷
ä
€) ED
(5.14)
C
«&t²
0
=
à
"
8
°
­
²
±
²
=
ä
Õ
=
0
r
°
à
A
F
4J
à
°
­
B=
­
ä
€) °
?
½&t²
3= @= A?
ä
J
0
=
for all
8
<
±
A
Õ
à
r
8 =
=
, whence it] is a symmetric and strictly
b
positive operator.
]
]
]
]
G subdomains, one can define
G
R EMARK] 5.6. For more
in
just
an
analogous
way.
­
¯
¯
¯
0
0
]
Õ
] Õ
ä
ä
Ó
Namely, for simplicity, let
. Let
be
denote the common boundary of Ò
and Ò
]
G
?
?
b . We consider
b and solve the Dirichlet
defined on Ò , such
that
±7±e±
Ò
problem on Ò with this boundary condition (which can be reduced to previous
subproblems in a recursive way, just as is the Schur complement reduced to previous
Schur
]
G
complements),
and finally calculate the jump of the corresponding normal derivatives
on Ó .
]
±e±7±
Here the bilinear form that replaces (5.14) will thus include a term on Ò
Ò
and a
term on Ò . For instance,Õ in the case
of 0 three subdomains,
we° have
°
°
°
Õ
Ò
P
Ò
P
A
# 3G IH
LK
<
(5.15)
JH 8
K
8
K
ä
Ú d
d
÷
d
0= >= ?
­
ä
5) à
­
M
?
A
²
4
d
±
²
=
K
r
­
à
è
d
±
=
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=
C
,&<²
ND
0
Õ
#= G
187
BLOCK MATRIX ­ PRECONDITIONERS
0
Õ
=
ä
&'²
OF
²
ä
A
Ò d
P
Ò d
P
b
, where d denotes the harmonic
for all
è
Ò .
extension of d A to Ò R EMARK 5.7. For problems with jumps in the diffusion coefficients, the conditioning
properties observed in Section 3 are in accordance with their analogues on the continuous
level. This will be outlined here. Namely, we have observed in Section 3 that the condition
numbers of the Schur complements are sensitive to jumps in the first approach but not in
the second approach. Accordingly, one can indicate for the same example that the standard
Poincaré–Steklov operator is sensitive to the jumps whereas the modified Poincaré–Steklov
operator is not.
0
Let us
0 therefore consider the model problem of Section 3. The domain Ò is decomposed
Ò Ô Ò in three subdomains Ò , Ò and
­
° Ò d ,0 such that there
0 are common boundaries Ó P
á
â
ä
á
Ò and Ò ­ d ¯
and Ó P Ò Ô Ò d , J but
have
no
common
boundary.
We
consider
an
elliptic
å
] 0
Œ “
áIä
á
á
á
á
b , with weak form (3.13), where
problem, formally as
with
is
-B$’÷Ù
Š
Š
á
á
(o
).
We
assume
a weight function on Ò , such that
d and,
\
varying the coefficients, we are interested in the case .
³Rí
The standard Poincaré–Steklov
operator
can
be
extended
directly
to such piecewise con]
á
stant coefficient problems,
such that one considers weighted normal derivatives on the interfaces with weights . Considering the bilinear form for our model problem with three
subdomains, the form (5.9) is replaced by
0
°
°
°
°
°
°
(5.16)
á
á
­
á
­
á
­
K
#
5P RQ
/0,
21
~ Å
²
Ià
á
?
á
±
²
²
òà
r
A
á
,
á
Å
±
²
Å
²
dHà
r
d
±
²
C
Å
d
è
Factoring out , we see that
is the constant multiple of an operator,\ á where the first
\
á
á
term
is proportional to and
the other two terms are bounded as , i.e.,
³üí
behaves similarly as Ø
in Corollary 3.2.
The modified Poincaré–Steklov operator can be extended similarly to piecewise constant
coefficient problems, using the same weighted normal derivatives as above. The bilinear form
for our model problem with three subdomains is the proper modification of (5.15):
,
<
8
á
ä
:
÷
Õ
á
Õ
3= >= A?
ä
€) 0
­
°
°
á
­
²
M # G
=
±
²
=
á
°
­
°
=
±
SD
=
=
C
for all
F , where
denotes the G
#
harmonic
extension of
to
if and only if
and
K , that is,
M
, and further
UT
UT
UT
T
(5.18)
VK
M
= C # as required for (5.17), and denote
Let= us now consider an= arbitrary test function
$ and =
=
=
. Then coincides with the
by an extension of to = , such that
$
-harmonic
on
K & since
T extension
= = equalsonzero,onandthealso
= both
= vanish onK the.
latter. Hence
entire
,
i.e.,
K
= = in (5.18) and using =
Setting
, we obtain
=
=
=
M
BM
(5.17)
Ï&½²
­
Õ
­
ä
Ú d
?
õ
A ö
0
d
A
­
á
­
Õ F&”²
ä
P
°
A
d
å
°
?
ÒK
á
Òd
Ú
°
0
²
P
²
­
à
‚J
Ú
­
á
?
b
­
°
°
²
0
?
Ú
0
à
­
­
á
°
­
ä
Òd
Ò
b
°
?
A
Ú
á
0
Õ
A
d
&<²
ß
Ò
Ò
€C
å
0
H±
%
b
Ò
Ò
Ò
d
ä
d
'0
Ú
å
°
4
d
dà
r
è
A
4
²
±
,&'²
ä
A
²
å
d
b
°
Hà
Ó
I±
?
­
Ú
LJ
A
­
r
?
0
4
á
Ú
4
0
Õ
è²
°
Ò
²
ä
P
ä
á
A ­
±
Ià
A
?
Ò
­
±
à
à
0
Òd
b
d
b
Ú
gÓ)
²
Hà
,&N²
Ú
°
­
á
‚J
ß
ÒK
Ò
°
H±
*C
Ú
A
á
Since by definition d
, we have just obtained an equality for the first
term
of (5.17).
Substituting this into the whole expression in (5.17), we obtain a form for Ú d
that contains
8
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O. AXELSSON AND J. KARÁTSON
á
<
integrals only on Ò
8
(5.19)
Ú d
and Ò
á
with
respective
weights
0
Õ
Õ
d
0= >= ?
ä
÷
d
á
ä
5) d
á
and °
°
­
=
I±
>à
A
:
d
á
°
dà
r
Ú
°
­
d
=
±
*C
è
To sum up, the behaviour of the Schur complements under jumps in Section 3 follows
that of their continuous analogues.
5.3. Approximate modified Poincaré–Steklov operator on a model problem. In this
subsection we consider a continuous analogue of the procedure (2.5)-(2.6), and show on a
model problem that it can be carried out in a similar way as on the discrete level. This
gives an alternate illustration of the fact that the condition numbers in Theorem 2.1 are mesh
independent.
Let us consider the 3D model problem 0 â
"
(5.20)
where
J
T
Þ
;
d
#VW
ä
0
b
T
in
|
0
is the unit ball.
Let us fix a positive integer
\[ [
0
Z
8
Using notation P
mains
(5.21)
,
+
ß
+
0
and
numbers
,YX
-BC
&
for the Euclidean norm of vectors 8
ND
l
G
],;X
&<T
|ˆ0
8
P
,YX
+1±e±7±÷+
0
l
Ò
,
b
G
+
¾V0
P
+
,
’
d
, we define annular subdo-
0
l
G
Z ,;X
0
;
F
|
-BeC7C7C7
C
¾0Q/
o
-{\O’
f
/
/
G
µ
f
¾
First, for simplicity, let 4-@7ú CeC7C€and
I
. Then (2.6) becomes $
e-¿
for the constant vector
.
Its
continuous
analogue,
with
the
notation
of
subsecµ
ú
0
tion 5.2, is to find an operator , such
that
^
aD
b$
%$
c&
^ 8_D ^
µ
å
(5.22)
-
0
(5.23)
0 Ò
8
&NT
0
T
P
Z b$ F
-®\B’I+
&L;
8
for the constant function
. Here Ó,P
and the procedure0 before that. We have
+1-
0
(5.24)
*
J
()
ä
ä
Õ
#VW
0
b
0
P
b
-C
Ò
aD
F , and 8
-®\B’
0
8
&NT
is defined in (5.12)
+
8_^
Pgb
Z
F
+1-{\O’
C
. Then
can be calculated explicitly.
, where is the solution of
in Ò
0
P
fe
0
Ó
`Z
d
and0
d&
A^
Further, Ó>gP
and ӂP
ÒK >Ó
Ò
²
First, the harmonic extension of to Òg is 0 ^
on Ó
)+
#
V
#
h
#
#
Here we use the form of the Laplace operator in 3D spherical coordinates, which reduces to
g g Z g for radially symmetric functions. Then an elementary calculation yields
Z # Z
#V# h ##ji
'k
##
# 'k
Hence by (5.12) and using that now
.
g on , we obtain 8_^
g g g
^ on , which means that the operator
That is, 8_^ is constant on , i.e., we can write 8_^
'k
required in (5.22) can be defined as
0
A
0
-
J,-@C
0
0
J
=
Ó
0
J
r
Ó
Ó
ú
(5.25)
µ
0
‚P
ä
ë
@¸
ì
0
Î
#$
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189
BLOCK MATRIX PRECONDITIONERS
¸
k
where is the identity
@¸ operator on Ó .
/
Our goal now is to verify a continuous
analogue of condition (2.7). According to the
µ
above,
corresponds to , further, as seen before, the analogue of is
0 the operator
J
the operator
,
such
that
homogeneous Dirichlet boundary conditions are considered on
T
Ò
. Hence the required
of (2.7) readsE as
@¸ analogue
E
$
%$
k
1J
(5.26)
+«-®\B’C
for some
X
E
X
J
Denoting by the smallest eigenvalue
of
with the given boundary conditions,
and taking
+
+R-®\B’
into account the condition
,
inequality
(5.26)
is
equivalent
to
. Here the
J
T
eigenfunctions of
are the restrictions
to ÒK of the eigenfunctions on with homogeneous
T
0
X
0
Dirichlet boundary
conditions
on
. The first eigenfunction
is the first three-dimensional
á
Œ„Ž
l
$
nm Z
m :l . Therefore (5.26) is satisfied.
mZ
aD
X
Z , jF
(5.27)
a
D
In order to determine the operator 8oX , problem (5.24) has now to be solved with replaced
]Z , F . The solution is
by LX
Z pg
p , and accordingly, the above property
Hence the constant 4 in (5.25) is replaced by p
m
l
is replaced by condition
m ,
(5.28)
,
p p satisfies the required analogue of (2.7),
If this holds then the operator X
for some
. Analogous calculations
i.e., X
k can be carried out to find
X .
rq sqs . Concerning
Inequality (5.28) is satisfied if, up to four digits,
l
,
k
the case of several %
subdomains,
one may define for technical convenience ,
X in
k
k
(5.21) to have equal volume of the subdomains. Then the condition
l ,
X
Z
Bessel function
P
, with eigenvalue æ
Now let us consider more subdomains.
Here by (5.21),
0
Ò
0
G
Ó
&<T
8
P
&<;
8
d
0
JU-
?
õ
ú
ú
µ
7CeC7C€ µ
C
E
ú
G
E
J
µ
+
’ò+
0
“
G
? õ
P
-®\B’
C
JU-
?
ú
“
Ó
P
+
+
0
ã
µ
+
P¹b
G
? ö
-J
€C
“
¸
? ö
b
C’
’
+
I
+
|ˆ0
b
is satisfied up to
b
Cm’
l
C !-
’
+
l
0
P
H
0
#d
#$d
, i.e., (5.28) is satisfied for any reasonable number of subdomains.
’Ó’
6. Element by element preconditioners for matrices partitioned in
block form.
As the method described in Section 3 uses a recursive computation, its parallelism is restricted; on the other hand, the method in Section 4 is parallelizable. Now a highly parallelizable method to construct preconditioners for the Schur complement matrix is presented,
which has also shown nice results in numerical tests [10, 12]. First the method is described
briefly, then its robustness with respect to coefficient jumps is shown.
6.1. Construction of the method. Let us consider an elliptic problem with piecewise
constant coefficients. We start with a coarse mesh of triangles (tetrahedra) or rectangles
(cubes), which has been constructed such that all coefficient jumps occur … across element
edges only. Each of these macroelements is then subdivided in ’ó
a number
(say
) of minieleÃN’
ments. The corresponding global matrix / is then
block form
ý/ partitioned in
2
(6.1)
Ytjt ]tju
Yuvt ]vu u
/
/
6
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uvu
/
/
O. AXELSSON AND J. KARÁTSON
/
corresponds to the coarse (macroelement) vertex nodes. To form a preconditioner
where
T
¸
/ G
/ of
to , we will be guided /«
by0 the2 /block matrix factorization
,
2
t tjt tVu
YtVt
uvt u
u
(6.2)
u
Yuvu ]uvt tjt ]tju
u
Ytjt
where
is the corresponding Schur complement matrix.
u
Y
j
t
t
In general,
is a full matrix and
has a very large size. Therefore, to form ,
we replace
and
by some approximations.
These will be based on macroelement
tjt
tjt
by element constructed matrices.
tjt , we first take the restrictions Rw to each macroeleFor the global assembled matrix
tVt
u by
tjt Rw , and assemble them to a global matrix denoted
ment, form their exact inverses
, which will replace
in (6.2). Similarly, each element version Rw of
is
computed exactly from the exact form of the corresponding
element matrix
j
t
t
V
t
u
x uvt uvuzy
Rw Rw
(6.3)
Rw
Rw u Rw
uvu uvt tVt tju
i.e., {w
in (6.2) is replaced by the assembly of
{w
Rw Rw {w . Then
t
tjt thetju form
to takes
tjt
Rw , denoted by , and the preconditioner
u
uvt
(6.4)
tjt
tjt
Note that the actions of
can take place fully in parallel, from the local actions of Rw .
/
/
b
/
0n/
/
/
G
/ J
M
@/
/
M
6
6
b
/
¸
/
M
T
M
/
/
G
/
/
G
T
ö
õ
ö
õ
/
M õ
/
0
ö
M õ
M õ
ö
/
/
J
ö
Ú õ
ö
Ú õ
õ
M
ö
Ú õ
T/
T
/
2
ö
Ú õ
¸
/
¸
T
b
C
b
6
M
ö
Ú õ
/
M
T
G
2
/
ö
Ú õ
0
ö
Ú õ
/
ö
Ú
M
/
/
ö
õ
G
/
/
0
ö
6
/
T
ö
õ
/
G
The above method can be extended to a multilevel version, but in this paper
we only study
T
the two-level version. Our goal is to show that the condition number of
is bounded
independently of coefficient jumps in the given elliptic operator.
We will use
that B/
¸
/ G
/
¸
@/
2
2
2
(6.5)G /10
¸
/
¸
@
/
/
¸
G
G
JKT
T
t
T
tVt ]tju
u
b
tjt ]tjt
Yuvt |t tjt ]tjt
6
M
J»T
b
M
t
u
tjt Vt u
u
M
6
C
6
b
6.2. Independence of coefficient jumps in a model
problem. For0 simplicity,
we
fol0
û
û
low
the
model
problem
in
Section
3,
and
study
the
case
of
three
macroelements
,
and
û
û
û
û
û
û
d . Accordingly, we have common boundaries ӂP
Ô
and ÓIP
Ô
d , but
û
and d have no common boundary. These macroelements are defined to match the coefficient
¯ å
]
0
of problem (3.13), i.e.,
á‚ä
á
w
/
G
-@$’$Ù÷C
o
á
We assume as in Section
3 that the relations (3.14)-(3.16) á hold.
Our goal is to show that the
¨T
\
condition number S
remains bounded as the ratio grows unboundedly.
The stiffness matrix then
has a form
as in (3.17), where the five rows/columns now
û
û
û
/ in
/ :f Õ
correspond to the nodes
,
and
d , on Ó> and on Ó)
á
á
^`
/10
(6.6)
`
`a
`
á
b Õ
/K
b
f
?
b
/
á
$
b
á
á
/Õ f
b
/Õ
f
á
A
d /KÕ $
d df
b
d
A
ç
d
?
fÕ

b
á
?
/
á
b/
ç
?
/ Õ
?
?
b

?
ç
A
fÕ
/ b
á
fÕ
b
A
/
á
fÕ
A
d
A
/d Õ
K
fÕ
h i
i
k i
b
çãè
A
C
i
A
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191
BLOCK MATRIX PRECONDITIONERS
/
we have
With the notation of
(6.1),
á
YtVt
/
ò0
a
^
á
$
ò0Â2 á
á
á
f
/Õ
f
? b
u
/Õ
A
A
d
(6.7)
é
J
M
where
is the diagonal matrix
0î2
?
ç
0
T
P
where
ments. Hence
õ
¯
á
A k
d
d
A
fÕ
A
ç
?
Õ
$
/
á
J
h

?
d
d
fÕ
A
çãè
/
Õ
/Õ
C
b
A
ç
/
G
b
/KÕ
fÕ
/
G
d$d
A 6
Õ
d

A
ç
ç
/
G /
$
$
^
ç è 6
fÕ
G
a /
C
b
A
fÕ
?
?
/
G
/
G /
dd
/
fÕ
b
fÕ
A k
d
A
h
. Further, by construction, we have
tjt
G
a
^
ç
T
?
,
ç
b
-Bc’$Ù
o
@/
b
T
$
b
b
/
(0
k
h
T
dd
çãè
^
fÕ
/
a T
T
/
fÕ
$
T
?
$
T
b
fÕ
/b
T
?
]
á
b
A
, since the latter act independently on the three macroele-
tjt ]tju
]
$
b
0
ö
/
Õ
b
H0
tjt
Rw
J
b
/
/
T
01/
/
á
]
á
]ƒ]
Õ
is independent of the
?

é
tjt tju
G
?
ç
/
P
/
fÕ
/ b
b
G
(6.8)
First we observe that
(0Â2
á
/
?
d 6
/ Schur
0né”complement
/ ÕBÕ
/ Õ is /
From (3.18), the
á
fÕ

b
^ á
Yuvu
/
f
fÕ
/
dd
/KÕ
/
á
a
h
d
á
(0
k
b
b
f
?
b/
á
$
b
Yuvt
/
/
b
b
/KÕ
Ytju
/
fÕ
$
A k
d$d
d
A
h
u
á
is also independent
of the . That is, the left and right
matrices
in the product in (6.5) are
/
G
independent of . It remains to study the matrix in the center.
G
?
0
/
0
/
0
P ROPOSITION 6.1. The condition number of M
is bounded as ç AH³ G í .
M
ç Ú
Proof. Let ؇
and M Ú P
P
A M
A M . Then S
S
Ø Ú . Here,
M
M
MÚ
ç
ç
following (3.20),
/ Õ
fÕ
á
2
/Õ
/ G
/
Õ
/KÕ
/ G
/
Õ
(6.9)0î2 ?
/ Õ
fÕ
u
ç A
ØÚ
ç
Ø
b
r
?
/KÕ b
/Kb Õ 6 f Õ
b0
f
?
/
G
J
b
f rÕ
/
çãAè
J
u
A
A 6
ç
/
$
J
á
d
d
d$d
d
?
? 
? ? . As seen after (3.20), here ØK is a Schur complewhere ØóP 0 0
á
ment, corresponding
to
the
positive
definite
matrix Ú $ from (3.7), modified by setting the
?
A
weights and ç ç
in
the
integrals.
Hence ØK is still positive definite. Denoting
by
the second to fourth terms of0 (6.9),
we
have
á
2
,Y}
á
ØÚ
Øøb
b
b 6
r
,}
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192
O. AXELSSON AND J. KARÁTSON
,}
Rw
?
¯ has bounded coefficients as ç A ¯ ³
í
.
where
/ ç
0
Let us now similarly rewrite M Ú . By definition, M is the assembly of the Schur comple-Bc’Ù
ö
ö
(o
). To form the latter, we note that
by
ments M õ û for the û element matrices õ
û
assumption and d only have interior vertices on Ó or Ód , respectively, whereas has
/
/
/
vertices on both of Ó>/ and Ó)d . Therefore
fÕ
the element matrices take
the following
f Õ form:
Rw
/
0
{w
á
2
? ö
õ
/
á
Õ
á
f
?
/
Rw
õ
é
Rw
Rw
M õ
]uvu
? ö
?
A
0«é”
/ ç
ç
A ö
M õ
J

2
0
?
ç
0Â2
/KÕ J
?
á ?
¯
f
A
Õ
/ ç Õè
/
M
$
M
á
M 5
0
J
0
J
M $
Therefore
2
0
MÚ
?
r
0
?
J
ç
?
Õ
?
b 6Õ
/K
G
Õ
/
r
fÕ
?
A

?
A
J
A
G
/
fÕ
/
Õ
f
/
G /
fÕ
/KÕ
f
/
á
/K
J Õ
?
/ Õ
?f
fÕ
A
ç è
J
A
/
?
/Õ
$
f
/
G
/
G /
A
A
/
A
A/ f Õ
G
? á
f
d
Rw
d
dd
fÕ
A
d
fÕ
fÕ
A
A 6
/
G
fÕ
$
/
G
C
A 6
J
fÕ
?
/Õ
f ÕJ
?
where
M 76
á
/
4
M
M c
/
G
fÕ
f Õ
f
/
?
fÕ
/
/
G
$
/Õ
á
J
?
$
f
A
?
/Õ
J
A
/
/
A
fÕ
/
G
d
d$d
?
$
f
d
/
G
? á
f
fÕ
fÕ
d
C
A
b
/
b 6
/KÕ
?
/Õ
/
G
fÕ
4-
?
,;~
/
/
M
G
{w
2
0
/
f
bG
/
/
ç
fÕ
G /
A
/KÕ f Õ
? á
á A J
A
A
d
f
?
?
f
/
øb
b 0
r
J
?
f
A
M
/
/
/KJ Õ
/KÕ
fÕ
?
2
ç A
Õ
ç A
Õ ç fÕ
/
2
/
?
r
á
f /
/KÕ
/
f
/KÕ
á
A
çãè
6
block matrix, where the first and second rows/columns
? ö
ö
and Ó , respectively. That is, M õ
and M õ è are added to
A ö
:
/KÕ
á
á
f
fÕ
A 6
A
A
f
?
fÕ
A
’<Ò
ö
á fÕ
Õ
/KÕ
P
/KÕ
Õ
d
çãè
/ Õ
K
f
0î2
A
/
d
A
ç
d
9
ç
/KÕ
?
M
A /6
fÕ
? Rw
0
J
/ A G
/
2
A
u

ö
è
M õ
á
d$d
‚P
Rw
?
á
f
d
0n/
0
fÕ
f
/
G
The assembly of M õ
is a
correspond to the boundaries Ó
the (1,1) and (2,2) blocks of M õ
0
Yu
fÕ
b
Rw
á
/
G
/
A
ç
fÕ
/
Õ
d
where

?
Õ
J
$
f
u
?
/
G
?
A
ç
/
Õ fÕ
b /
?
ç
/
/KÕ
ö
è
/
á
/
6
á
J
/?
A
ç

]u
?
á

is
0 from (6.8).
/Õ Then
fÕ
and
/
$
Rw
õ
? 6
u
]uvu
é”/
á
0ý2
?
á
/
fÕ
?
/
á
A ö
Õ
A
ç
Yu
/
2 á
0
?
ç
/
ç Aè
r
ç
A
fÕ
Õ
A
/K
J Õ
J
A
/
f
f
/
? $
/
G
G
/ fÕ
fÕ
A
J
/KÕ
A
ç Aè
f
A
ç
d
/
/
G
dd
d
fÕ
A 6
/KÕ
f
/
/
G
fÕ
/
?
?
? ? is a Schur complement of the matrix Ú $ in (3.7).
where M óP
$
Hence M is positive
definite,
further,
denotes the second
matrix
above,
which has bounded
?
+
½coefficients as ç Aò³Rí . (Recall that, by assumption, b
.)
¼
ç Aè
ç
,_~
ç
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193
BLOCK MATRIX PRECONDITIONERS
[
0
[
Now it is easy
to derive the spectral equivalence of M Ú and Ø Ú . Let us consider vectors in
? "
8
A
the form
, where the decomposition into the vectors 8 and 8 corresponds to the
block form0 of á M Ú . Then
MÚ
8
á
± 8
M
8
± 8
,;~
r
Hence
MÚ
Here, by assumption, b
M
8
ç ?
Here the matrices Ø
ØÚ 8
± 8
and M
8
8
± 8
A
ç ?
ç A
ç ?
r
,}
,~
ç
± 8 8
± 8 8
± 8
8
ØÚ
± 8
8
MÚ
8
Ø
± 8
M
8
0
MÚ b
,}
r
b 6
Ø
r
8
± 8
C
,Y}
r
± 8
8
± 8 C
and
MÚ 2
øb
M
P
b
b 6
0
=
,Y}
were seen to be positive definite. Let us now define
á
á
These matrices
are also positive definite since they coincide with
the case XDY] . G Then (6.10)
X Yimplies
]
X YZ5[
X
G
G
± 8
C
± 8 8
8
Ø
r
á
± 8
8
ØÚ
± 8
Øüb
P
and
. Hence
,~
r
0î2
Ø
M
ç
Ø
± 8
«-
A
± 8
8
± 8
8
0
± 8
8
ØÚ
(6.10)
á
0
=
MÚ
ØÚ
€
G
/
G
MÚ
ØÚ
C
and M Ú , respectively, in
YZ5[
ØÚ
,~
r
G
M
Ø
Ú 5
?
ØÚ
which yields the desired boundedness of S M Ú
as ç A ³Rí .
T
?
ç
C OROLLARY 6.2. The condition number
of
is bounded as ç Aó³Rí .
]
á
Proof. We have seen just before Proposition
6.1 that the left andç right matrices in the
product in (6.5) are independent of the . ? Hence it remains to prove that the matrixBin
/ the
center has
bounded
condition
number
as
.
Since
this
matrix
is
block
diagonal,
its
ç
A
ô
³
í
/
G
¨T
ç
eigenvalues
coincide
with
those
of
its
diagonal
blocks,
therefore
it
suffices
that
S
?
and S M M
have bounded condition numbers as ç Aò³ í . The latter has been proved
ç
in Proposition 6.1, whereas in the former case
/
tjt Ytjt
u
tjt tjt
/
T
0
^
a T
/
$
á
b
$
]
b
T
b
b /
$
b
T
k
b
d$d
h
dd
is even independent of the .
]
R EMARK 6.3. The above result can be extended to theá case of more than three subdomains under corresponding assumptions on the coefficients .
Acknowledgements. The authors acknowledge the generous provisions at the Mathematisches Forschungsinstitut Oberwolfach, which enabled writing the first version of this
paper during their ‘Research in Pairs’ stay at the Institute from 24 February–8 March 2008.
The first author was also supported by the grant IET40030045 of the Academy of Sciences
of the Czech Republic (Program Information Society), and the second author was also supported by the Hungarian Research Grant OTKA No.K 67819 and by HAS via Bolyai János
Scholarship.
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Kent State University
http://etna.math.kent.edu
194
O. AXELSSON AND J. KARÁTSON
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