ETNA
Electronic Transactions on Numerical Analysis.
Volume 20, pp. 164-179, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
A BDDC ALGORITHM FOR A MIXED FORMULATION
OF FLOW IN POROUS MEDIA
XUEMIN TU
Abstract. The BDDC (balancing domain decomposition by constraints) algorithms are similar to the balancing
Neumann-Neumann methods, with a small number of continuity constraints enforced across the interface throughout
the iterations. These constraints form a coarse, global component of the preconditioner. The BDDC methods are
powerful for solving large sparse linear algebraic systems arising from discretizations of elliptic boundary value
problems. In this paper, the BDDC algorithm is extended to saddle point problems generated from the mixed finite
element methods used to approximate the scalar elliptic problems for flow in porous media. Edge/face average
constraints are enforced and the same rate of convergence is obtained as for simple elliptic cases. The condition
number bound is estimated and numerical experiments are discussed. In addition, a comparison of the BDDC
method with an edge/face-based iterative substructuring method is provided.
Key words. BDDC, domain decomposition, saddle point problem, condition number, benign space, edge/facebased iterative substructuring method
AMS subject classifications. 65N30, 65N55, 65F10
1. Introduction. The BDDC algorithms, introduced by Dohrmann in [5], see also [13,
14], are nonoverlapping domain decomposition methods, which are similar to the balancing
Neumann-Neumann (BNN) algorithms. In BDDC, the coarse problems are given in terms of
a set of primal constraints. An important advantage with such a coarse problem is that the
Schur complements that arise in the computation will all be invertible. The relation between
the BDDC and BNN is similar to that between the FETI-DP and one level FETI. Recently,
the BDDC and FETI-DP algorithms for elliptic problems were rederived and a much shorter
proof of the main result in [14] was given in [11].
Mixed formulations of elliptic problems, see [2], lead to large, sparse, symmetric, indefinite linear systems. Such methods have extensive applications, as in flow in porous media,
where a good approximation to the velocity is required.
Overlapping domain decomposition methods for this kind of problem were developed in
[6, 15, 16, 17]. These additive or multiplicative overlapping Schwartz alternating methods
reduce the problem to a symmetric positive definite problem for a vector, divergence free in a
finite element sense. Then two-level overlapping methods are applied to the reduced positive
definite problem in the benign, divergence free subspace. The algorithms converge at a rate
independent of the mesh parameters and the coefficients of the original equation.
In [9], two non-overlapping domain decomposition algorithms were proposed. They are
unpreconditioned conjugate gradient methods for certain interface variables and are, to the
best of our knowledge, the first iterative substructuring methods. The rate of convergence is
independent of the coefficients of the original equations, but depends mildly on the mesh parameters. The consequence of the singular local Neumann problems that arise was addressed
in [9]. Other non-overlapping domain decomposition methods were proposed in [8] and [4]
with improved rates of convergence. A BNN version of the Method II of [9] was proposed in
[3], see also [20]. The same rate of convergence is obtained as for simple elliptic cases.
Using mixed formulations of flow in porous media, we will obtain a saddle point problem
which is closely related to that arising from the incompressible Stokes equations. We note
Received May 12, 2005. Accepted for publication August 31, 2005. Recommended by Y. Kuznetsov.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012,
USA. (xuemin@cims.nyu.edu). This work was supported in part by the US Department of Energy under
Contract DE-FG02-92ER25127.
164
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
165
that, in a recent paper [12], the BDDC algorithms have been applied to the incompressible
Stokes equation, where the constraints enforced across the interface satisfy two assumptions.
One assumption forces the iterates into the benign subspace in which the operator is positive
definite and the other ensures a good bound for the condition number. In general, both these
assumptions are required.
In this paper, we extend the BDDC algorithms to mixed formulations of flow in porous
media. This work is directly related to [12], but our situation is also different. First of all,
our problem is not originally formulated in the benign, divergence free subspace, and it will
therefore be reduced to the benign subspace, as in [6, 15, 16, 17], at the beginning of the
computation. In addition, only edge/face constraints are needed to force the iterates into
the benign subspace and to ensure a good bound for the condition number, since RaviartThomas finite elements, see [2, Chapter III], are utilized. These elements have no degrees of
freedom associated with vertices/edges in two/three dimensions. Also, the condition number
estimate for the Stokes case can be simplified since the Stokes extension is equivalent to the
harmonic extension, see [1]. However, this is not the case here, and different technical tools
are required. We also note that our BDDC method is closely related to an edge/face-based
substructuring iterative method. We will give a detailed description later.
An iterative substructuring method with Raviart-Thomas finite elements for vector field
problems was proposed in [24, 21]. We will borrow some technical tools from these papers
in our analysis of the BDDC algorithms.
The rest of the paper is organized as follows. The mixed formulation for the elliptic
problems and its finite element discretization are described in Section 2. We reduce our
system to an interface problem in Section 3. In Section 4, we introduce the BDDC methods
for our mixed methods. We give some auxiliary results in Section 5. In Section 6, we provide
of the condition number for the system with the
an estimate of the form
BDDC preconditioner; these
and are the diameters of the subdomains and elements,
respectively. We also compare the BDDC methods with an edge/face-based algorithm in
Section 7. Finally, some computational results are given in Section 8.
2. An elliptic problem discretized by mixed finite elements. We consider the following elliptic problem on a bounded polygonal domain in two or three dimensions with a
Neumann boundary condition:
. "!$#%'&)(+*-, inin 34 65
/!$#%'&)(0*21
Here . is the outward normal to 37 and is a positive definite matrix function with entries
#
in 89 satisfying
! (
(2.2)
:; #7!=<>( :@?BADCE:FC G for a.e. <IH G
for some positive constant A .
The functions
,JH 8 ! ( andOQ1KP H L4MN O% ST! 34P ( satisfy the compatibility condition
,7R"< 1%R"U'*-V 5
The equation (2.1) has a solution which is unique up to a constant. Without loss of
&
generality, we assume that
and that has mean value zero. We also require that the
I
1
W
*
V
,
solution has mean value zero over ; therefore we have a unique solution.
&
We assume that we are interested in computing
X#Q'& directly as is often required in
flow in porous media. We then introduce the velocity Y :
Y * X#%'& G
(2.1)
ETNA
Kent State University
etna@mcs.kent.edu
166
X. TU
&
and call the pressure. We obtain the following system for the velocity
(2.3)
[ \ Y * X#%'&
Z .] Y ^* ,
Y *_V
in
in
in
3465
Y
&
and the pressure :
` !$<>(+*_#7!$<>( LaM and define a Hilbert space by
cb !dRegf G (+*hiH 8 ! ( or 8 ! (kj"lm]niH 8 ! ( and io . *_V on 34'p G
with the norm
Pyx
PFx
PFx
C i C qrtsvuTw * C i C z|{ q } ~ C i C z|{ q G
where ~ is the diameter of .
!dRegf G ( , it is possible to define its normal component Y . on 34 ,
Given a vector Y
H
as an element of L4MN 34 , and the following
PFx
STPFxƒ inequality holds
! ( .
(2.4)
CnY C X/€‚ { q „CnYC XqrtsuTw G
with a constant that is independent of ~ , the diameter of , see [24, Section 2]. The trace
operator that maps a vector in component in }L4MN 37 is thus
G ( into itsseenormal
!$R"beegf surjective;
! (
continuous,
and
it
can
be
shown
to
[7,
Ch.
I, Th. 2.5 and Cor. 2.8].
P
The weak form of (2.3) is as follows: find Y
H b !dRegf G ( and &H 8 b ! (*…h‡†KˆF†‰H
8 ! ( GmŠ †‹R"<o*-V p such that,
P
Y
B

G
G
P #7 ! Y G iŒ( !=i &)(Ž* V Š P )iH 8 b b !$ R"egfG G (
! †‘(
* ,7†‹R"< 7†’H ! (
where Y G
#7! • iŒ(+* Š Y ; ` !=<>(ki>R"< and  ! Y G †‘(0* Š !‚] Y (“†‹R< .
Let ” be the lowest order Raviart-Thomas finite element space with a zero normal
component on 34 , see [2, Chapter III, 3], and let – be the space of piecewise constants
with a zero mean value, which are finite dimensional subspaces of b
G ( and 8 b ! ( ,
d
!
R
g
e
f
•
respectively. The pair ” , – satisfy a uniform inf-sup condition, see [2, Chapter IV. 1.2]. The
• ” and – such that,
finite element discrete problem is: find Y & H
H
P
•
#7 ! Y Y G G i ( B !=i G & (—* V Š )i H – ” G
! † (
* ,7† R"< 7† H
Let
and the matrix form is:
˜„™ š ; ˜ Y * c˜ ž 5
š œ
V › & ›
›
The system (2.5) is symmetric indefinite with the matrix symmetric, positive definite. For
™
details on the range of negative and positive eigenvalues of (2.5), see [19].
3. Reduction to an interface problem. We decompose into Ÿ nonoverlapping subdomains with diameters ,
s
s e@* G n G Ÿ , and with * „¡‹¢ s s . We assume that
each subdomain is a union of shape-regular coarse rectangles/hexahedra and that the number
(2.5)
of such rectangles/hexahedra forming an individual subdomain is uniformly bounded. We
note that the algorithm can be extended to different types of subdomains. In a more general
case, we can still define faces, regarded as open sets that are shared by two subdomains. Two
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
167
nodes belong to the same face when they are associated with the same pair of subdomains.
We then introduce quasi-uniform triangulations of each subdomain. The global problem (2.5)
is assembled from the subdomain problems
x
x
x
v
q
s
Y
˜ ™ qvvq ss š q s
qvs ¥ * ˜ ž vq s › 5
š
V I› ¤ & ¦
x
(3.1)
x
x$£
The degrees of freedom of the Raviart-Thomas finite elements are the normal components on the boundary of each element only.
is
Let be the interface between the subdomains. The set of the interface nodes
defined as
, where
is the set of nodes on
and
is the set of nodes on
. We decompose the discrete velocity and pressure spaces
and into
§ §
X
®
¬
)
­
­
§ ¨
4
3
4
3
4
3
7
3
4
3
s=w
s •”
* !d© s«ª s$34w w (°¯
34 –
• ” •±³² •µ
” ´ G – * – ±°² –¶b5
(3.2)
*x
x
•µ
´
•
•
±
±
” is the space of traces• on± § of functions of ” . and – are direct± sums of subdomain
qvs , and subdomainx interior pressure spaces
interior velocity spaces
x – qvs , i.e.,
• ± ² • ±qs G – ± ² – ±qvs 5
*¸s ¬· M
*¸s ¬· M
x
x
• ±
The elements of qs are supported in thex subdomain and their normal components vanish
PF§ ¹ ® 34 s , while thes elements of – ±qvs are restrictions of
on x the subdomain interface §
s
*
± qs . –ºb is the subspace of – with constant values
elements in – to which satisfy Š
s
†
*2V
b† qvs in the subdomain s that satisfy x
» qvs (3.3)
¬s · M † b½¼ ! s (0*-x V G
x which maps functions
where ¼ is the measure of the subdomain . ¾ b qvs x is the operator
s
s
!
(
in the space –¶b to its constant component
x of the subdomain
• qs • s . ±À
¿ • ´ , the space of thex
v
q
s
We denote the subdomain velocity space by
* space by • ´ ¬ • ´qvs .
• ´ qvs , and the associate product
interface velocity variables by
*^Á s· M
x as
x point
x$£ problems
xd£ (3.1) can be written
The subdomain saddle
x ÅÇÆÆ
ÅÇÆÆ Y qvs x ± ÅÇÆÆ
±tqvs ± x ±mqs± ´qvs ± x
™ ±tqvs± x š x$£ ™ ±mqvs´ x V x$£ ÆÆÆ qvs wx ± ÆÆÆ Vž qvs ± ÆÆ
(3.4)
ÂÃà š ´qvs ± ±qV s´ š ´‘qvs ´ x qvV s´ ÂÃà Y& qvs w x ´ * ÂÃà w x È G
b È Ãà w È ÃÃÄ Vž qvs ´
Ãà ™ š
™
š
´
Ä
Ã
x x V x x V š b qvs x x V x ÃÄ & qvx s w b
w
x x
• x ±qvs G – ±qvs G • ´qvs G – bqx s . We note that, by the divergence
± ± ´
where Y qvs G qvs G Y qs G qvs x b
! w & w w & w (6H2!
(
±
theorem, the lower left block of the matrix of (3.4) is zero since the bilinear form  qvs G b qs
=
!
i
† (
•
±
±qvs and a constant b qvs in the subdomain .
always vanishes for any qvs
s
i H
†
ETNA
Kent State University
etna@mcs.kent.edu
168
X. TU
3.1. Obtaining a divergence free correction. First of all, we seek a discrete velocity
Y H • ” such that,
Yš * ž 5
(3.5)
•
Let ” be the lowest order Raviart-Thomas finite element space on the coarse triangulation, associated with the subdomains, with zero normal components on 34 and let – be
the space of piecewise constants with vanishing mean value. ¾ b; is the natural interpolation
•
• É – . We also use the same interpolation operator on the
operator from ” ÊÉ – to ”
corresponding right hand side space. Let
˜c™ bb š b; * ¾ b ˜„™ š ; ¾ b; G
(3.6)
š Vœ›
š Vœ›
and
˜ Y bb * ¾ºb; ˜™ bb š b; L4M ¾b ˜Ëž  5
š V ›
›
& ›
We note x that the coarse grid solution Y b does not necessarily
x x satisfy (3.5), but that š Y b ž has mean value zero over each subdomain , see
x [16, 6].
x the local Neumann problems,
x Then
s
qs Y b wÌqvs G G G Ÿ , are all well´
with Y qvs
hand sides ž ™
q
s
w *ËV andx the right
qvs x Y b wÌqvs x ¥ eX* nn
¤
x
š
$
x
£
posed. We can solve
x±
x x ±x
±tqvs ± x ±tqvs±
v
q
s
Y
™ ±mqs± š ¥ qvs w ± ¥ * ž Íqvs ! ™ qvs Y qvs b wÌY qvs b wÌ( qvs ± ¥ G eÎ* G n G Ÿ G
(3.7)
¤š
(
V ¤& w x¤ ! š
and set
Y s * Y qvs w ± ¥ G e°* G n G Ÿo5
¤ 
Y* b ÏY M n ÏY which satisfies (3.5). We then write the solution of (2.5) as
Let Y · ˜ Y * ˜ Y ˜ Y G
& ›
V › & ›
where the correction Y G ; satisfies
! &)(
˜@™ š ; ˜ Y * ˜ ™ Y 5
(3.8)
x x$š £ VÐx$£ › &˛ V x ›
x
This problem can be assembled from the subdomain problems:
ÅÇÆÆ Y ±qvs x ÅÇÆÆ Ñ ± Å Æ
x±mqvs ± ±tqvs± ´qvs ± x
qvs x ÆÆ
™ ±tqvs± x š x$£ ™ ±mqvs´ x V x$£ ÆÆÆ ±qs x ÆÆÆ
ÂÃà š ´qvs ± ±mqvV s´ š ´‘qvs ´ x qV s´ ÂÃà &Y ´qvs x * Âà VÑ ´ qvs È G
(3.9)
x x ÃÃÃÄ ™ x V x š V x ™š b qvs´ x š b V x È x ÃÃÃÄ & bqs È x ÃÃÄ V x xkÔ
x
ѱ
Ñ´
x ± x Ô± ´
• ± ± • ´
where Y qvs G qs G Y qs G b qvs
! &
& (aHÒ! qvs G – qvs G qvs G – bqs ( and qvs *]_Ó ™ qvs Y qvs ± and qvs *
Ó ™ qvs Y qvs ´ .
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
x
x
x
169
3.2. A reduced interface problem. We now reduce the global problem (3.8) to an interface problem.
We define the subdomain Schur complements
as: given
, determine
such that,
x x
Õ ´ qvs Ö ´qvs
´ qvs
´qvs • ´qs
Õ
Ö
H
x xd£ xd£
x
ÅÆ
±tqvs ± x ±tqvs± ´qvs ± x Å ÆÆ Ö ±qvs x x Å ÆÆ
x
x
x
d
x
£
x
™ ±tqvs± š
™ ±mqvs´
±qvs È * VV
(3.10)
È .
È
ÂÃÃÄ š ´qvs ± ±mqvV s´ š ´‘qvs ´ ÂÃÃÄ Ö & ´qvs
ÃÄÂ xÕ ´ qs Ö ´qvs
™ š
™
´
We know, from the definition in (3.10), that the action of Õ qvs can be evaluated by solving a
Neumann problem on the subdomain . We note that these Neumann problems are always
s
well-posed, even without any constraints on the normal component of the velocity since we
have removed the constant pressure component constraints. Furthermore, since the local
x x$£
matrices
±tqs ± x ´"qs ± x
™ ™
¤ ™ ´"qs ± ™ ´qs ´ ¥ x
are symmetric, positive definite, we have, by an inertia argument,
x Schur complements Õ ´ qvs defined in (3.10) are symmetric,
L
3.1. The subdomain
positive definite.
´
Given the definition of Õ qvs , the subdomain problems (3.9) are reduced to the subdomain
x
x x$£
x
interface problems
x
x
´qvs
Y ´qvs
Õ ´ qvs ´ š b qvs´
55v5 Ÿ
¤ š b qvs V ¥ ¤ & bqvs ¥ * ¤Ø× V ¥ G e°* GmÙ|G G G
x x$£
x
where
x x
x x$£
x±m± ±t± LaM Ñ ±
qvs
´qs Ñ ´ qvs ´qvs ± x ±mqvs´ÊÛ ™ ±tqvs ± x š qvs
* …Ú ™ š ¤ š qvs V ¥ ˜ V › 5
×
x
´
´
´
We denote the direct sum
x of Õ qvs by•ÜÕ ´ . Let ¾ qvs be the operator which maps functions
• ´qvsin.
the continuous interface velocity space ”
to the subdomain components in the space
´
´
The direct sum of the ¾ qs is denoted by ¾ . Then the global interface problem, assembled
• ” ´ É – b , such
´
from the subdomain interface problems, can be written as: find Y G b
!
&
6
(
H
that
´
´ ´
´
´
(3.11)
Õ Ý xd£˜ Y b x * Ý Õ Ý b ´ š Ý b; x ¥ ˜x Y b * ˜ G
& › ¤š
& › ×V ›
V
´
´ ´ ´
´ ´ £ x x
where
*^Þ s· ¬ M ¾ qvs qs , š Ý b *-Þ s· ¬ M š b qvs ¾ qvs , x$and
×
× ´ ´ ´ ´ » ´ ´ ´
Õ Ý * ¾ ; Õ ¾ *߬· ¾ qvs Õ qs ¾ qvs 5
(3.12)
s M
•Ü´ É –¶b . But by Lemma
Thus, Õ Ý is an interface saddle point operator defined on the space ”
´ ´ .
3.1, this operator is symmetric positive definite on the benign subspace where Ý b Y
š
*WV
EMMA
ETNA
Kent State University
etna@mcs.kent.edu
170
X. TU
Y! ´ G &7( ;
lies in this benign subspace. We will proFrom (3.8), we know that the correction
pose a preconditioner for (3.11) which keeps all the iterates in this benign subspace. Therefore, the iterates remain in the benign subspace in which the preconditioned operator is positive definite and a preconditioned conjugate gradient method can be applied.
•à ´
x
•µ
” á@² •â * •µ
” á@² ã ä ¬· • âqvs å 5
à ´ * •µ
s M
4. The BDDC methods. We follow [12, Section 4] closely in this section. We introduce
a partially assembled interface velocity space
by
•” á
x
Here,
is the coarse level, primal interface velocity space which is spanned by subdomain
interface edge/face basis functions with constant values at the nodes of the edge/face for
two/three dimensions. We change the variables so that the degree of freedom of each primal
is the direct sum of the
, which
constraint is explicit, see [11] and [10]. The space
is spanned by the remaining interface velocity degrees of freedom with a zero average over
each edge/face. In the space
, we have relaxed most continuity constraints on the velocity
across the interface but retained all primal continuity constraints, which has the important
advantage that all the linear systems are nonsingular in the computation. This is the main
difference from an edge/face-based iterative substructuring domain decomposition method,
where we will encounter singular local problems; see Section 7.
We need to introduce several restriction, extension, and scaling operators between different spaces.
restricts functions in the space
to the components
related to
the subdomain .
maps functions from
to
, its dual subdomain component.
is a restriction operator from
to its subspaces
and
is the operator which
maps vectors in
into their components in
.
is the direct sum of
àæ ´
x
•â
• âqvs
x
x
•µ
¾ ´qs â
à •´ â
x • ´qvs
µ
•
´
s ¾ qvs
” x •µqvs á
•µ
´
´¾ ‘á x
áqvs
”
”
¾
x
•µ
á
´èç镵´
á qvs ¾ ´ •µ
”
à
æ
ˆ
•
ç
•
´
´
´
‘
´
á
¾ ´qvs and ¾ ê ˆ ”
à is the direct sum of ¾ and ¾ âqvs . We define the positive scaling
factor ë‘ì
s !=í)( as follows: for î Hï ‡ð Ù|Gtñ ( ,
ë sì !=í7(+* ­ôó`EõDòs ö!$í)`E( ­ò G íoH 34 s=w ® § G
Þ
!=í7(
37 ­ . We assume that ` s !=í)(
where ÷Òø is the set of indices ù of the subdomains such that
è
í
H
x
x
is a constant in each subdomain. We then note that ë"ì
s !$í)( is constant on each edge/face since
the nodes on each edge/face are
shared by the same pair of subdomains. Multiplying each
x
´ is the
â
â
row of ¾ qvs with the scaling factor ëì
gives us ¾ ~ qs . The scaled operators ¾ ê ~
w
=
!
7
í
(
w
´á and the ¾ ~qvs â . Wes also use the notation
direct sum of ¾
w
´ ú
´ ú
~
¾
ê
¾
ê
~
w
¾ê *
¥ and ¾ ê * ¤
¥5
¤
´ ´
´
•ü´ ,
We also denote by û , û Ý , and û ê , the right hand side spaces corresponding to
•µ
´
µ
•
´
” , and à , respectively.´ We´ will use´ the same•µrestriction,
extension, and scaled restriction
´ , •µ
” ´ , and •µ
operators for the spaces û , û Ý , and û ê as for
à ´.
´ •µ
We define the partially assembled interface velocity Schur complement Õ ê
à ´Øç ûê ´
ˆ
by
(4.1)
Õ ê ´ * ¾ ´; Õ ´ ¾ ´ 5
x
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
171
´ • à $x £ ´ , Õ ê ´ Ö ´ x ûê ´ satisfies
Ö
H á4± x ÅÇÆ Hx ÅÇÆ
±tq M ± x ±tq M±
x ÅÇÆÆÆÆ
™ ±tq M± x š x £
™ ê ê q ±mMq Máx £ ÆÆÆÆ Ö ±q ±q M M x ÆÆÆ 
š âÎq M ± ±tq V Mâ
š á4q M â ÆÆÆ Âà & âq M ÆÆÆ Âà V Õ ê ´ Ö ´ âq M ÆÆÆ
Ã
Â
Ã
ê
ÃÃ .! ( 5
ÃÃ Ö
(4.2)
Ãà ™ x š xd£
™
*
Ã
ÃÃ ..
.
.
.
ÃÃ
..
..
.. È
Ã
È
Ã
ÃÃÄ Õ ê ´ Ö ´ á È
ÃÃÄ ê á4q M ± ê ±mq Má
á
Ã
Ä
7
á
á
Ö
55n5 ™ ê
! (
™ š
xd£ x x
x
x$£ x x x x x
Here, x
á4qs â G ê á7á » ¾ áqvs á)qvs á ¾ áqvs G ê ±mqvsá ±mqvsá ¾ áqvs 5
ê™ á4qvs ± * ¾ á qvs ™ á4qvs ± G ™ ê á7qvs â *
™ * s ¬· M ™
š * š
• ´ , we can also
´
Given the definition Õ ê on the partially assembled interface velocity space à
´
´
obtain Õ Ý from Õ ê by assembling the dual interface velocity part on the subdomain interface,
i.e.,
x
(4.3)
Õ Ý ´ * ¾ ê ´; Õ ê ´ ¾ ê ´ 5
´
´
We can also define the operator ê b , partially assembled from the subdomain operators b qvs ,
š
š
which maps the partially assembled interface velocity to the subdomain constant pressures.
´
´
Then Ý b can also be obtained from ê b by assembling the dual interface velocity part on
´
´
š
š
the subdomain interface, i.e., Ý b
ê b ¾ê ´ .
š
š
*
Therefore, we can write the global interface saddle point problem operator Õ Ý , introduced
in Equation (3.11), as
´ Ý b; ´
´ ; Õ ê ´ ¾ ê ´ ¾ ê ´; ê b ; ´
Õ
Ý
¾
ê
š
š ¥5
(4.4)
ÕÝ * Ý b ´
´
´
*
¥
ê
b
¾
ê
¤š
¤ š
V
V
Õê ´
x
xd£
xd£
âÎq M ± x
™ ±tq Mâ x
š âÎq M â
™ x
á4â
™ê q M
x$£
¾ áqvs ™
can also be defined by: for any given
then
The BDDC preconditioner for solving the global interface saddle point problem (3.11) is
´ ú Õê ´
~
¾
ê
;
a
L
M
ý *
(4.5)
w ¥ ê b´
¤
¤š
We use the notation
´
ê
Õ
Õê * ê b ´
¤š
š ê b;
V
´
š ê b;
V
´ aL M ¾ ê ~ ´ ú
¥ ¤ w ¥5
¥G
´
• ´
µ
! Y G þ b c( H ” É –ºb , such
´
¾* ê ~; Õ ê 4L M ¾ ê ~ ˜ 5
›
´› of ”æ ´ and àæ ´ ,× Vÿ
respectively, as in [12, Defiw
•Ü
Ý b´ Ö ´ pG
”•Ü´
š
à ´ š ê b ´ Ö ´ *-*-VV p/5
then the preconditioned BDDC algorithm is of the form: find
that
´
¾ ê ~; Õ ê L4M ¾ ê ~ Õ Ý ˜ Y b
&
´
We define two subspaces æ”
w and æà
nition 1]:
•µ
”•µ´´ w * h Ö ´´ H
à w * hÖ H
(4.6)
ETNA
Kent State University
etna@mcs.kent.edu
172
X. TU
•” ´ É – b •à ´ É – b
•” ´ É – b •à ´ É – b
w
w
ÕÝ ´ Õê ´
•µ
” ´ É –ºb •µ
à ´ É –ºb µ
” ´ w É –¶b •Ü´
Ö H • à ´ w É –ºb ¾ ê •µ~; Ö ´ H •µ
Ö ´ H ´ à w É –¶b ¾ ê ~; Ö H ” w É –ºb
´
µ
•
´
Ö *]! Ö G & b (DH à w É –ºb
ê b Ö *-V
š
´
´ ú
´ ´
(4.7)
¾ºê ~; Ö * ¾ ê ~; w ¥ ˜ Ö b * ¾ ê ~; w b Ö ¥ H •Ü
” ´ É –ºb5
¤
& › ¤ &
´ ´ ´ and we find that
We only need to show that Ý b ¾ ê ~; Ö
š
´Ý b ¾ºê ~; ´ Ö ´ w ê b ´ *-¾ ê V ´ ¾ºê ~; ´ Ö ´ ê b á Ö á 5
š w ´* š ´ w * š
*2V
Here we use the definitions of Ý b and ê b for the first equality. For the second, we use
š
š
x degrees of freedom
x onx
the fact that the Raviart-Thomas finite element functions only have
edges/faces.
BDDC algorithm, we choose the continuous primal interface velocity
•µx á andIntheoursubdomain
• âqs such that, if Y âqs • âqvs ,
space
dual interface velocity spaces
H the
´ ´ computes
â
then Y qvs has a zero edge/face average for each edge/face. In fact, ¾ ê ¾ ê ~;
â
w
average of the dual interface velocities Ö
, then distributes them back to each subdomain
á
and leaves Ö
the same. We recall that the weights at these nodes are the same for each
edge/face since these nodes are shared by the same pair of subdomains. The averaged dual
interface velocity still has a zero edge/face average for each edge/face. For the third equality,
´ ´ ê b á Ö á , since Ö • à ´ É – b .
we use that ê b Ö
š
* š
*2V
H w
Therefore, we can conclude that the preconditioned BDDC operator, defined in (4.6), is
•
´
positive definite in the benign subspace à
w É – b.
We call
and
the benign subspaces of
and
,
respectively. With Lemma 3.1, it is easy to check that both operators
and , given in
(3.12) and (4.1), are symmetric, positive definite when restricted to the benign subspaces
and
, respectively and we also have
,
.
L EMMA 4.1. For any
Proof. We need to show that for any
,
. Given
, we have
and
5. Some auxiliary results. We first list some results for Raviart-Thomas finite element
function spaces needed in our analysis. These results were originally given in [24, 21, 23].
We consider the interpolation operator
from
onto
. Recall that
is the
Raviart-Thomas finite element space on the coarse mesh with mesh size , which is defined
in terms of the degrees of freedom
, by
;
•”
•” •” O
! ; Y (ˆÌ* Y . R
G 5
We consider the stability of the interpolant in the next lemma.
; depends only on the aspect ratios of
exists a constant , which
L and of5.1.the There
elements of , such that, for
x ƒ all Y H • ” , x
H C Regf7! ; Y ( C z { q C Regf YC z { q G
xƒ x
xÔ
C ; YDC z { q Ï
v" Ó CôYDC z { q Ï C R"egf YDC z { q 5
EMMA
Proof. See [24, Lemma 4.1].
We define
as the the space of functions that are constant on each element of the
edges/faces of the boundary of
and its subspace
, of functions that have mean
Ÿ ! 43 s (
s
Ÿ’b ! 37 s (
ETNA
Kent State University
etna@mcs.kent.edu
173
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
34
s s
Ÿ §
s• ” §
. Let
be the space of functions defined on , such that, for each
value zero on
subdomain
and each edge/face of , is constant on . We note that
is the space
of normal components on of vectors in
.
The stable extension operator, defined in the next lemma, provides a divergence-free
extension of boundary data given on
.
L EMMA 5.2. There exists an extension operator
, such that, for
any
,
H ’Ÿ b ! 34 s (
and
(5.1)
Ÿ x
34 s
ŸÍb 43 ç • qvs
s ˆ ! s (D
G for G
R"e‚f s *-V
íoH s
P ¹x ƒ
SP ¹ x
C +C z { „C+C { 5
s q
/€‚ q
Here is independent of , , and .
Proof. See [24, Lemma 4.3].
Given a subdomain , we define partition of unity functions associated with its edges/faces.
Let
be the characteristic function of , i.e., the function that is identically one on and
zero on
. We clearly have
34 s ¯ s
» STP ¹ !$í)(+* G almost everywhere on 34 s ¯ 3465
Given a function H Ÿ ! 34 s ( and a face 34 s , let
ˆÌ* H Ÿ ! 37 s ( 5
We have the following estimates for the edge/face
of the particular functions in
STP ¹ components
Ÿ ! 34 s ( with a vanishing average on the subdomain
edges/faces.
Ÿ ! 34 s ( with Š R ‰*ÊV , and for any 34 s , Š !R Ò*
L
5.3. Let H
. There then exists a constant , independent of and , such that, for any
"
STP ¹ x
Š H ŸR @ *], V
ƒ C "6 C # /€‚ { q STP ¹ x
STP ¹ x
(5.2)
$ ! Ï
v" ( CØ% C /€d { q -C+C /€d { q 5
EMMA
Proof. See [24, Lemma 4.4].
The following lemma compares norms of traces on the subdomain boundaries that share
an edge/face.
L EMMA 5.4. Let
and
be two subdomains with a common edge/face . Let
be
a function in
, that vanishes outside . Then, there is a constant that depends
only on the aspect ratios of
and , such that
s ­
LaM“N ! 34 s (
s
C(' C STP*)“x
­ STP ¹ x ƒ
„C('XC /€‚ { q 5
/ €d { q
Proof. See [21, Lemma 5.5.2].
´ É – b.
We next list some results for the benign subspace àæ
w have
´
´
´
Let CEÖJC , +
Ö* ; Õ°ê Ö and CôÖ C ,.+ ´ - * Ö ´; Õ ê Ö . We then
L
5.5. Given any Ö
H àæ w É –ºb , we´ have
CEÖJC , + * CôÖ C ,.+ - 5
EMMA
&
ETNA
Kent State University
etna@mcs.kent.edu
174
X. TU
Proof.
´; 2b; 1 Õ ê ´ ´ š ê b; ´ ˜ Ö ´ Ö ´; Õ ê ´ Ö ´ CôÖ ´ C ,.+ - 5
CEÖJC , + * Ö ; °Õ ê Ö 0
Ö
* / † ¤ šê b V ¥ † b › *
*
define the average operator by 3 ~
¾ ê ¾ ê ~; . We see that, for any vector Ö *
*
•
´Ö We
´
! G † b (H à É – ´ b ,
´Ö ´
´ ú ¾ ê ~; ´ ú Ö ´
~
Ö
3
¾
ê
~
w
3 ˜ b * ˜
˜ w
˜ b * ˜ b G
(5.3)
†
›
›
›
† ›
† ›
´
´
´
~
~
where 3
w * ¾ ê ¾ ê ; w , which computes the average of the interface velocities across the
subdomain interface. Lemma 4.1 shows that after averaging a benign vector across a subdomain interface the result is still benign.
•µ´ É –ºb
An estimate of the norm of the 3 ~ operator restricted to the benign subspace à
w
is given in the next lemma.
L
5.6. There exists a positive constant , which is independent of and , and
the number of subdomains, such that,
ƒ C 3 ~ ÖJC , + +}
CEÖJC , + G  Ö *]! Ö ´ G † b (H •Üà ´ w É –ºb"5
´ G b • à ´ É – b , we know, from Lemma 4.1, that ¾ ê ~; Ö
Ö
Proof. Given any Ö
•Ü
”H ´ w É –ºb . Therefore,*Ë3 ! ~ Ö † * ( ¾ ê H ~ ¾ ê ~; Ö w H •µ
à ´ w É –ºb . We have, by Lemma 5.5, that
Ô
ƒ C 3 ~ ÖJC , +
ƒ Ù Ó CEÖJC , + -CôÖ 3 ~ ÖJC , + Ô
Ù Ó CEÖJC , + -CôÖ ´ 3 ~ w ´ Ö ´ C , + - Ô
´ ´
´ ´
* Ù Ó CEÖJC , + -C ¾ ! Ö x 3 ~ w Ö ( C ,4- 8¹ 7
ã CEÖJC , + » C ¾ ´qvs Ö ´ 3 ~ ´ Ö ´ C ,6- 5 å 5
(5.4)
Ù
x *
s ¬· M ! w (
´x
Let Ö
s * ¾ qs Ö and´ set ´ ´
» 둭 ì Ö
´qvs Ö 3 ~ Ö
(5.5)
¾
i s !=í)( ˆÌ* ! w (n!=í)(* ­ôóõ ö ! s !$í)(° Ö ­ !=í)(( G íoH 34 s ® §05
Here ÷ ø is the set of indices of the subdomains that have on their boundaries. Since a fine
­ 34 that is also
í
edge/face only belongs to exactly two subdomains, for an edge/face s
s
shared by ­ , we have
­
(5.6)
i s * ë ­ ì Ö s ë ­ ì Ö ­ G on s 5
We note that the simple inequality ƒ
(5.7)
` s 둭 ì { : 9;³! ` s G ` ­ ( G
EMMA
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
175
î Hï ‹ . ð |Ù Gtñ ( .
vanishing mean value on each face of , we can define, by Lemma
s
i s has. a. Then
s
s
i=s< * !=i (
(5.8)
Regf/PFi ¹$s< x *-V G for íoH s G
ƒ
SPy¹$x
and,
C i s< C z|{ q „C i s . C /€‚ { q 5
(5.9)
¹7
Py¹=x ƒ
STP ¹ x
We then obtain
C i s C , -5 *ƒ ` s C i s< ¹ )C zF{ q ¹ º) ` s C i s . C X/€‚ ST{ Pq ¹ x
º` s ¹ ) » SPy¹ >C !=i s . ( C /€d { q 5
(5.10)
­
¹ ) . the average
T
S
F
P
d
¹
x
over s ,
Using (5.6), we have, with Ö
s
! (
STPF¹=x
` s >C ¹ ) !$i s . ( C /€‚ { q
*ƒ ` s >C { 둭 ì ! Ö ¹ )s Ö ­ . (° . C /€‚ . { q ¹ ) Ô STPy¹$x
Ù ` s 둭¹ì ) Ó >C ! Ö s ! Ö ¹ )s ( ( C ST P ¹ /x €d { q
ƒ >C ! Ö ¹ ) ­ . ! Ö ­ . ( ( ¹ C ) /€d { q STPy¹=x
Ù ` s C> ¹ ) ! Ö s . ! Ö s . ( ¹ ) ( C /€d { q STP*)“x
Ù ` ­ ?C ! Ö ­ . ! Ö ­ . ( ( C /€‚ { q 5
Here we use Lemma 5.4 and (5.7) for the last inequality.
We only need to estimate the first term since the second term can be estimated similarly.
. has vanishing mean value on 34 s . By Lemma 5.2,
Since Ö is in the benign space, Ö
s
we can construct
Ö s< * s ! Ö s . ( G
such that,
R"egf Ö s< *_V G for íèH s 5
•
Let Ö b
< H ” be defined by
in s 5
Ö b< * Ö s< otherwise
¹)
V
¹ ) Ö b and Y . . By the definition of , we know that Let Y
* ;
*
* ;
! Ö s . ( ¹ ) , and for any 34 ¹ ) s , ?Š ! Ö s ST PF. ¹dx (“R!
º*-V . Using Lemma 5.3, we have
(5.11)
>C ¹ ) ! Ö s . ! Ö s . ( ( STC P ¹ x/€d { q
STP ¹ x
STP ¹ x
*ƒ >C ! Ö s . ( C X/€‚ { q .
.
ƒ Ï
v" ! ( CEÖ s C X/€‚ { q STP ¹ x ^CEÖ s STP C ¹x /€d { q Ï
v" ! ( CEÖ s . C /€‚ { q ^C C /€‚ { q G
holds for
Since
5.2,
ETNA
Kent State University
etna@mcs.kent.edu
176
X. TU
where we use the triangle inequality for the last inequality.
By Lemma 5.1, we know that
P ¹x ƒ
P ¹x
P ¹x
C R"egf Y C z { q C R"egf Ö b< C z { q * C Regf Ö s< C z { q *_V 5
and
P ¹x ƒ P ¹x
P ¹xÔ
CôY C z { q Ï
v" : Ó CEÖ b< C z { PFq ¹=x } C Regf Ö b< C z { q
* Ï
v" : CEÖ s< C z { q 5
(5.13)
¹ ) (5.12), and (5.13),¹ ) we obtain:STP ¹ x
Using (5.11), (2.4),
ƒ >C ! Ö s . ! Ö s . ( ( C /€d { q
STP ¹ x
STP ¹ x
.
ƒ Ï
v" ! ( CEÖ s C X/€‚Py ¹${ x q ^C C PF/¹d€ x { q ƒ Ï
v" ! ( CEÖ s< C qrtP svuT¹ xw ^CôY C XqrtsuTw P ¹ x
ƒ Ï
v" ! P ¹ x ( CEÖ s< C z { q ! Ï
v" ( CEÖ s< C z { q ƒ Ï
v" CôÖ s< C z { ¹ 7q
Ï
v" CôÖ C ,6-5 5
`s
s
Here we use that
R"egf Ö s< *_V for the third inequality.
Finally, we obtain
¹)
8¹ 7
STPF¹=x ƒ ` s C? !$i s . ( C /€‚ { q
Ï
CEÖ s C , - 5 5
(5.14)
´ C , + - ; then by Equations
Since Ö is benign, we have, from Lemma 5.5, that CEÖJC , +
E
C
Ö
*
(5.4), (5.5), (5.10), and (5.14), we have
ƒ C 3 ~ ÖJC , + Ï
v" CEÖ ´ C ,.+ - * Ï
v" CôÖJC , + 5
(5.12)
6. Condition number estimate for the BDDC preconditioner. We are now ready to
formulate and prove our main result; this follows directly from the proof of [12, Theorem 1]
by using Lemma 4.1 and Lemma 5.6.
T HEOREM 6.1. The preconditioned operator
is symmetric, positive definite with
respect to the bilinear form
on the benign space
and
ýWL4MQ•µÕ Ý ´
” w
d@ Y
(6.1)
Here, is a constant which is independent of and .
,B
@
G
C
A
ƒED
ƒ @dY G Y A , B ý L4M ÕÎÝ Y G YGF , B Ï
É –ºb
• ” ´ É –ºb"5
G Y A ,B G  Y H µ
w
ETNA
Kent State University
etna@mcs.kent.edu
177
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
TABLE 8.1
Condition number bounds and iteration counts, for a pair of the BDDC and the FBA algorithms, with a change
and
.
of the number of subdomains.
HJIKMLON P(QSR
Num. of sub.
T Vø É VT(U Iter. BDDC
Cond. Num.
5
1.66
ÉW W
8
2.95
É
9
3.08
Ù É Ù
9
3.13
CX É CX
ÙV É ÙV
8
3.15
Iter.
5
8
7
7
7
FBA
Cond. Num.
2.43
2.90
2.75
2.72
2.71
TABLE 8.2
Condition number bounds and iteration counts, for a pair of the BDDC and the FBA algorithms, with a change
subdomains and
.
of the size of subdomain problems.
NYZN
4
8
12
16
20
Iter.
8
8
9
9
9
P(QSR
BDDC
Cond. Num.
2.17
2.95
3.47
3.88
4.20
Iter.
7
8
9
9
9
FBA
Cond. Num.
2.12
2.90
3.45
3.83
4.15
7. Comparison with an edge/face-based iterative substructuring domain decomposition method. We define an edge/face-based iterative substructuring domain decomposition
method as a hybrid method (see [22, Section 2.5.2]). Similar to the BNN method, as defined
in [18, Section 4], the coarse problems and the local problems are treated multiplicatively
and additively, respectively, in this preconditioner. We use a different coarse component, i.e.,
a different choice of the matrix
for the coarse problem, but the same local problems as
in [18, Section 4]. Here, each column of
corresponds to an edge/face on the interface
of and is given by the positive scaling factor
. It is clear and we can prove that the
condition number with this preconditioner is also bounded by
. We will call
this method the FBA.
The size and sparsity of the coarse problems of the BDDC and the FBA are the same.
However, the two algorithms are different. The FBA is a hybrid algorithm and a coarse
problem has to be solved before the rest of the iterations. In contrast, only the variables have
to be changed at the beginning of computation with the BDDC, to accommodate the edge/face
constraints. In addition, the FBA requires two Dirichlet local problems and one singular local
Neumann problem in each iteration, whereas the BDDC requires one local Dirichlet problem
and two nonsingular local Neumann problem. In the latter algorithm, singular problems are
avoided. Numerical experiments show that FBA is somewhat slower than BDDC.
8+b
8b
ë"sì !$í)(
* ï V G [ *Ë
‡ ðôŸ
]
8. Numerical experiments. We have applied our BDDC and FBA algorithms to the
model problem (2.1), where
. We decompose the unit square into
subdomains with the sidelength
. Equation (2.1) is discretized, in each subdomain, by
the lowest order Raviart-Thomas finite elements and the space of piecewise constants with
a finite element diameter , for the velocity and pressure, respectively. The preconditioned
conjugate gradient iteration is stopped when the -norm of the residual has been reduced by
a factor of
.
We have carried out two different sets of experiments to obtain iteration counts and con-
V L^]
Ÿ É Ÿ
\
ETNA
Kent State University
etna@mcs.kent.edu
178
X. TU
TABLE 8.3
Condition number bounds and iteration counts, for a pair of the BDDC and the FBA algorithms, with a change
and is in a checkerboard pattern.
of the number of subdomains.
HJIKMLN P
Num. of sub.
T Vø É VT(U Iter.
3
W ÉÉ W
3
3
Ù É Ù
3
?X É ?X
ÙV É ÙV
3
BDDC
Cond. Num.
1.03
1.06
1.07
1.08
1.08
Iter.
5
7
7
7
7
FBA
Cond. Num.
2.20
2.44
2.49
2.51
2.53
TABLE 8.4
Condition number bounds and iteration counts, for a pair of the BDDC and the FBA algorithms, with a change
subdomains and is in a checkerboard pattern.
of the size of subdomain problems.
4
8
12
16
20
NGY_N
P
Iter.
3
3
4
4
4
BDDC
Cond. Num.
1.04
1.06
1.10
1.11
1.12
Iter.
7
7
8
8
8
FBA
Cond. Num.
2.00
2.44
2.69
2.88
3.02
`a`]
dition number estimates. All the experimental results are fully consistent with our theory.
In the first set of experiments, we take the coefficient
. Table 8.1 gives the iteration
counts and the estimate of the condition numbers, with a change of the number of subdomains.
We find that the condition number is independent of the number of subdomains for both
algorithms. Table 8.2 gives the results with a change of the size of the subdomain problems.
In the second set of experiments, we take the coefficient
in half the subdomains
and
in the neighboring subdomains, in a checkerboard pattern. Table 8.3 gives the
iteration counts, and condition number estimates with a change of the number of subdomains.
We find that the condition numbers are independent of the number of subdomains for both
algorithms. Table 8.4 gives the results with a change of the size of the subdomain problems.
Acknowledgments The author is grateful to Professor Olof Widlund for suggesting this
problem and all the help and time he has devoted to this work. The author also thanks Professor Jing Li for personal communications and useful notes. Thanks also go to Professor
Marcus Sarkis for his suggestion of the comparison of our BDDC algorithm with the FBA.
`* ` * V V
REFERENCES
[1] J. H. B RAMBLE AND J. E. PASCIAK , A domain decomposition technique for Stokes problems, Appl. Numer.
Math., 6 (1990), pp. 251–261.
[2] F. B REZZI AND M. F ORTIN , Mixed and hybrid finite element, vol. 15 of Springer Series in Computational
Mathematics, Springer Verlag, Berlin-Heidelberg-New York, 1991.
[3] L. C. C OWSAR , J. M ANDEL , AND M. F. W HEELER , Balancing domain decomposition for mixed finite
elelements, Math. Comp., 64 (1995), pp. 989–1015.
[4] L. C. C OWSAR AND M. F. W HEELER , Parallel domain decomposition method for mixed finite elements
for elliptic partial differential equations, in Fourth International Symposium on Domain Decomposition
Methods for Partial Differential Equations, R. Glowinski, Y. A. Kuznetsov, G. A. Meurant, J. Périaux,
and O. Widlund, eds., Philadelphia, PA, 1991, SIAM.
[5] C. R. D OHRMANN , A preconditioner for substructuring based on constrained energy minimization, SIAM J.
Sci Comput., 25 (2003), pp. 246–258.
ETNA
Kent State University
etna@mcs.kent.edu
BDDC ALGORITHM FOR FLOW IN POROUS MEDIA
179
[6] R. E. E WING AND J. WANG , Analysis of the Schwarz algorithm for mixed finite element methods, RAIRO
Modél. Math. Anal. Numér., 26 (1992), pp. 739–756.
[7] V. G IRAULT AND P. R AVIART , Finite Element Method for Navier-Stokes Equations, Springer Verlag, New
York, 1986.
[8] R. G LOWINSKI , W. A. K INTON , AND M. F. W HEELER , Acceleration of domain decomposition algorithms
for mixed finite elements by multi-level methods, in Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, held in Houston, Texas, March 20-22, 1989, T. Chan,
R. Glowinski, J. Périaux, and O. Widlund, eds., SIAM, Philadelphia, PA, 1990.
[9] R. G LOWINSKI AND M. F. W HEELER , Domain decomposition and mixed finite element methods for elliptic
problems, in First International Symposium on Domain Decomposition Methods for Partial Differential
Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., Philadelphia, PA, 1988,
SIAM.
[10] A. K LAWONN AND O. B. W IDLUND , Dual-Primal FETI methods for linear elasticity, Tech. Report TR2004855, Department of Computer Science, Courant Institute, September 2004.
[11] J. L I AND O. B. W IDLUND , FETI–DP, BDDC, and Block Cholesky Methods, Tech. Report TR2004-857,
Department of Computer Science, Courant Institute, December 2004.
, BDDC algorithms for incompressible Stokes equations, Tech. Report TR-861, Department of Com[12]
puter Science, New York University, 2005.
[13] J. M ANDEL AND C. R. D OHRMANN , Convergence of a balancing domain decomposition by constraints and
energy minimization, Numer. Linear Algebra Appl., 10 (2003), pp. 639–659.
[14] J. M ANDEL , C. R. D OHRMANN , AND R. T EZAUR , An algebraic theory for primal and dual substructuring
methods by constraints, Appl. Numer. Math., (2004). To appear.
[15] T. P. M ATHEW , Domain Decomposition and Iterative Refinement Methods for Mixed Finite Element Discretizations of Elliptic Problems, PhD thesis, Courant Institute of Mathematical Sciences, September
1989. TR-463, Department of Computer Science, Courant Institute.
[16]
, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,
part I: Algorithms and Numerical results, Numer. Math., 65 (1993), pp. 445–468.
, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems,
[17]
part II: Theory, Numer. Math., 65 (1993), pp. 469–492.
[18] L. F. PAVARINO AND O. B. W IDLUND , Balancing Neumann-Neumann methods for incompressible Stokes
equations, Comm. Pure Appl. Math., 55 (2002), pp. 302–335.
[19] T. R USTEN AND R. W INTHER , A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix
Anal. Appl., 13 (1992), pp. 887–904. Iterative methods in numerical linear algebra (Copper Mountain,
CO, 1990).
[20] M. V. S ARKIS , Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous
coefficients using non-conforming elements, Numer. Math., 77 (1997), pp. 383–406.
[21] A. T OSELLI , Domain decomposition methods for vector field problems, PhD thesis, Courant Institute of
Mathematical Sciences, May 1999. TR-785, Department of Computer Science.
[22] A. T OSELLI AND O. B. W IDLUND , Domain Decomposition Methods - Algorithms and Theory, vol. 34 of
Springer Series in Computational Mathematics, Springer Verlag, Berlin-Heidelberg-New York, 2004.
[23] B. I. W OHLMUTH , Discretization Methods and Iterative Solvers Based on Domain Decomposition, vol. 17
of Lecture Notes in Computational Science and Engineering, Springer Verlag, 2001.
[24] B. I. W OHLMUTH , A. T OSELLI , AND O. B. W IDLUND , Iterative substructuring method for Raviart-Thomas
vector fields in three dimensions, SIAM J. Numer. Anal., 37 (2000), pp. 1657–1676.