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Electronic Transactions on Numerical Analysis.
Volume 16, pp. 165-185, 2003.
Copyright  2003, Kent State University.
ISSN 1068-9613.
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ANALYSIS OF TWO-DIMENSIONAL FETI-DP PRECONDITIONERS
BY THE STANDARD ADDITIVE SCHWARZ FRAMEWORK
SUSANNE C. BRENNER
Abstract. FETI-DP preconditioners for two-dimensional elliptic boundary value problems with heterogeneous
coefficients are analyzed by the standard additive Schwarz framework. It is shown that the condition number of the
preconditioned system for both second order and fourth order problems is bounded by
, where
is the maximum of the diameters of the subdomains, is the mesh size of a quasiuniform triangulation, and the
positive constant is independent of , , the number of subdomains and the coefficients of the boundary value
problems on the subdomains. The sharpness of the bound for second order problems is also established.
Key words. FETI-DP, additive Schwarz, domain decomposition, heterogeneous coefficients.
AMS subject classifications. 65N55, 65N30.
1. Introduction. The Finite Element Tearing and Interconnecting (FETI)
method [14, 20, 21, 19, 32, 18, 15, 34, 35, 1] is a nonoverlapping domain decomposition method that has been implemented for large scale engineering applications. In the FETI
approach the system of finite element equations for the nodal variables (primal variables) is
enlarged to a system where the nodal variables on any subdomain are independent of the
nodal variables on the other subdomains, and the continuity of the finite element functions
across the interface of the subdomains is enforced by Lagrange multipliers (dual variables).
By eliminating the primal variables, a system of equations for the dual variables is obtained,
which can then be solved by the conjugate gradient method. In the case of many subdomains,
preconditioning is necessary for fast convergence and FETI preconditioners have been
studied in [29, 31, 23, 24, 5, 4].
Recently, a new FETI approach for two-dimensional problems was introduced in [16, 17,
33], where the continuity of the finite element functions at the cross points is retained in the
enlarged system and Lagrange multipliers are introduced only to enforce the continuity at the
other nodes on the interface of the subdomains. After the primal variables associated with
these other nodes and the nodes off the interface have been eliminated, a system involving
both primal variables (nodal variables associated with the cross points) and dual variables
(Lagrange multipliers associated with the nodes on the interface that are not cross points)
is obtained. Two notable features of the FETI-DP approach are: (i) the evaluation of the
operator associated with the dual-primal system no longer involves the solutions of singular
problems on floating subdomains, and (ii) a coarse problem is built into the evaluation of the
reduced operator associated with the dual variables.
Dirichlet preconditioners for the FETI-DP approach were studied in [30] where polylogarithmic bounds for the condition numbers of the preconditioned systems were obtained
for second and fourth order problems on two-dimensional domains. The goal of this paper is to give an alternative derivation of the results in [30] and demonstrate that the
bound for second order problems is sharp, using the standard additive Schwarz framework
[11, 2, 39, 38, 12, 22, 37, 6]. This paper is therefore a continuation of the earlier work
[5, 4, 8]. Note that this approach can also be applied to three-dimensional FETI-DP methods
[17, 33, 25] and such an analysis of 3D FETI-DP methods will be carried out in a separate
paper.
Received March 17, 2003. Accepted for publication May 3, 2003. Recommended by O. Widlund.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. E-mail bren-
ner@math.sc.edu. This research was supported in part by the National Science Foundation under Grant No.
DMS-00-74246.
165
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Analysis of Two-Dimensional FETI-DP Preconditioners
The rest of the paper is organized as follows. The description of the FETI-DP algorithm
for two-dimensional second and fourth order elliptic boundary value problems is given in
Section 2, followed by the definition of FETI-DP preconditioners and some preliminary estimates in Section 3. Condition number estimates for the FETI-DP preconditioners are then
given in Section 4. The sharpness of the condition number estimate for second order problems
is established in Section 5.
be a bounded polygonal domain
2. Two-Dimensional FETI-DP Methods. Let
subdivided into nonoverlapping (open) polygonal subdomains
(cf. Figure 2.1),
with diameters
. The maximum of
will be denoted by .
Ω1
Ω2
Ω3
Ω6
Ω4
Ω7
Ω5
Ω8
F IG . 2.1. A nonoverlapping subdivision with an underlying triangulation
')(+*-,
,
. ! &/0
For concreteness we will consider two model problems. We take
!"#%$&
(2.1)
1234!657 89"
for the second order model problem, and
C C !
C 0?> C 0DA C 0E> C 0FA /0 124!G57 89
"
(2.2)
for the fourth order model problem, where the $ are positive constants.
Let H be a quasiuniform triangulation of with mesh size I such that each is a union
of the triangles in H (cf. Figure 2.1), and J8"KMN L 8" be the OP finite element space
associated with H when QR#TS and the Hsieh-Clough-Tocher macro finite element space (cf.
[9]) when QR#%U . The discrete problem that we want to solve is:
Find VW5XJ8" such that
' (6[
(2.3)
?VY4Z"# !/0 12\5]J8"
[
where 5]^ 8" and
:
?VY4Z"# _ `V "=
(2.4)
(D*
(+*
with VD2#aVcb and 2#aYb .
b 2.1. The results
b of this paper also hold for other elements for second and
R
' ( *;:
9
"#%$&
=<?>@ AB< EMARK
fourth order problems, such as the bilinear finite element and the reduced Hsieh-CloughTocher macro element [10].
The description of the FETI-DP approach requires some terminology from domain decomposition: The set of all the vertices of the polygonal subdomains
will be
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Susanne C. Brenner
C
denoted by . For example, there are 16 such vertices in for the domain decomposition
depicted in Figure 2.1. The subset of consisting of vertices not on
will be denoted by (the set of cross points). For example, there are 5 cross points on the boundary of the subdomain in Figure 2.1. An open line segment on the boundary of a subdomain between two
consecutive vertices in will be referred to as an edge of the boundary of the subdomain. For
example, there are 5 edges on the boundary of the subdomain
in Figure 2.1. The interface
of the subdomains is the set is the part of
disjoint
, where ?# C 9 C
C !
C . The set will be denoted by and is the set X# _ .
from
We will also use to denote the set of the nodes of the finite element space belonging
# _ to the geometric object and to denote the number of elements in a set .
Let
be the subdomain finite element space associated with the triangulation
on
induced by whose members vanish up to the derivatives of order on
.
Let be the subspace of the product space
defined by
J\ 9
"
Q S C C
J8 " ... XJ8 "
(2.5)
J # "P65XJ89?" for S!#"%$ and > #aC ' & up toC the
> &*) derivatives of order Q( S at all the cross points on
Note that the definition of ? . . " in (2.4) can be extended to the space J . Furthermore,
?E4Z"2#,+ implies that is a polynomial of degree Q-aS on ! for S %"./$ . The
continuity at the cross points then implies that is a global polynomial of degree Q C aS
on and hence \#0+ , because of the homogeneous Dirichlet boundary condition(s) on
.
Therefore, the bilinear form ? . . " remains symmetric positive definite (SPD) on J .
9
J
H
R EMARK 2.2. Sometimes a cross point is defined to be a point in belonging to the
boundaries of at least three subdomains. Let 1 be the set of cross points according to this
definition. In the case where all the subdomains are convex we have 1 2 . But in general 1
is a proper subset of . Note that in the case where a subdomain is surrounded by another,
the absence of cross points according to the stricter definition creates a singular problem on
the inner subdomain in the FETI-DP approach. This would not happen if the larger set is
used.
We can identify the global finite element space
with the subspace of
whose
members (or more precisely their components) are continuous up to the derivatives of order
( along the interface . Moreover, the continuity of the derivatives of the finite element
functions on up to order 3
can be enforced by Lagrange multipliers.
For each 4 5687 we introduce the Lagrange multiplier space 9#: as follows. Let 4
;. & . In the second order case 4 is a vertex and 9 : is spanned by the linear functional
< : & = defined by
M#
Q
>
S
@ >@ 5MJ
J8"
Q S
5
#% & C4?"D >ZC4?"
5
1 # 3...& +" 5]J
where > . .E? A represents the canonical bilinear form defined on F
J D J . In the fourth order
case 4 is either a vertex or a midpoint. If 4 is a midpoint, then 9G: is spanned by <I: H @ > @ & 5WJ defined by
C
C
(2.7)
> < :H @ >@ & 4? A # CKJ & & C4?"IL CMJ > > N4E"
12 # ...&4 " 5XJ9
J
J
where & (respectively > ) is the outer unit normal of & (respectively > ). If 4 is a vertex,
<
then 9O: is spanned by : @ >@ & , < : H @ >@ & , <Q:P @ >@ & 5M=
J . The linear functionals < : @ >@ & and < :H @ >@ & are
Q
<
P
defined as in (2.6) and (2.7), and : @ >@ & is defined by
C
C
(2.8)
> < :P @ >@ & 4 ? A # KC ' R & & C 4?I" L MC R > > C 4E"
12 # ... 4 "c5MJ
(2.6)
> < : @ >@ & @?BA
J
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Analysis of Two-Dimensional FETI-DP Preconditioners
R
R
>
C > C
where &J (respectively J ) is the unit tangent along
& obtained by rotating the unit
normal & (respectively ) counterclockwise through a right angle.
p
Ωl
>
Ωl
p
µ p,k,l
p
µ p,k,l
µ p,k,l
t
n
p
p
Ωk
Ωl
p
Ωk
Ωk
F IG . 2.2. Lagrange multipliers along the interface of two subdomains
The three types of Lagrange multipliers are depicted graphically in Figure 2.2. The
Lagrange multiplier space 9 is taken to be
9
: 7
aJ 9O:
J
J "
5 J
we can characterize the subspace of
corresponding to
as + for all < 9,) .
The first step in the FETI-DP approach is to replace (2.3) by the following problem:
9 such that
Find 1 > < @?BA
Using
9
#
#
5
V ?" 5XJ
:
: ')( * [
(2.9)
_ V 1 " L%>
4? A # _ /0 125XJ9
> < QV 1 ? A
# +
1< 5 9 where V 1 # V 1 V 1 " and # 4 " . It is easy to check that (2.9) is non(
(
singular and the solution of (2.3) is related to the solution of (2.9) through the relation
V71 # Vcbb V!bb " .
We can also rewrite (2.9) more concisely as
: ')( * [
< "c5]/
V 1 ?"=`F < " #
/
0
1
E
J 9 (2.10)
_ where
:
E"=`F < " # _ ! "IL%>
& @? A L > < ;? A 1 `6E"=E < " 5]J/ 9 Let J # # C "X5 J the nodal variables of ,C S# "% $ , vanish
at all the nodes in ! ) (i.e., vanishes at the cross points on c for theC O finite
element and vanishes up to the first order derivatives at the cross points on
for the
Hsieh-Clough-Tocher macro element). The space J 9 admits the decomposition
J 9 # ETNA
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Susanne C. Brenner
where 5XJ
# J # E < " 5 J 9 `F < "=`6 + "#
. . " is nonsingular, we have
dim J/ 9 "# dim L dim and . Indeed, since the bilinear form )
Moreover, if
+ )
E + " 5
+
for all
, then
+6#
4`FB+"=E + "4"9#?E4Z"=
which implies # + , because ?4. . " is SPD on J .
We can therefore write
VY1 ?"# V 1 + " L% V 1 ?"
(2.11)
V 5]J
V ?" 5
where 1
and 1 .
The second step in the FETI-DP approach is to reduce (2.10) to the following dual-primal
problem:
Find
V 1 ?" 5
such that
: ' (+*-[
V 1 3E"` < " # _ /0 1 ` < "c5 R
2.3. Since V 1 in 2.11" is determined by
:
: ' ( *-[
V
4
"
#
_ 1 _ /0 12 5]J and the components of V
are independent of one another, the reduction in the second step
of the FETI-DP approach involves solving SPD problems on the subdomains in parallel.
Let J aJ be the orthogonal complement of J with respect to ?4. . " . Note that
(2.13)
J #J J J .'+) and has the decomposition
# J O + ) J O + ) where J 2 +) # E"X5TJ G9 `6 E"=E + " # + for all a5 J ) . Note
also that E" 5 J + ) is completely determined by and therefore can be written as
- E" , where is the linear map from 9 to J defined by
(2.14)
- E"=E +" # +
12\5]J9
(2.12)
EMARK
Hence, we can write
(2.15)
V 1 3E"!# V 1 + " L% ?"=
V 5]J
.
where 1 In the final step of the FETI-DP approach the dual-primal problem (2.12) is further reduced to the following problem:
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Analysis of Two-Dimensional FETI-DP Preconditioners
W5
9
Find such that
: ')( *-[
?"= < < " # _ < " &/0
(2.16)
1< 5 9 The role of a FETI-DP preconditioner is to improve the conditioning of the system (2.16) so
that it can be solved efficiently by a preconditioned conjugate gradient method.
R EMARK 2.4. Since 1 in 2.15 is determined by
V
"
:
: ' ( *&[
_ E V 1 "# _ /0
1 5MJ F
the reduction in the final step of the FETI-DP approach involves solving a SPD coarse problem whose dimension is
dim
J #
dim
J
J #
dim O
for the
finite element,
for the Hsieh-Clough-Tocher macro element.
5
The description above of the FETI-DP approach follows the actual solution process given
in [16, 17]. It shows that the evaluation of the operator defined by (2.16) on a given <
9
involves solving SPD problems on the subdomains in parallel and also solving a SPD coarse
problem that provides global communication among the subdomains. But the analysis of
FETI-DP preconditioners (cf. [30]) is best carried out through an alternative formulation of
(2.16) that is based on a decomposition of different from (2.13).
Let , #"%$ , vanishes up to the derivatives of order
on ) . The space admits the decomposition
J
J # C # ` 4 "c5]J S
Q S " 9
J
J # J J6
where J 6 is the orthogonal complement of J with respect to ?4. . " , i.e.,
J 6 # 5]J E`F 2"9# + 12T5XJ )
(2.17)
Note that J 6##9 # J +) # `6 E" 5 J G9 E"=`FB+" #-+ for all
5MJ ) .
R
2.5. For 5 ( J 6 , the
nodal variables of can take arbitrary values along
(
6 7 and share common but arbitrary" values at the cross points. The rest of the nodal
" are then determined by the ? . . " orthogonality of J 6
variables at the nodes in to J . In particular, is a discrete harmonic function on in the second order case and a
discrete biharmonic function in the fourth order case.
R
2.6. Since > < 4
? A # + for all a5TJ and < 5 9 , we will treat 9 as a
subspace of J 6 .
EMARK
EMARK
We can write
V 1 # V 1 L V 1 where
V 1 5XJ
and
V 1 5XJ6 , and reduce (2.9) to the following problem:
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Susanne C. Brenner
Find
(2.18)
V 1 ?" ]5 J 6 9 such that
: V
1 "IL > ? 6
_ > < V1 ? 6
: ' (+* [ # _ /0
#%+
1 ]5 J 6 1< 5 9 . .E? 6 is the canonical bilinear form defined on J 6 ]J 6 .
J 6 J=6 be defined by
: (2.19)
> P ? 6#%?` 4 "#
_ "
for all # 4 "=4
# 4 "c5MJ 6 , and 5]J 6 be defined by
: ' (+*&[ (2.20)
>Z4 ? 6\#
/
0
2
1
#
`
4
"c5MJ 6 _ where >
Let We can now rewrite (2.18) as
> 1
?
(2.21)
< V 6
> V1 ? 6
4 ? 6 # >Z4 ? 6
# +
%
L >
Note that the operator is SPD, i.e.,
(2.22)
>
? 6 > # ? 6
> P 4 ? 6 +
12< M5 J 6 1 55 9 12 4 5]J 6&
12 5XJ6 +) From (2.21) we obtain the following problem for :
Find 9 such that
M5
> < B (2.23)
? 6#> < B
? 6
V
1< 5 9 The two problems (2.16) and (2.23) are identical since they both come from (2.9) by
eliminating 1 . Indeed, we have, by (2.14), (2.17), (2.19), (2.20) and (2.22),
< # <
< < " #=> < B ?6
E"
: ' (D* [ <
"
/
0
=> < B ? 6
#
_ 1< 5 9 1< 5 9 1< 5 9 Our analysis of FETI-DP preconditioners will be based on the formulation (2.23).
3. FETI-DP Preconditioners and Preliminary Estimates. Let
defined by
< ? # > < B
< ? 6
1 < < 5 9 where > . .E? is the
canonical bilinear form defined on 9 " .9 Y#
(3.1)
from (2.22) that
><
is SPD, i.e.,
><
< ?R#> < < ?
1 < < 559 9
9
9 be
.9 . It follows
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Analysis of Two-Dimensional FETI-DP Preconditioners
><
< ? 1< 5
+
9 '+)
We see from (2.23) that is the operator that needs to be preconditioned in the FETI-DP
approach. The preconditioner for will be constructed using Schur complement operators
associated with the subdomains.
Let
) or the space
be the space of discrete harmonic functions (
of discrete biharmonic functions (
) that vanish up to the derivatives of order at
the cross points on
. In order words,
is characterized by the following conditions:
(3.2)
(3.3)
JZ %J 9 "
Q # S
Q
T
#
U
Q
C !
5]J)
*
9"!#+ 12!5MJ\ 9 " N L 9"=
vanishes up to the derivatives of order Q(S at every 4]5 6 S
Note that
J . .. ]J aJ6 The Schur complement operator ?!
J) J is defined by
> 4 ? # ` 4 "
12 4 5MJ (3.5)
where >4. . ? is the canonical bilinear form on J XJZ . It is clear that F
(3.4)
is SPD.
In order to define the FETI-DP preconditioner we need connection maps
=J 5
9
We will treat the second order case and fourth order case separately.
Let 4 6 7 . Then 4 belongs to the common boundary of two subdomains
) by
We define the function : ) ?
(3.6)
:Z+"9#
?
and
2>
and
&.
+ "#?
Q: PG5]J ,
: * $ (3.7)
2#
>
: @ *? < : @ @ $
L
$
: 7
where the $& ’s are the coefficients appearing in (2.1) and : @ G5XJZ satisfies
S # 4- *
(3.8)
: @ 3"#
+
5 6 7 4I)
For the second order model problem we define, for arbitrary $ ` $& $ "
+ "
$ " " # S " @ @
" "
$
"
"
$ "
@ L @
@ 5 JZ
R EMARK 3.1. The scaling L in 3.7 and 3.9 below enables us
to obtain an estimate for that is independent of the ’s. This technique is wellknown in the literature cf. [36, 28, 13, 35, 34, 24, 25, 4, 5] . In fact, the results in this paper
remain valid if R"! respectively R in 3.7 and 3.9 is replaced by P respectively
P for any
in 3.7 and 3.9 for simplicity.
. We choose
The definition of for the fourth order model problem follows the same principle. First
we introduce the discrete biharmonic functions * : , : H , : P and H . Note that a function
in
is determined by its values and the values of its normal and tangential derivatives at
the vertices of on , and also the values of its normal derivatives at the midpoints of the
edges of on . For a vertex 4 6 7 , we define (i) :
to be the function that
$ $ "
J)
H
H
S U
5
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Susanne C. Brenner
S
@ 5 J to
takes the value at 4 and takes the value zero for all other nodal variables, (ii) : H
be the function whose normal derivative at 4 is and takes the value zero for all other nodal
variables, and (iii) : P
to be the function whose tangential derivative at 4 is and takes
the value zero for all other nodal variables. For a midpoint
D , we define H
to
be the function whose normal derivative at is and takes the value zero for all other nodal
variables.
We can now define, for arbitrary = ,
S
@ 5MJZ
S
Q 5 L @ 5 J)
Q S
G5]J :
# : $ $L $ >
: @ *? < : @ @ L >
:H @ ? < :H @ @ 7
L >
: P @ ? < :P @ @ (3.9)
:
$ >
H @ ? < H @ @ L
*
L 7 $ L $ L L *
where 6 7 @ is the set of the vertices of the triangulation H on and 6 7 @ is the set of the
midpoints of the edges of H on .
Let c=JZ! J ... MJ a
J 6 be the embedding map defined by
\# "
"4>6# 12 5MJ (3.10)
+
# "
and ! 9 J be the restriction map defined by
> < 4;
? # > < ; ? 6 12T5]J (3.11)
Then the maps and form a partition of unity on 9 :
:
< # <
< 5 9 (3.12)
1
_ which can be easily verified using (2.6)–(2.8), (3.7) and (3.9).
The FETI-DP preconditioner 9 Q 9 is given by the formula
:
# _ P where P 9 Q J is the transpose
of with respect to >4. . ? and > . .E? .
9 9 is based on the following well-known
Our analysis of the operator
" from the additive Schwarz theory [11, 2, 39,
characterizations of D " and +
38, 12, 22, 6]:
< < ?
>
(3.14)
"# N
_ ** * * * : _ A 7 _ >
B ? (3.13)
(3.15)
"# N _
$#%
"!
!
>< < ?
* * * * * : >
B ? _ A 7 _ !
!
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Analysis of Two-Dimensional FETI-DP Preconditioners
In the rest of this section we provide
characterizations of
and derive a lower bound for .
9 , we have
L EMMA 3.2. Given any <
5
=> <
(3.16)
"
< ? #
A ><
< ? and > ? ,
? 4 "IL U> < ? 6 Proof. We can, by (2.19), rewrite the right-hand side of (3.16) as
4 ? 6 L U> < 4 ? 6 Therefore the minimum occurs at # < and the value of the minimum is, by (3.1),
> < < ? 6 U > < < ? 6# => < B
< ? 6#=> < < ? A >
M5 J * *
? # A ` 4 "IL U >
4 ? The proof of the following lemma is similar.
, we have
L EMMA 3.3. Given any =>
(3.17)
" can be derived as a corollary of Lemma 3.2 and Lemma 3.3.
A lower bound for L EMMA 3.4. For both the second order model problem and the fourth order model
problem, it holds that
" ! S (3.18)
# < 5MJ . It follows from (3.12) that
:
< #
(3.19)
_ Let 5MJ be arbitrary. Then # 3 D"9# _ 5WJ 6 (cf. (3.4) and (3.10)),
and in view of (3.11),
:
: <
:
>< ?6 # >< ?6 #
> ? #
_ _ _ >
4 ? :
`
4
IL U >
- ? #%?` 4 I" L U > < ? 6 "
_ Proof. Let <
559
be arbitrary and It follows that
(3.20)
: * *
_ A ` 4 "IL U >
4 ? !
A ?` 4 "IL U> < ? 6 ETNA
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We deduce from (3.16), (3.17) and (3.20) that
:
>
B ? _ (3.21)
< ? => <
!
The estimate (3.18) follows from (3.14), (3.19) and (3.21).
The estimates for in the next section requires another characterization of
> < < ? and >
? as dual norms.
L EMMA 3.5. Given any < 59 , we have
< ?6
< >
<
(3.22)
>
?
#A % $
"
5
where
_N :
#? 4 " # _ 4 " (3.23)
#
#- < . From (2.19), (3.1), (3.23) and a standard duality formula we
< ?6
> 4 ? 6
< >
<
<
? # > ? 6 #> P 4 ? 6 # A > ? # A 6
_ N
_ N
Proof. Let
have
><
# %
$#%
Y M5 J * * * >
* ? >
B ? # A _ N
*
#4" The proof of the following lemma is similar.
L EMMA 3.6. Given any = , we have
(3.24)
$#%
where
(3.25)
+ "
4. Condition Number Estimates. We first derive an upper bound for for the
second order model problem.
In order to avoid the proliferation of constants, from here on we use (or )
to represent the inequality constant , where the constant is positive and independent
of , and $ . The statement is equivalent to and .
Let <
, where = for
" ,$ . In order to apply (3.15) we need to
I #
_ Y 5 J S
in terms of _ >
- ? . This will be accomplished by exploiting
the characterizations (3.22), (3.24) and well-known estimates for discrete harmonic functions.
Let # ` 4 " 5KJ 6 be arbitrary and # 4 " , where is the
discrete harmonic function on c with the following properties:
C4?"9#% C4?" 14X5 C 9
and is piecewise linear on C ! ,
bound > <
< ?
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Analysis of Two-Dimensional FETI-DP Preconditioners
C at all the vertices in
i.e., agrees with
two consecutive vertices.
Our first observation is that
(4.1)
.
and is linear on the edge between any
2#a 5MJ)
and
:
:
> > ? 6# > > > ? 6&
>_ @> _ where we have used the fact that the components of are continuous across the interface of
the subdomains and therefore > < 4 ? 6# + .
R
4.1. The relation > < 4 ? 6 # > < 4 % ? 6 allows us to use well-known
><
(4.2)
? 6#> < ? 6# > < EMARK
estimates for the BPS preconditioner [3] in the study of the FETI-DP preconditioner. In
this sense the FETI-DP algorithm is dual to the iterative substructuring algorithm while the
original FETI algorithm is dual to the balancing domain decomposition algorithm [27].
Our second observation is that
> > ? 6# + unless and > share a common edge.
>@ the discrete harmonic function on >
Let be an edge on > . We will denote
C > by , i.e.,
that agrees with > on and vanishes on
:
(4.4)
>@ #
> C 4?" : @ > >
(4.3)
: :
Clearly, we have
> #
(4.5)
>
>
> @ where
is the
on .
set, ofthethesetedges
Let of edges on . We denote by
subdomains sharing . From (4.2)–(4.5), we have
5 "
><
(4.6)
4 ? 6#
:
6
:
@ > >
the set of the indices of the two
> >@ ? 6-
Note that the inner sum on the right-hand side of (4.6) is given by
>
>) 2> >@ L @ ? 6 > >@ ? 6 , where is
where " K) #
. Below we will focus on the term > @ L
a common edge of & and > (cf. Figure 4.1), since a similar result also holds for the term
> > > > >@ L @ ? 6 .
We have, by (2.6), (3.7), (3.10) and (4.4),
:
> @ *? 6 # $& $ L > $ >
>
: @ ? > < : @ > @ @ ? A
: :
>
$
# $& L $ > >
@ N4E" : @ ? # $&
$ L > $ > >
4 @ ? : >
@ L 2>
>@ ? 6
L%>
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Susanne C. Brenner
Ωj
Ωk
F IG . 4.1. A common edge of and *
and hence it follows from (3.24) that
@ ? 6 >
B ? @ Let be the mean of over ! , i.e.,
')( * S
(4.8)
# 9 /0-
>
(4.7)
We have by scaling
(4.9)
= and it follows easily from the definition of that
(4.10)
= = (*
We can then apply the discrete Sobolev inequality, the Poincaré-Friedrichs inequality,
(4.1), (4.8)–(4.10), and well-known estimates from the theory of nonoverlapping domain
decomposition (cf. [3, 6]) to obtain
*
(*
@ #$ $ $ $ @ @ @ L SL -` IF" = @ " ` "* (4.11)
L ( S * L -86 IF" "D ( * " $ ( * L S L -8 IE" ( *
(*
$ L S L -8 (+* IE" 8 * L0 SL -` IF" $ # S L ` IE" Let 1 @ be the discrete harmonic
on c that agrees with > @ at the nodes in C function
and vanishes at the other nodes of
. Then we have, by (3.7) and (3.10),
:
>
$
> >
>@ *? 6 # $ L $ >
>
: @ ? > < : @ >@ > >@ ?BA
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:
# $ $ L > $Y> >
>@ C4E" : @ ? 2# $ $ L > $ > >
- 1 @ ? :
and hence it follows from (3.24) that
*
(4.12)
> > >@ ? 6 $ $ L > $Y> >
B ? 1 @ *
S
U >
- ? `$ > $ " 1 @ Moreover, the analog of (4.11) for
* >@ implies that
8$Y>$ " 1 @ $ > 1 @ (4.13)
#$ > >@ 8 4S L `\> IE" > We then obtain from (4.6), (4.7) and (4.11)–(4.13) the estimate
* :
:
:
> < 4 ? 6 S L -8 IE" >
B ? * 6 :
:
S L -8 IE" _ >
B ? _ (4.14)
:
# SL -8 IE" _ >
B ? 12 5XJ6&
178
Analysis of Two-Dimensional FETI-DP Preconditioners
Combining (3.22) and (4.14) we find
:
SL -` IF" >
? _ _ and 5 J for .S ,"2-$ . The following lemma is then a
> < < ? (4.15)
#
whenever <
simple consequence of (3.15) and (4.15).
L EMMA 4.2. For the second order model problem, it holds that
" S L -8 IE" where the positive constant is independent of I , and $ .
The derivation of an upper bound for " for the fourth order model problem
follows a similar line, with the necessary modifications of some of the definitions.
Let < # _ , where 5TJ . For an arbitrary # "5 J 6 , we
define #R` 4 " to be the discrete biharmonic function on c with the following
properties:
C 2
agrees with up to theC first order
derivatives at
is piecewise cubic on and C KC J is piecewise linear on C Note that the components of are continuous up to the first order derivatives along the
interface of the subdomains and therefore (4.1)–(4.6) remain valid, provided we take @ to be the discrete biharmonic function on c that is identicalC with up to the first order
derivatives on and vanish up to the first order derivatives on
.
(4.16)
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Susanne C. Brenner
be an common edge of and > . Again (4.7) holds because of (3.9) and (3.24).
# 0 L 0 be defined by the property
')(D* C ')(+* C C 0 ` ! "+/0 # C 0 !"Z/07#+D
(4.17)
Let
Let
We can then apply the discrete Sobolev inequality, the Poincaré-Friedrichs inequality, (4.17)
and well-known estimates from the nonoverlapping domain decomposition theory for fourth
order problems (cf. [39, 26, 7]) to obtain the following analog of (4.11):
*
(D*
#%$ , @ %$ @ ( * (*
,
(4.18) %$ (+* L SL -` IE" " (D *
(D*
, %$ L SL -` (D* IE" !"* L S L -8 IE" $ # SL -` IF" If we define @ to be the discrete biharmonic function on c such that @ agrees with
>C @ up to the first1 order derivatives on and 1 @ vanishes up to the first order1 derivatives on
9
, then the estimates (4.12) and (4.13) remain valid. Finally (4.6), (4.7), (4.12), (4.13)
@ and (4.18) imply (4.14) and (4.15), and we have proved the following analog of Lemma 4.2.
L EMMA 4.3. For the fourth order model problem, it holds that
SL -` IE" is independent of I , and $ .
"
(4.19)
where the positive constant
Combining Lemma 3.4, Lemma 4.2 and Lemma 4.3, we have the following theorem for
two-dimensional FETI-DP preconditioners.
T HEOREM 4.4. For both the second order model problem and the fourth order model
problem, it holds that
- "9#
(4.20)
+ "
D "
4SL -` FI " I "
where the positive constant is independent of , and $ .
R EMARK 4.5. A closer inspection of 4.16 and 4.19 reveals that the constant
in 4.20 depends on the quasi-uniformity of the triangulation, the shape regularity of the
subdomains, and the numbers and positions of the cross points on the boundaries of the
subdomains.
"
"
5. Sharpness of the Condition Number Estimate for Second Order Problems. In
this section we will show that the bound in Theorem 4.4 is sharp for second order problems.
We will take to be the unit square and the subdomains to be nonoverlapping squares with
side
obtained by a uniform subdivision (cf. Figure 5.1). The triangulation
is then
obtained by a uniform subdivision of the subdomains into triangles. Furthermore, we take the
coefficients
to be 1 identically. We need to show that
$-
H
Z " 4 SL ` FI " Without loss of generality, we may assume ` IF" S .
D " S
and
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Analysis of Two-Dimensional FETI-DP Preconditioners
F IG . 5.1. A uniformly subdivided square
J # # ` 4 " 5 J ) S
Let 6 -
62 , for " ($ , vanishes at every node on
whose distance from the nearest cross point is less than or equal to ) (cf. Figure
5.1,
where we have marked every node whose distance to the nearest cross point is ). It is
clear that dim 6 6 be the orthogonal
dim 6 and dim 9
dim 6 . Let 6
complement of 6 with respect to
. Since the dimension of the subspace 6 of 6 defined by 6 /
6 D> ? 6 ,+ for all 6 ) equals the dimension of 6 ,
9 6 be a nontrivial Lagrange
the intersection of 9 and 6 is nontrivial. Let <
multiplier. Then we have, by (3.22),
>< ?6
(5.1)
> < < ? J
J
J #
5 J
J
? . . "
?4 #
# S U
J
J J
J
J
5 J
J
5
J
# 4 for some # ` "c5MJ 6 aJ ... ]J .
Given any decomposition
:
< #
(5.2)
_ J , we have, in view of (4.3),
where 5]=
:
: :
(5.3)
> < 4 ? 6
# > < > > > ?9#
> > >@ ? 6&
_
6 @ > is the discrete harmonic function on > that agrees with > on and vanishes on
where
C > >. @ Note
: that
(5.4)
> > >@ ? 6
# > c @ L > >@ ? 6 L > > > > @ L c @ ? 6 @ > where " K)2# , and it suffices to analyze the first term on the right-hand side of (5.4).
Let @ be the discrete harmonic function on that agrees with > @ (or equivalently
> ) on 1 and vanishes on C . We can then write, by (2.6), (3.7) and (3.10),
> @ L > >@ ? 6 # US >
4 @ 1 @ ? (5.5)
Observe that since (respectively > ) vanishes at all the nodes within a distance of *
(+* > ), we have
(+*
from the corners* of (respectively
(5.6)
@ ,
# @ , @ , # J
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Susanne C. Brenner
*
and similarly
1 @ 8 ,
>@ 8 (5.7)
It follows from (3.24) and (5.5)–(5.7) that
>
(5.8)
@
L
>
>@ *? 6
, >
*
*
? @ 8 * L0 1 @ "
? L > 3
>
- >
Combining (5.3), (5.4) and (5.8) we find
4 ? 6
><
:
>
? _ which together with (5.1) implies
><
(5.9)
< ? :
>
- B ? _ whenever (5.2) holds. We conclude from (3.14) and (5.9) that
" S
(5.10)
"
Now we turn to the derivation of a lower bound for , which involves the special piecewise linear functions constructed in [8] for proving the sharpness of the condition
number estimates for the BPS and the Neumann-Neumann preconditioners.
By Corollary 3.6 and Corollary 3.9 in [8], there exists a continuous piecewise linear
function on the real line with respect to the uniform subdivision with nodes " (" )
such that
)I
D`0E"9# +
D`0E"9#+ 0E"
D +"#
S L 8 EI "=
S L ` FI "
=
K @ N # !
(5.11) (5.12) (5.13)
(5.14)
(5.15)
0D
for
120]5X!
!
5
N@
SL -8 EI " where is the piecewise linear nodal interpolation operator with respect to the nodes
for " .
L EMMA 5.1. Let be the continuous piecewise linear function on + defined by
D +
+ 5 +
(5.16)
5 +
"E`
" \5
F0?"
D`0E"9#
Then we have
(5.17)
F0?" D "
D "
0 W" "
N@
0 0 0 ]
SL -8 EI " +
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Analysis of Two-Dimensional FETI-DP Preconditioners
#R
"
L . Note that is linear
Proof. We can write between the four nodes + , ,
and , equal to 0 at + and , and equal to + at
and
. Therefore, in view of (5.13) and a scaling argument, we have
(5.18)
M !
(5.19)
"
S
L Y` FI " agrees with = on ++ , vanishes on
. " on . Therefore, it follows
N@
On the other hand, = and agrees with from (5.15) that
+ +" M N @ S
L -8 IE" @ N L0
N@
The estimate (5.17) follows from (5.18) and (5.19).
Our derivation of a lower bound for involves the 9 subdomains depicted in
Figure 5.2. First we define to be the discrete harmonic function on
that is identical with
and and vanishes on the other
the function defined by (5.16) on the common side of
three sides of . From (5.17) we have
(5.20)
, L "
# (
(
5XJ S
8 IE" Moreover, there exists, by (3.24) and a simple dimension argument (or the Hahn-Banach
theorem), such that
>
(5.21)
? # >
4 ? and we define
< (5.22)
Ω6
Ω5
Ω4
Ω7
Ω9
Ω3
Ω8
Ω1
Ω2
# F IG . 5.2. The definition of
e3
e4
( c, d )
Ω1
e1
e2
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Susanne C. Brenner
5 J
#
CS
+ except for = " ; (ii) ,
Next we define a special
6 as follows: (i) " , is the discrete harmonic function on whose trace on
for
is identical
with
around the corners indicated in Figure 5.2 and zero at all other nodes (so each of
, ,
and are nontrivial around two corners, and each of
, , and are nontrivial
around only one corner); (iii) is the constant function + .
A more precise definition of, say
, is as follows. Let
be the lower left corner of
(cf. Figure 5.2). Then vanishes on . On the
and the four edges of
be edges and , we have
S
C + "
/ "
+
L M /;L 0 "9# /;L 0 "9# `0 W "
and on is defined by
+0 "
+ 0 L 0 ;/ L W"# +
+0 W"
+
0
0 ] 0 0 ]
Now we make two crucial observations. The first is that, by (3.7), (5.16), (5.22) and the
definitions of and , we have
4 ? 6 # US >
? 3
The second observation is that (5.12), (5.14) and the definition of (+*
:
#
(5.24)
_ S L 8 IE"=
>< (5.23)
imply
It follows from (3.24), (5.20), (5.21), (5.23) and (5.24) that
(5.25) > < ? 6 >
? L >
B ? L 4 S -8 IE" 4S Y` IF" >
B ? Combining (3.22) and (5.25) we find
(5.26)
>< < ?
SL -8 IE" >
B ? Finally (3.15), (5.22) and (5.26) yield
Z "
(5.27)
SL Y` FI " Therefore, for the second order model problem we have, by (5.10) and (5.27),
(5.28)
- 9" # is independent of I , and $ .
"
"
!
SL -` FI " where
In summary we have established the following theorem.
T HEOREM 5.2. The condition number estimate 4.20 is sharp for the second order
model problem.
"
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Analysis of Two-Dimensional FETI-DP Preconditioners
"
"
R EMARK 5.3. The sharpness of 4.20 for the fourth order model problem can also
be established within the additive Schwarz framework. Indeed 5.10 can be obtained in a
similar fashion with only minor modifications. However, the derivation of 5.27 requires the
construction of special piecewise quadratic polynomial functions whose symmetry properties
are different from the piecewise linear functions constructed in [8]. Such piecewise quadratic
polynomial functions will also be useful for showing that the condition number estimates in
[39] for iterative substructuring algorithms are sharp. The investigations in this direction
will be pursued elsewhere.
"
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