ETNA
Electronic Transactions on Numerical Analysis.
Volume 26, pp. 103-120, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
A FETI-DP PRECONDITIONER FOR MORTAR METHODS
IN THREE DIMENSIONS
HYEA HYUN KIM
Abstract. A FETI-DP method is developed for three dimensional elliptic problems with mortar discretization.
Mortar matching conditions are considered as the continuity constraints in the FETI-DP formulation. Among them,
face average constraints are selected as primal constraints in our FETI-DP formulation to achieve an algorithm as
scalable as two dimensional problems. A Neumann-Dirichlet preconditioner is used in the FETI-DP formulation and
it gives the condition number bound
#%
"!$#%"&&"')(+*
#%
where
and
are sizes of domain and mesh for each subdomain, respectively. When the subdomain with the
is independent of
, ,
smaller coefficient is chosen as the nonmortar side across the interface, the constant
and the coefficients of the elliptic problem. The proposed algorithm can be applied to two dimensional elliptic
problems with edge average constraints only as primal constraints and it can be generalized to geometrically nonconforming subdomain partitions. Numerical results present the performance of the algorithm for elliptic problems
with discontinuous coefficients.
Key words. FETI-DP, non-matching grids, mortar methods, preconditioner
AMS subject classifications. 65N30, 65N55
1. Introduction. FETI-DP methods were introduced by Farhat et al. [7] and applied
to solving elliptic problems with conforming discretizations both in two and three dimensions [8]. In three dimensions, subdomains intersect with neighboring subdomains on faces,
edges, or at corners, while they intersect on edges or at corners in two dimensions; the continuity of solution is imposed on faces and edges with dual variables and at corners with primal
variables in the dual-primal FETI (FETI-DP) methods. However, numerical results in [7, 8]
show that we need additional primal constraints for three dimensional problems to attain the
same efficiency as two dimensional problems. For these primal constraints, additional Lagrange multipliers are introduced and they are treated as primal variables in the FETI-DP
formulation. FETI-DP methods with various redundant constraints have been studied and
their condition number bound was analyzed by Klawonn et al. [16, 17] for elliptic problems
with heterogeneous coefficients. Numerical results were further provided in [14].
FETI-DP methods have been also applied to mortar finite elements methods [5, 6, 11, 20,
4]. In [5, 6], the condition number bound of FETI-DP operator was analyzed for various types
of preconditioners but it depends on ratios of mesh sizes between neighboring subdomains.
In [11], a Neumann-Dirichlet preconditioner was proposed and analyzed for elliptic problems
with discontinuous coefficients. In this case, the condition number bound does not depend
on the mesh sizes and the coefficients when the subdomain with the smaller coefficient is
chosen as the nonmortar side. Moreover, numerical results show that the Neumann-Dirichlet
preconditioner works much more efficiently than other FETI-DP preconditioners for elliptic
problems with highly discontinuous coefficients. Recently, a preconditioner, with its weight
factor depending on mesh parameters and the problem coefficients, was introduced and analyzed to be independent of coefficients and mesh parameters for two dimensional elliptic
problems, see Dokeva et. al. [4]. For three dimensional problems, FETI methods with mortar
discretizations were developed and their numerical results were provided in [19].
,
Received November 11, 2005. Accepted for publication November 22, 2006. Recommended by Y. Kuznetsov.
This work was supported by BK21 Project.
National Institute for Mathematical Sciences, 385-16, Doryong-dong, Yuseong-gu, Daejeon 305-340, South
Korea (hhk@nims.re.kr).
103
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104
H. H. KIM
The primary contribution of our work is the extension of the FETI-DP method in [11] to
three dimensional problems and to the second generation of mortar methods. In [11], vertex
continuity constraints are introduced as primal constraints. However, for the three dimensional case we need primal constraints other than the vertex constraints to obtain a method
as scalable as the two dimensional case in [11]. We select as constraints that averages of the
solution across subdomain interfaces are the same, which are the so-called face constraints in
[17]. Similarly to the previous work in [11], we propose a Neumann-Dirichlet preconditioner
for the FETI-DP formulation and show that the condition number bound
-
.0/21
3546$78887 9;:=<?>A@CB5DFEG<H 3I%JK3MLL?NPO
holds for elliptic problems with discontinuous constant coefficients. Here,- H 3 and JK3 are sizes
of domain and mesh for each subdomain, respectively, and the constant is independent of
H 3 , J 3 , and the coefficients of elliptic problems. In our FETI-DP formulation, we follow a
change of basis formulation introduced in [15, 18]. The change of basis makes the analysis of FETI-DP algorithms easier when primal constraints other than the vertex continuity
constraints are used. Moreover it gives an efficient and robust implementation of FETI-DP
algorithms [12, 13].
We note that edge average constraints can be considered as primal constraints for two
dimensional problems. The continuity constraints at vertices can not be selected as primal
constraints for the second generation of mortar methods [1]. We are able to extend the result
in [11] to the second generation of mortar methods by introducing edge average constraints
and using the change of basis formulation. Furthermore, the condition number bound estimate
of this case can be carried out similarly to three dimensional case presented in this paper.
This paper is organized as follows. In Section 2, we introduce finite element spaces
and norms and in Section 3, we derive the FETI-DP formulation with the Neumann-Dirichlet
preconditioner. Section 4 is devoted to analyzing the condition number bound of the FETI-DP
algorithm. Numerical tests are
Section 5.
- presented- inL denotes
Throughout the paper, or Q <?R
a generic positive constant that does not
depend on any mesh parameters or on the coefficients of the elliptic problems.
2. Finite element spaces and norms.
2.1. A model problem and Sobolev spaces. Let S be a bounded polyhedral domain
in TVU and W N < S L be the space of square integrable functions defined in S equipped with the
XZY[X
Y
norm
N\^]$_
`Kacbdfe ` NhgFij
6
The space H < S L is the set of functions, which are square integrable up to the first weak
derivatives, and the norm is given by
XYXZkml
Yts Y
_5`na bdpoe
`rq
q
gui @ g > ` e ` Nhguinv
N
where g ` denotes the diameter of S .
We consider6 the following model elliptic problem:
Find {}| H < S L such that
s
(2.1)
~ q
Y
y
y
d  y < i L i |ƒy S
<€[< i L q { < i LxLL ‚
d „ i |‡† S
{ <i
6xw Nzy
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A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
ˆZ‰
105
where  < i L is a square integrable function and [< i L is a positive and bounded
9 function in S .
Let S be partitioned into non-overlapping polyhedral subdomains Š)S 3x‹ 3546 . We assume
that the partition is geometrically conforming, which means that each subdomain intersects
its neighboring subdomains on a full face, a full edge or at a vertex. Each subdomain S 3
is equipped with a quasi uniform triangulation SG3Π, which consists of tetrahedrons. These
triangulations need not be aligned across subdomain interfaces.
Y
Y ‘
Each subdomain S 3 is equipped
with a conforming
linear finite element space
y
6
Y  3 bŽd Š | H Y < S 3 L b
|“’ 6 <€” L ” |ƒS 3Œ
6
6
df„ on †S—–˜†S 3 ‹ and ’ 6 <€”
where H  < S 3•L Žb d Š | H < S 3L b
of degree Rf> in ” . We assume that
yš™
y
i s |“s S 3
[< i L d  3
y L  3Vœ  3ž
3
›
where  3 is a positive constant. A bilinear
form
<
Y
sb Y
y
› 3 < { 3 3 L bŽd  3 e `KŸ q { 3 q 3 gFij
(2.2)
H
‹y
L
T
is a set of polynomials
is defined as
introduce Sobolev spaces
6 w N We now
6 defined on the boundaries of subdomains. The space
< †S 3•L is the trace
X¡ X
X¡ space
X k¢l£ of H < S 3• ¡ L equipped
 k¢l£ with the norm
3 N ] _5¤¥` Ÿ a bŽd 3 N ] _5¤¥` Ÿ a @ `K> Ÿ 3 N\^]_5¤¥` Ÿ a y
g
¡
¡

 ¡  k¢l£


3 N ] _5¤¥` Ÿ a bd e ¥¤ ` Ÿ e ¤¦` Ÿ 3 < i L ~ 3 <€§ L N gP¨ < i L g©¨ <"§ L j
i § U
~
6 w N 3Ž« L
3
«
3
|;†S , H 6 ¬w ¬ < ª is the set of functions in W N < ª 3« L whose zero extension
For any face ª
3
3L
into †[S is contained in H
XZY[X k l£ N < †[S and
 Y k islM£ equipped with theY norm
N y< i L gP¨uj
N ­­ ] _5®FŸŽ¯ a bŽd N ] _
®uŸŽ¯a @ e ®FŸ°¯
i
dist < Y †[ª 3Ž«)L
6xw
X$±Y[X we
XZY[X k l£ relation for
X±YX k | l£ H ¬¬ N < ª 3Ž« L :
k l£ have the following
From Section 4.1 in [25],
y
]Z_²¤¦Y `KŸ"a R
]
]
5
_
¥
¤
K
`
"
Ÿ
a
_
®
a
°
Ÿ
¯
(2.3) ±Y
Q
±
Y
Y
±
Y
R
­­
where is the zero extension of to †S 3 , i.e., d
on ª 3Ž« and d‚„ on †S 3K³ ª 3« .
where
2.2. Mortar matching conditions. Let us define
 bŽdµ´
Y
and
¸
¡
bd ´
Y
9
¶
| 3546  3 b
9
¶
| 3
4=6
¸
3y b
is continuous at subdomain vertices ·
¡
is continuous at subdomain vertices ·
y
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106
H. H. KIM
¼$»¼$» ¼»¼»
¼»$¼»$¼»¼»
¼»$¼»$¼»¼»
¼$» ¼»
Ωj
º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹
º¹$º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹
º¹$º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹
º$¹º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹
º¹$º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹
º¹$º$¹ º$¹º$¹ º$¹º$¹ ¾$¾$½ º$¹º$¹ ¾$¾$½ º$¹º$¹ ¾$¾$½ º$¹º$¹ ¾¾½ º$¹º$¹ º¹º¹
º¹$º¹$º$¹º$¹ º$¹º$¹ ½$º$¹º$¹ ½$º$¹º$¹ ½$º$¹º$¹ ½ º$¹º$¹ º¹º¹ h
º¹$º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ Ω
º¹
º$¹º¹$º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º$¹º$¹ º¹º¹º¹ j |Fij
¹$º ¹$º ¹$º ¹$º ¹$º ¹$º ¹$º ¹º
Ωi
h
Ωi |Fij
¸
¸
F IG . 2.1. Mortar and nonmortar sides of



¿ À
3 ¤¥`nŸ . We will approximate
where 3 is the trace space of 3 , i.e., 3 d
solution

 is not contained in 6 S the
of the problem (2.1) in . We note that the space
H < L . In order
to approximate the solution of the problem (2.1) in the nonconforming finite element space
 , we impose the mortar matching condition on  , for which jumps of a function inY 
Y s¦sÁs Y
across
face (interface) are orthogonal to a Lagrange multiplier space, i.e., d
y a y common

6
Â
9
L
| satisfies Y Y
<
™
y™ y
e ® ŸŽ¯ < 3 « Lxà 3Ž« g©¨Âd…„ à 3Ž« |ƒÄ 3Ž« ª 3Ž«
~


where Ä 3« is a Lagrange multiplier space given on the common interface ª 3Ž« bd †S 3 –0†S « .
On ª 3Ž« , we distinguish S 3Œ ® Ÿ°¯ and S « Œ ® Ÿ°¯ as in Figure 2.1 and choose one as a mortar
side and the other as a nonmortar side. On each nonmortar side, we define a finite element
¸
Y[
Y
space
6
3« bŽd Š ® Ÿ°¯ | H ¬ < ª 3«)L b | 0Å _ 3Ž« a‹ y
(2.5)
where Æ <"Ç€È L is the nonmortar side (nonmortar subdomain) of ª 3Ž« .
(2.4)
To get the optimal order approximation, we need the following abstract conditions on the
space Ä 3Ž« ;
3Ž« 9 ŸŽ¯
(A.1) The basis Š¥É Ê ‹ Ê 46 are locally supported, that is, the number of elements
in Sm3Ž3Œ « ¸ ® ŸŽ¯ , which have nonempty intersections with the simply connected support
of É Ê , is bounded independently of mesh sizes and ª 3Ž« .
(A.2) 3Ž« and Ä 3« have the
- same dimension. ËKÔ
(A.3) There is a constant such that
¸
X¦ËVX
X ÔÕ® ŸŽX ¯ g©¨ ™ Ë
\ ] _5® ŽŸ ¯ a R -ÍÐ[ÌxÑuÎnÒ Ï Ÿ°¯ÂÓ \ ] _5® ŸŽ¯ a
|
3Ž« j
Ê¥× x6 w N 3 «)L
X < ª , X there exists Ö Œ |˜Ä  3Ž«  kcsuch
ØÙ©lM£ that
y
)
Ê
×
6
] _5® ŸŽ¯ a
Ö ~ Ö Œ N\ ] _
® Ÿ°¯ a R - J 3N Ö N

where Ú is the order of finite elements in 3 .
The condition (A.4) implies that > |ƒÄ 3« . In the following, we assume that the Lagrange
multiplier space Ä 3Ž« satisfies the above conditions; the standard Lagrange multiplier space
(A.4) For ֗|
H
in [2] and the Lagrange multipliers with dual basis in [9] are those examples.
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A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
ˆZ‰
107
In our FETI-DP formulation, we will use the mortar matching condition (2.4) as continuity constraints. These continuity constraints can further be written into
9
¡
y
Û
3 3 Ý
d „
Y 
3
4=6žÜ
3 ¤¥`nŸ . We note that the matrices 3 are not boolean matrices as in the original
where 3 d
Ü¡
FETI (or FETI-DP) methods.
In the following, we will use the same notation for finite element functions and the corre¸
¸
sponding vectors of nodal values. For example, 3 is used to denote a finite element function
or the vector of nodal values of that function. The same applies to the notations for function

spaces such as 3 , , , etc.
¡
3. FETI-DP formulation.
3.1. FETI-DP operator. In this section, we formulate the FETI-DP operator for the
problem (2.1) with the mortar matching condition as continuity constraints. For Þuß elliptic
problems, it was shown that using the primal variables at vertices is not enough to get the same
condition number bound as à%ß problems; see numerical results in [7, 8]. Hence, additional
primal constraints are introduced to accelerate the convergence of the FETI-DP method.
For the Þuß elliptic problems with conforming discretizations, Klawonn et al. [16] developed FETI-DP methods with various redundant constraints. They introduced edge average or
face average constraints as primal constraints to achieve the same condition number bound
as à%ß elliptic problems. The continuity constraints on edges (faces) are that the averages of
functions across a common edge (face) are the same. In [17], they extended the results to a
case with face constraints only.
In the mortar discretizations, two sets of primal constraints are possible; the one that
contains continuity constraints at vertices and the face average constraints, and the other one
with only the face average constraints. Using the second set of the primal constraints we can
generalize our algorithm and theory to the second generation of mortar discretizations [1]
and to geometrically non-conforming partitions [10], since we can relax the continuity at
subdomain vertices. In our work, we consider the first case with the vertex continuity and
the face averages as primal constraints. We may impose the face average constraints by
introducing additional Lagrange multipliers and then treat them as primal variables in the
FETI-DP formulation; see [7, 8, 16]. In our FETI-DP formulation, we follow the change of
basis formulation introduced in ¡ [15, 18]¡ that
 leads to
¡ much¡ easier
 analysis and more robust
implementation; see [12, 13].
3 ® Ÿ°¯ ( or 3Ž« d « ® Ÿ°¯ ) we consider a change of
On each interface ª 3« , for 3Ž« d
¡
variables so that
¡
¡ _ 3« a
3Ž« d…á ®FŸ°¯ãâ ä _ 3« a y
嚡 æ
¡
_
a
Ž
3
«
where á ®uŸŽ¯ retains unknowns at the boundary of ª ¡ 3« , å
is the average of 3Ž«
¡
®uŸŽ¯ 3Ž« gP¨ y
_ 3Ž« a
å d Ó
® Ÿ°¯ > gP¨
¡
Ó
_ 3Ž« a
and the function ä
has the average value
¡ zero on ª 3Ž« , i.e.,
_ 3Ž« a
e ®uŸŽ¯ ä gP¨çd‚„^j
on ª
3Ž« , i.e.,
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¡
108
H. H. KIM
_ 3« a
We note that ä
is a function in the above equation
¡ and it can be represented using the
change of basis given by the transform á ® ŸŽ¯
3 into
¡
After the transforms, we express the
¡ unknowns
¡ _3a
3 d â ä_ 3 a y
åèæ
where é stands for the unknowns of primal variables, i.e., the averages on faces and unknowns at vertices, and ê stands ¸for the¸ remaining unknowns. These notations will be used
throughout this paper.
¸
We now consider
a subspace
¡ ¸ ë of ¡ that¡ satisfies the primal constraints
ë bŽdÝ´
(3.1)
¸
ä
Let
bAe¡ ®FŸ°¯ < 3
~
|
«)L gP¨çd‚„
is continuous at subdomain vertices ‹
and
be a space of the vectors,
¡
¡
ä
j
_ 6 a Áð ññ
y
ä í
d ìîïî ¡ ... ò
_9 a
¸
¸
¡ ä
and let å be the space of primal unknowns
¸
¸ å . ¸ We then decompose the space ë
dual and the primal parts,
ë d
õ
We define
äôó
¸
_3a
å b å 
¸
å j

å `KŸ
into the
s s
that restricts
Y
y
_ 3 a the primal unknowns to the local primal unknowns.
Let ö _ 3 a be the Schur complement matrix obtained from the bilinear form › 3 < L in (2.2)
and let ÷
be the Schur complement forcing vector obtained from ` Ÿ  3 gui . After the
_3a
_3a
change of variables, the matrix ö
and vector ÷
are written into Ó
_ 3 a d â ö äz__ 33 aäa
ö
ö å ä
_3a
ö ä _ 3 åa y ÷ _ 3 a d â ÷
ö å[å æ
÷
We recall the mortar matching condition
¡
¸
Û
¡
¸
9
3546 Ü
_3a
ä_ 3 a j
å æ
¡
3 3 d݄nj
Since |øë satisfies the face average constraints and > |øÄ 3Ž« , the above continuity constraints are redundant for |ƒë . We consider a subspace Ä 3« of Ä 3« that has one less basis
than Ä 3Ž« . We impose the mortar matching condition (2.4) with the Lagrange multiplier space
Ä 3Ž« instead of Ä 3Ž« and obtain its matrix representation
Û
9
3
46 Ü
¡
3 3 d…„^j
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¸
¡
A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
The above constraints are then non-redundant constraints for
Û
(3.2)
¡
9
ä
3 4=6 < Ü
Let
¡
¡
¸
¸
¡
_3a _3a
_3a _3a
ä @ å å L d݄^j
Ü
|ùë
ˆZ‰
109
. We rewrite it as
¶
Ä ä d Ž3 « Ä 3Ž« j
We then obtainy the following
mixed formulation of the problem (2.1) with the constraints (3.2):
å y ÃKL | ä œ ¡ å œ Ä ä satisfying
¡
Find < ä
ö zä ä ¡ ä @ ö ä å
¡
ö å ä ä @ ö åå
ä
Ü
(3.3)
where
¡ å
ä Ã d ÷
@ r
Ü
¡
å @ úå Ã d ÷
ä @ åÜrú å
d „
Ü
ä y
å y
y
_3a y
ö zä äcü
_ 6 a _ 6 a ð ññ
ö ä å å
y
ö ä å d ìîîï _ 9 a ...õ _ 9 a ò
ö yä å å
ö å ä d ö äú å õ
õ
9
_
a
_
a
_3a y
3
3
Û
ö å[å d 3
4=6 < å L ú ö å[å å
sÁs¦s
9 _3a õ _3a y
_6 a y y _9 a y
Û
ä d û ä
ä ü å d
å å
¡
3546 Ü
Ü
Ü
Ü
Ü
_ 6 a ð ññ
_6 a ð ñ
¡
õ
ä ñ
9
֊
_3a _3a y ä
y
Û
dýîîïì ¡ ... ò j
÷ ä díìîîï _ ...9 a ò ÷ å d 3546 < å L ú ÷ å
_9 a
ä
¡ ÷ä ¡
After eliminating ä and å from (3.3), we obtain
ª ÿþ à dÝgKj
¡
We note that
Á±
¡ y à ¡ N y
y
0
.
2
/
1
ª ÿþ+à à d Ñ Ü y ö
ö äzä d
where
diag354678õ 88?7 9fû
¡
± d ä
å
y
¡
¡
Ü äzä Ü ä å y
Ü
ä
ö
ö
d o å v j
ö d o ö å ä ö å[å v
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110
H. H. KIM
±
More precisely, we compute
× 6 d äzä
ä å ª å× å 6 ª å ä y
ª ÿþ d Ü ö r
ª
ª
@
Ü ú
where
9
äª zä d ä ö äz× ä6 ä d Û
Ü
Ü ú 3 4=6
× 6
ª ä å d ~ < Ü å ~ Ü ä ö äzä ö
y
ª å ä d ª äú å õ
9
_3a
_3a
Û
ª å[å d 3546 < å L ú < ö å[å ~
_3a _ a
ä < ö äz3 ä L × 6 <
Ü
Ü
9
ä å L d Û <
~ 3546 Ü
_3a y
ä L
ú
õ
_3a
_3a _3a × 6 _3a _3a y
ä äzä L ö ä å L å
å
~ Ü <ö
õ
_3a _3a L × 6 _3a L _3a
ö å ä < ö äzä ö ä å å j
From the above formula, we can see that the computation ª ÿþ à ×can
6 be done by applying
matrix-vector multiplications in each subdomain except the term ª åå .
3.2. Preconditioner. We derive a preconditioner from the similar idea to [11], in which
a Neumann-Dirichlet preconditioner is built from a dual norm on the Lagrange multiplier
space using a duality pairing between¸ the Lagrange multiplier space and the finite element
space on nonmortar sides. In the following, the idea is provided in more detail.
¸
¸
¸
¸
¸
¸
We further decompose the space ë into
(3.4)
ä ó
ë d
¸
ä 7Å ó
å d
å y
ä 7 ó
Æ stands for the space of vectors for the unknowns at the interior of
where the subscript
¡
nonmortar faces and the subscript stands for the remaining unknowns. In other words, we
ä into
¡
split a vector ä |
¡
¡
¡
Å
ä d o ää 77 v y
Å
¡
where ä 7 are unknowns at the interior of nonmortar faces and
¡ condition
¡
unknowns. We recall the mortar matching
ä 7
are the remaining
ä ä @ å å d…„nj
Ü
¡
¡
¡
ä 7Å ä 7Å @ ä 7 ä 7 @ å å
d „nj
Ü
Ü
Ü
Ü
It is then written into
Å
Here, the matrix ä 7 is square and invertible.
Ü the Neumann-Dirichlet preconditioner
±
We will propose
of the form
× 6
ª ÿ þ d Ü ß özß Ü
ú
where
Ü
and ß
Ü
are given by
d
Ü
ä 7Å
Ü
ä 7
y
ß Å%Å
y
å ß d ï
ì
Ü
ß
ð
ß å[å
ò j
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A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
ˆZ‰
111
The Neumann-Dirichlet preconditioner provides the weights
× 6 y
y
d‚„
ä 7Å ¸ ä 7Å ß
ß åå d…„nj ¸
¸
Ü ú Ü
¡
This preconditioner is originated from a dual norm on the Lagrange multiplier space
Ä ä ; see [11]. We recall the space ë and ä Á±7 Å in (3.1) and (3.4). For | ë , we define a
X¡tX
¡ ¡
norm
y
¸
¡
N d ö ¸ j
±¡
ä 7 Å | ä 7 Å has the zero average on each face ª 3Ž« and has zero values
¸
Since
a function
±¡
Å
at subdomain vertices, its zero extension ä 7 to
satisfies the primal constraints, i.e.,
Å
ä 7 |ƒë . We may write
¡
¸
±¡
ä 7Å ð
ä 7 Å d ï „ ò |“ë j
ì „
¡
Å
Á± ¡ ±
±¡
We then define a norm for ä X7 ¡ by X
y
ä 7 Å N d ö ä 7 Å ä 7 Å y
¡
and a dual norm on the space
X X Ä ä by
X¡ ä 7 Å X ä 7 Å y à /%1
à d .0
(3.6)
Ñ Ü ä 7 Å j
× 6
The Neumann-Dirichlet preconditioner ª ÿþ X isX given by
y
ª ÿþAà à d à N j
(3.7)
ß %Å Å d
(3.5)
Similarly, the matrix ª
+þ
X norm
can be obtained from a X dual
y
y
ª ÿþAà à d à N ¡
¡
where the dual norm is given
X X by
± ¡ y à ¡ X¡tX y à N d . %/ 1
N j
à N d .0Ñ /% 1 Ü
y
Ü
Ñ
N
ö
The preconditioner is
× 6 originated from the idea that these two dual norms will be sufficiently
close so as to get ª ÿþ as a good preconditioner for ª ÿþ . The lower bound estimate can be
±¡
¡
done from X X
X X
(3.8)
±¡
à N d
Å
Á± ±¡ ä 7 Å ¡ ± y Ã
XZ¡tX y Ã
N
N d
.0/21
.
0
2
/
1
Ñ ¸ Ü ä 7 Å y ä 7 Å R Ñ Ü
N
ö
à N y
because ä 7 is contained in ë . In the following
section, we will provide an upper bound
× 6
ÿ

þ
of the Neumann-Dirichlet preconditioner ª
.
From (3.6) and (3.7), we find the following form of the preconditioner
(3.9)
9
׏ÿþ 6 d Û û ä _ 3 a Å L L × 6 „ ü äz_ 3 aä â < ä _ 3 a 7 Å L × 6 y
7
ª
ö
Ü „
æ
3 46 < < Ü
ú
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112
H. H. KIM
that provides the weights in (3.5). The computation ª
Dirichlet problem in each subdomain, i.e.,
_3a
_3a
ö äzä d < äzä ~
where
_3a
_ 3 a d _3 a
îïì ä
_ 3 a
å ÿ × þ 6 Ã
_3a
can be done by solving a Neumann-
_3a L × 6 _3a L y
$ä
ä
< _3a
äz_ 3 aä
å ä
_ $3 ä a
_3a ð ñ
_ 3 åa
ä _ 3 aå ò j
Y
åå Y
y L
y
_3a
Here is the stiffness matrix of the bilinear form › 3 < {
for {
|  3 and the subscripts , é , and ê stand for the subdomain interior unknowns, the unknowns for the primal
variables, and the remaining unknowns, respectively.
When we compute
_3a
ä 7Å L × 6 v à y
ö äzä o < Ü ú „
we solve the problem
_3a _3a
{ d
_ 3 a â < ä _ 3 a7 Å L × 6 Ã y
Ü „
æ
_3a
ä 7Å L × 6 Ã
ä
where Neumann boundary condition, <
, is given on the nonmortar faces and zero
Ü on the remaining
Dirichlet boundary condition is provided
part of the subdomain boundary.
¸
4. Condition number bound estimation. On the interface ª 3« , we assume that S 3 is
the nonmortar side and S « is the mortar side. We denote the mesh sizes¸ in eachY subdomains
S 3 and S « by JK3 and J« , respectively. We recall the space 3« in (2.5).
3« for |—W N < ª 3«2L
D EFINITION 4.1. We define a mortar projection 3Ž« b W N < ª 3«)L 
by
Y
Y
e ® ŸŽ¯ <
~
3Ž« L Ãn3Ž« gP¨…d…„ ™ ÃK3« |ƒÄ 3« j
For the space Ä 3Ž« satisfying the conditions (A.1)-(A.4)
2.2), we can show that
6 w N (see3Ž«)Section
3
«
L
¬
¬
the mortar projectionX
onXthe
Y[isX k continuous
YX k space
l£
l£ H Y < ª (see [9] or [24] );
™
y
6 w
| H ¬ ¬ N < ª 3Ž«)L
­­ ] _
® ŸŽ¯ a
where is a constant not depending on H 3 and Jn3 . Moreover, the projection is continuous
on the space
¡ W N < ª 3Ž«)L .
L EMMA 4.2.
S 3 is the nonmortar side of the interface ª 3« d †[S 3 –}†S « , any
6 w N When
Ž
3
)
«
L
function | H
< ªX ¡tsatisfies
X k l£
X¡tX k l£
N
y
3
3Ž« N ­­ ] _
® Ÿ°¯ a R - o >A@ log H JK3 v
N ] _
®FŸ°¯xa
3Ž«
where H
3IFJn3
­­ ] _
® °Ÿ ¯ a R
denotes the number of elements across the nonmortar subdomain S
3.
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etna@mcs.kent.edu
113
ˆZ‰ ¸ 
be the W N -projection onto the finite element space 3 ®FŸ°¯ , that is the
Proof. Let !
3
¡
X interface ª 3« XZ.¡tItX then
X k¢lChapter
kml£ satisfies,X see¡t[3,
£
restrictionXof
to¡tthe
II], X¡X kmlM£
y
N\ ] _
® Ÿ°¯ a R - JK3 N ¡ ] _
® ŸŽ¯ a
N ] _
® ŸŽ¯ a j ¡
(4.1)
!
! ¡ N ] _
® Ÿ°¯ a R
¸ ~ ¡
For the function ! , we
¡ denote by ®uŒ ŸŽ¯ < ! L and ¤¥Œ ® ŸŽ¯ < ! L the nodal interpolants of ! to
F
®
Ž
Ÿ
¯
3
the space
that vanish at the nodes on †ª 3Ž« and at the nodes interior to ª 3« , respectively.
¡
¡
We then decompose !
into ¡
d ® Œ ŸŽ¯ < ! L @ ¤¥Œ ® Ÿ°¯ < ! L j
!
¡
A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
¸
X consider
¡ X k lM£
We now
X
¡
¡ X k l £
X
¡ X k l£
3« < L N ­M­ ] _
®uŸŽ¯xa R à 3« <
L N ­­ ] _ ®FŸ°¯xa @ à 3Ž« < ! L N ­­ ] _
®FŸ°¯xa j
~ !
Using an inverse inequality, the continuity of 3« in the W N -norm, and (4.1), the first term is
¡ X k l£
X ¡
¡ X
X¡X kmlM£
estimated Xby ¡
×
6
L N ­­ ] _5® ŸŽ¯ a R - J 3 3Ž« <
L N\^]Z_
®FŸ°¯xa R 3Ž« <
N ] _
® ŸŽ¯ a j
(4.2)
~ !
~ !
X consider
¡ X k the
X
¡ X k l£
X
¡ X k lM£
l£ second term,
We
3« < ! L N ­M­ ] _
®uŸŽ¯xa R à 3« < ®FŒ ŸŽ¯ < ! LxL N ­M­ ] _
®FŸ°¯ a @ à 3« < < ¤¥Œ ®FŸ°¯ < ! LxL N ­M­ ] _
®uŸŽ¯xa
X ¡tX k lM£
X
¡ X
N
3
N ]_
®uŸŽ¯xa @ - J 3× 6 3« < < ¤¥Œ ® ŸŽ¯ < ! L L N\ ] _
®uŸŽ¯ a
!
R o >A@CB5DFE H JK3 v X¡
X k lM£
X
¡ X
N
3
×
6
R o >A@CB5DFE H JK3 v X¡X NkmlM£ ] _
®FŸ°¯ a @ J X 3 ¤¥Œ ® ŸŽ¡¯ < ! X L N\ ] _5®FŸŽ¯ a
3 N
- ¤¥Œ ® Ÿ°¯ ! L N\ ] _5¤¥®FŸ°¯xa
R o >A@CB5DFE H J 3 v X¡X Nk lM£ ] _
® Ÿ°¯ a @
<
3 N
N ]Z_
® Ÿ°¯ a j
(4.3)
R o >A@CB5DFE H J 3 v
Here we have used a face lemma [21, Lemma 4.24], an inverse inequality, (4.1), the continuity
¸ 4.17]. From (4.2) and (4.3), we obtain the
of 3Ž« in W N < ª 3Ž«)L , and an
¡ edge¡ lemma
sÁs¦s ¡ [21, Lemma
desired estimate.
y y 9 L
|“ë , we have
L EMMA 4.3. For d < 6
X
¡
¡
X kmlM£
 ¡  k¢l£
 ¡  k¢l£
N
3
3« < 3
¥« L N ] _ ®uŸŽ¯xa R - o >A@CB5DFE H JK3 v û 3 N ] _5¤¥`nŸ"a @ « N ] _5¤¥`^¯xa ü y
~
¡
¡
¡
where S 3 is the nonmortar side of the interface ª 3« .
Proof.¡ Since ¡ satisfies
¡ the primal
¡ constraints, 3 and « have the same average value
3Q Ž« on ª 3« . We then obtain the resulting bound from Lemma 4.2 and a Poincaré inequality by
replacing 3 and « by 3
Q 3Ž« and « ~ Q 3Ž« , respectively.
~
R EMARK 4.4. The face average constraints are important in applying a Poincaré inequal-
ity to the above analysis, while the continuity at vertices are not necessary. Since the continuity constraints at the vertices can be relaxed, it is possible to extend the theory and algorithm
to the second generation of mortar discretizations and to geometrically non-conforming subdomain partitions.
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114
H. H. KIM
To obtain a condition number bound that does not depend on mesh sizes and the coefficients, we need the following assumptions.
A SSUMPTION 4.5. On a common interface ª 3« , we choose the subdomain S 3 with
smaller  3 as the nonmortar side and the subdomain S « with larger  « as the mortar side.
The Assumption 4.5 is conventionally used in the analysis of the mortar methods; see
[22, Remark 3.1].
We now estimate the upper bound of the FETI-DP algorithm.
L EMMA 4.6. Under Assumption 4.5, for à |ƒÄ ä , we have
y
3 N
ª ÿþ à à R - 354. 67/%888?1 7 9 ´‡o A> @CB
DuE H J 3 v
where the constant does not depend on J[3 , H 3 , and  3 .
¡
Proof. We note that
X X
Z±
¡ y á N y
y
0
.
%
/
1
(4.4)
ª ÿþ à à d X à X N d Ñ Ü y ö
Xä ¡
y
0
.
%
/
1
ÿ

þ
Ã
Ã
Ã
d
N
d
(4.5)
ª
Ñ Ü
y y
· ª ÿ þ à à 7Å
ä
¡
ä X 7 Å y à N
7 Å N j
and 3Ž« , we have
Ü
¡
¡
¡
ð N
y Ã
Û
3« 3
« Lxà 3« gP¨ ò j
N d ïì
e
Ü
3Ž« ®FŸ°¯ < ~
From the definitions of
¡
Let
¡
3« d 3Ž« < 3
«Lj
~
From > |ƒÄ 3Ž« and the definition of 3Ž« , we have
¡
¡
« L g©¨Âd…„nj
e ® Ÿ°¯ " 3Ž« g©¨Âd e ® ŸŽ¯ < 3
~
Since " 3« has zero average on ª 3« and has zero values on †ª 3Ž« , after the transform introduced
in Section 3.1 we may
¡ write
¡
¡
y à Û
«¥L Ãn3Ž« g©¨Âd ä 7 Å " ä 7 Å y à N y
d
e ® ŸŽ¯ 3Ž¸ « < 3
(4.6)
~
Ü
3Ž«
Ü
Å
Å
ä 7 Å . We note that " ä 7 Å can be a function or a vector
where " ä 7 d " 3Ž« on ª 3« and " ä 7 |
¡
X we X get
of unknowns. From the above relation
(4.6) and (4.5),
y à y
N R ª ÿþ à à " ä 7 Å N j
(4.7)
Ü
We will show that
Z± ¡ ¡
X
X
y 3 N
-—.0/%1 ´ o
" ä 7 Å N j
(4.8)
3
R
>A@ log H J 3 v · ö
"
The desired bound then follows from (4.4) and (4.7).
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etna@mcs.kent.edu
±
A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
¸
±
Å ±
Let " ä 7 be the zero extension of "
Å
easy to see that " ä 7 |ôë . Let " 3 d " ä
X thenX obtain Á± ±
±
We
"
ä 7 Å N d ö " ä 7 Å y " ä 7 Å y
d Û ö _ 3 a " 3  " 3#
k l£
3
Û
3 " 3 N ]Z_²¤¦`
R
X Xk
3 
- Û Û 3 " 3Ž« N
R
3 «  X ¡
d - Û Û  3 3« <
3 «
ˆZ‰
¸
115
±
ä 7 Å  to the all interfaces, i.e.,
into the space . It is
y
7 Å ¤¥`nŸ the restriction of " ä 7 Å to the subdomain S 3 .
lŸ a£
­M­ ] _ ® ¡ Ÿ°¯ a X k lM£
3
¥« L N M­ ­ ] _
®uŸŽ¯xa
~
¡ ¡
¡ ¡
ð
N
y
y
3
3
Û
_
a
_
a
3
«
- .0/%1 ´‡o
3 $3 @  « ö
« %« v ò j
R 354678887 9
>A@ log H J 3 v · ïì 3 « o ö

  k l£ inequality
Here we used the well-known
  k lM£
y
y
_
a
3
 3 " 3 N ]Á_5¤¥` Ÿ a R ö " 3 " $3 R  3 " 3 N ]Z_5¤¥` Ÿ a
ª 3« . From
the relation in (2.3), and Lemma 4.3, and S 3 is the nonmortar side of the interface
I
3
«
Assumption 4.5,   R > so that we have shown (4.8) with the constant independent of
J 3 , H 3 , and  3 .
R EMARK 4.7. The above analysis can be applied to two dimensional problems with
edge average constraints only as primal constraints.
From the lower bound estimate in (3.8) and the upper bound estimate in Lemma 4.6, we
then have the condition number bound of the FETI-DP algorithm.
T HEOREM 4.8. Under Assumption 4.5, the condition number bound holds for the FETIDP algorithm,
× 6
-ù.0/21 ´ o
3
< + þ ª +þ L R
>A@
& ª
where the constant
-
does not depend on J3 , H
H 3v N· y
3
log J
3 , and  3 .
5. Numerical results. In this section, we provide numerical tests for the proposed
FETI-DP algorithm. We consider s the following model problem:
~ q
where S
y
d <„ > LU
y y
d 
<€[< i § " L q { L f
d „
{ ‚
is the unit cube, the exact solution is
in
on
S
y
†S
y
y y
{ < i § " L d i y y > " L('*) N Ì,+.- </ i L Ì,+.- </ § L Ì,+0- </ " L j
[ < §
We divide the domain S into 1 œ 1 œ 1 uniform cubical subdomains with side length
S 3 is discretized by conforming trilinear finite elements with
H d > I 1 . Each subdomain
K
J
3
uniform cubes of the size . The mesh size J3 can be different to different subdomains.
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116
H. H. KIM
l
TABLE 5.1
(Iter), corresponding condition numbers (Cond), and the broken
-norm er32The number of '; CG iterations
A
@
79
5:4 <>= Ÿ ? & ' , of the FETI-DP algorithm for the model problem with the constant coefficient
ror,
B 0C *AD%#* EÁ G& 4658
F ; left four columns (the subdomain problem size HJI increases with the fixed number of subdomains
KML
L
KPL
FON
), right three columns (the number of subdomains
)
Æ
4
16
24
32
Iter
14
16
18
19
Cond
6.1185
8.8967
10.9198
11.7914
6
increases with the fixed local problem size HJI
1ƒU
H -error
1.099819e-02
5.576953e-03
3.706825e-03
2.773728e-03
)
àuU
/
U
U
Iter
14
18
18
F9Q
).
Cond
6.1185
7.3615
7.5818
The corresponding Lagrange multiplier is given by the tensor product of two dimensional
multipliers considered in [23]. Even though the theory provided in the previous section was
developed for tetrahedral finite elements, it extends to the trilinear finite elements without
difficulty.
To see the performance of the
y y preconditioner, we perform two types of experiments;
y y
the case when the coefficient < i § " L d > and the case when the coefficient [< i § " L is a
positive constant  3 in each subdomain S 3 , they can be discontinuous across the interface. We
solve the FETI-DP
× 6 equation using the conjugate gradient method with the Neumann-Dirichlet
preconditioner ª ÿþ in (3.9).×The
R conjugate gradient iteration is performed up to the relative
residual norm reduced by > „
.
y y
We first consider the model problem with [< i § " L d > . All subdomains have a uniform
triangulation of cubical elements with the mesh size H I < ) Æ L , where H is the diameter of the
subdomains. In
6 Table 5.1, the number of CG iterations, condition numbers, and the errors in
the broken H -norm are shown as the increase of the local problem size with a fixed number
of subdomains 1}U < d à%U L and as the increase of the number of subdomains 1—U with a fixed
local problem size ) Æ < d / L . From the result, we observe the log N -growth of the condition
number in terms6 of the local problem size (see Figure 5.3), the optimal convergence of errors
in the broken H -norm, and a scalability iny terms
y of the number of subdomains.
We now consider the cases when [< i § " L is discontinuous
across the subdomain iny y
terface. In our example, we consider four values of [< i § " L , 1, 10, 250, and 1000, and
distribute these values to the subdomain partition. In Figure 5.1, we present a cube that
is divided
y y into eight uniform cubical subdomains. For the eight subdomains we distribute
of Figure 5.1. When we introduce more number of subdomains,
[< i § " L as in the right
for an example 1 U d ) U , we put the eight cubical partition in a periodic pattern and obtain
the coefficient distribution. In the following, we will present the results varying the choices
of the mortar and the nonmortar sides and varying mesh sizes depending on the coefficient
distribution.
F IG . 5.1. A subdomain partition (left) and the values
four subdomains on the top, respectively.
5000
(n)
10
(3n)
250
(2n)
1
(4n)
1
(4n)
250
(2n)
10
(3n)
5000
(n)
B .C *#D%*AEÁ&
(right) of four subdomains on the bottom and
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A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
ˆZ‰
5000
10
5000
10
1
250
1
250
! &
117
F IG . 5.2. An example of non-uniform meshes depending on the coefficients (left: the smaller mesh sizes for the
smaller coefficients, right: the reversed selection) when
HJI I FON is the smallest mesh size.
Z&
l
2
' uŸ & '
* %* Á&
AD $E and the subdomain partition
model problem with a discontinuous coefficient B
meshes), center (non-uniform meshes), right (reversed non-uniform meshes).
)
Æ
8
16
24
32
error,
! -norm
' K FVNZ& HJI & for the
; left (uniform
TABLE 5.2
The number of CG iterations
(Iter), corresponding condition numbers (Cond), and the broken
A
@
465S7T5:4 ; <U= ?
, of the FETI-DP algorithm
as the decrease of the largest mesh size
0C
Iter
12
14
15
16
uniform 6
Cond
H -error
4.39 5.5265e-03
5.74 2.8026e-03
6.61 1.8626e-03
7.29 1.3937e-03
Iter
12
14
14
15
nonuniform6
Cond
H -error
4.15 5.5494e-03
5.31 2.8130e-03
6.06 1.8698e-03
6.66 1.3991e-03
K
reversed nonuniform
6
Iter Cond
H -error
10
2.96 7.8789e-03
13
4.22 1.0320e-02
14
5.06 7.3698e-03
15
5.69 5.5064e-03
Table 5.2 presents the performance of the algorithm varying the mesh sizes. We have
used the uniform mesh size H I < ) Æ L for all subdomains, the non-uniform mesh sizes, from
on the coefficients. In addition we have selected the subdomain
H I Æ to H I < ) Æ L ,y depending
y
with smaller [< i § " L as the nonmortar. For the non-uniform case, we selected smaller mesh
sizes for smaller coefficients, see the right in Figure 5.1, with H I < ) Æ L the
6 wZY smallest. Such nonuniform meshes satisfy the optimal ratio of meshes J[3•IFJP«XW <" 3I  «¦L
that was observed
in [24] by applying an appropriate adaptivity strategy when the right hand side  < i L of the
model problem does not reflect the jump in the coefficient distribution. In addition, we have
tested the case when the selection for the mesh size is reversed, see Figure 5.2. For the
reversed case, the nonmortar side has the larger mesh size.
The results in Table 5.2 present that the condition numbers seem to be not affected by
the selection of mesh sizes. All three give the B5DFE N -growth of the condition number bound
with respect to the local problem size, see Figure 5.3. We can see that the errors are almost
the same for the uniform and the non-uniform cases. However the reversed non-uniform
case gives larger errors than the previous two cases at the same discretization level. We can
conclude that using nonuniform mesh sizes, that are finer for smaller coefficients, is more
practical for the model problem with discontinuous coefficients considering the presented
errors and the number of iterations.
In Table 5.3, we present the performance varying the selection of mortar and nonmortar
sides. Here we have used smaller
sizes for smaller coefficients. The first case is when
y mesh
y
the subdomain with smaller < i § " L is the nonmortar side. This selection satisfies Assumption 4.5. We also consider the reversed selection to see how it affects the performance. The
reversed selection shows poor performance. The errors are also larger compared to the regu-
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118
H. H. KIM
0.65
0.6
C1
cond/(1+log(4n))
2
0.55
0.5
0.45
0.4
C2
0.35
C3
0.3
0.25
5
* F* Á&
C4
10
M©`_ M& & '
!
* %* Á&
15
20
25
4n (the max. number of nodes)
* %* Á&
30
35
*#D%:*Aconstant
E¦& , non-
F IG . 5.3.
increase of the maximum number of nodes HJI ,
\a HJI ) as .the
.C Plot of ([Z\,I^]
B C #D AE , uniform meshes, : discontinuous B .C
AD #E bF , N : discontinuous
coefficient B
0
C
AD #E , reversed non-uniform meshes
uniform meshes, H : discontinuous B
c2
l
ˆ
! & -norm error,
for the model
TABLE 5.3
The number of CGAiterations
(Iter), corresponding condition numbers (Cond), and the broken
@
4657d5 : 4 ; <>= #?
, of the FETI-DP
.C algorithm as the decrease of the largest mesh size HJI
'
Ÿ & '
* %* ¦&
K
' K
Z&
FON ; regular selection (the
problem with a discontinuous coefficient
.C C B #D AE and the subdomain partition
D
is the nonmortar), reversed selection (the opposite).
subdomain with the smaller B
)
Æ
8
16
24
32
Iter
12
14
14
15
* F* &
regular selection
6
Cond
H -error
4.15 5.5494e-03
5.31 2.8130e-03
6.06 1.8698e-03
6.66 1.3991e-03
Iter
31
102
147
162
reversed selection
6
Cond
H -error
73.67
6.2726e-03
1.79e+03 3.7076e-03
2.14e+03 2.3687e-03
2.41e+03 1.8248e-03
lar selection at the same discretization level. We can conclude that the selection of nonmortar
sides is important in obtaining a good performance of the Neumann-Dirichlet preconditioner
while the selection of mesh sizes is not. A good mortar approximation can also be obtained
from the regular selection.
In Table 5.4, we present the scalability of the algorithm as the increase of the number of
subdomains when the local problem size is fixed. The number of iterations and the estimated
condition numbers are shown. We perform tests varying the type of meshes; uniform meshes,
non-uniform meshes, and reversed non-uniform meshes. In addition we perform tests varying
the selection of the nonmortar sides; the regular selection and the reversed selection. The
results also show that the performance depends on the selection of the nonmortar side and
seem to be not affected by the type of meshes. Except the reversed selection of the nonmortar
side, all gives a good scalability in terms of the number of subdomains.
In conclusion, the FETI-DP algorithm works efficiently for elliptic problems with jump
coefficients when the selection of nonmortar sides satisfies Assumption 4.5.y Ity even gives
smaller condition numbers and the number of iterations than the case with [< i § " L d > , see
Figure 5.3.
Acknowledgments. The author is grateful to Professor Chang-Ock Lee at KAIST in
South Korea for the valuable discussions and also thanks to the anonymous referees for the
helpful comments.
ETNA
Kent State University
etna@mcs.kent.edu
A FETI-DP PRECONDITIONER FOR MORTAR METHODS IN
ˆZ‰
119
TABLE 5.4
The number of CG iterations (Iter) and corresponding condition numbers (Cond) of the FETI-DP
.C algorithm
#D AE ; top
as the increase of the number of subdomains for the model problem with a discontinuous coefficient B
(varying the type of meshes), bottom (varying the selection of the nonmortar).
* %* Á&
1ƒU
àFU
)
/
U
U
uniform
Iter Cond
14
6.11
14
5.63
15
5.73
1ƒU
àFU
)
/
U
U
nonuniform
Iter Cond
12
4.15
14
5.03
14
5.10
regular selection
Iter
Cond
12
4.15
14
5.03
14
5.10
reversed nonuniform
Iter
Cond
10
2.96
13
4.00
13
4.04
reversed selection
Iter
Cond
31
73.67
52
92.26
56
91.60
REFERENCES
[1] F. B. B ELGACEM , The mortar finite element method with Lagrange multipliers, Numer. Math., 84 (1999),
pp. 173–197.
[2] F. B. B ELGACEM AND Y. M ADAY , The mortar element method for three dimensional finite elements, Math.
Model. Numer. Anal., 31 (1997), pp. 289–302.
[3] D. B RAESS , Finite Elements, Cambridge University Press, Cambridge, 1997.
[4] N. D OKEVA , M. D RYJA , AND W. P ROSKUROWSKI , A FETI-DP preconditioner with a special scaling for
mortar discretization of elliptic problems with discontinuous coefficients, SIAM J. Numer. Anal., 44
(2006), pp. 283–299.
[5] M. D RYJA AND O. B. W IDLUND , A FETI-DP method for a mortar discretization of elliptic problems, in
Recent Developments in Domain Decomposition Methods (Zürich, 2001), Lect. Notes Comput. Sci.
Eng., Vol. 23, Springer, Berlin, 2002, pp. 41–52.
, A generalized FETI-DP method for a mortar discretization of elliptic problems, in Domain Decom[6]
position Methods in Science and Engineering (Cocoyoc, Mexico, 2002), UNAM, Mexico City, 2003,
pp. 27–38.
[7] C. FARHAT, M. L ESOINNE , P. L E TALLEC , K. P IERSON , AND D. R IXEN , FETI-DP: a dual-primal unified
FETI method. I. A faster alternative to the two-level FETI method, Internat. J. Numer. Methods Engrg.,
50 (2001), pp. 1523–1544.
[8] C. FARHAT, M. L ESOINNE , AND K. P IERSON , A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl., 7 (2000), pp. 687–714.
[9] C. K IM , R. D. L AZAROV, J. E. PASCIAK , AND P. S. VASSILEVSKI, Multiplier spaces for the mortar finite
element method in three dimensions, SIAM J. Numer. Anal., 39 (2001), pp. 519–538.
[10] H. H. K IM , M. D RYJA , AND O. B. W IDLUND , A BDDC algorithm for problems with mortar discretization,
Technical Report 873, Department of Computer Science, Courant Institute, New York University, 2005.
[11] H. H. K IM AND C.-O. L EE , A preconditioner for the FETI-DP formulation with mortar methods in two
dimensions, SIAM J. Numer. Anal., 42 (2005), pp. 2159–2175.
[12] A. K LAWONN AND O. R HEINBACH , A parallel implementation of Dual-Primal FETI methods for three
dimensional linear elasticity using a transformation of basis, SIAM J. Sci. Comput., 28 (2006), pp. 1886–
1906.
[13]
, Robust FETI-DP methods for heterogeneous three dimensional elasticity problems, Comput. Methods
Appl. Mech. Engrg., 196 (2007), pp. 1400–1414.
[14] A. K LAWONN , O. R HEINBACH , AND O. B. W IDLUND , Some computational results for Dual-Primal FETI
methods for elliptic problems in 3D, in The 15th International Conference in Domain Decomposition
Methods, Berlin, 2003.
[15] A. K LAWONN AND O. B. W IDLUND , Dual-Primal FETI methods for linear elasticity, Comm. Pure Appl.
Math., 59 (2006), pp. 1523–1572.
[16] A. K LAWONN , O. B. W IDLUND , AND M. D RYJA , Dual-primal FETI methods for three-dimensional elliptic
problems with heterogeneous coefficients, SIAM J. Numer. Anal., 40 (2002), pp. 159–179.
[17]
, Dual-primal FETI methods with face constraints, in Recent Developments in Domain Decomposition
Methods (Zürich, 2001), Lect. Notes Comput. Sci. Eng., Vol. 23, Springer, Berlin, 2002, pp. 27–40.
[18] J. L I AND O. W IDLUND , FETI-DP, BDDC, and block Cholesky methods, Internat. J. Numer. Methods Engrg.,
ETNA
Kent State University
etna@mcs.kent.edu
120
H. H. KIM
66 (2006), pp. 250–271.
[19] D. S TEFANICA , A numerical study of FETI algorithms for mortar finite element methods, SIAM J. Sci.
Comput., 23 (2001), pp. 1135–1160.
, FETI and FETI-DP methods for spectral and mortar spectral elements: a performance comparison,
[20]
in Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM01) (Uppsala), vol. 17, 2002, pp. 629–638.
[21] A. T OSELLI AND O. W IDLUND , Domain Decomposition Methods—Algorithms and Theory, Springer Series
in Computational Mathematics, Vol. 34, Springer, Berlin, 2005.
[22] B. I. W OHLMUTH , A residual based error estimator for mortar finite element discretizations, Numer. Math.,
84 (1999), pp. 143–171.
, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal.,
[23]
38 (2000), pp. 989–1012.
[24]
, Discretization Methods and Iterative Solvers Based on Domain Decomposition, Lect. Notes Comput.
Sci. Eng., Vol. 17, Springer, Berlin, 2001.
[25] J. X U AND J. Z OU , Some nonoverlapping domain decomposition methods, SIAM Rev., 40 (1998), pp. 857–
914.