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Electronic Transactions on Numerical Analysis.
Volume 32, pp. 173-189, 2008.
Copyright  2008, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
NUMERICAL ANALYSIS OF STOKES EQUATIONS
WITH IMPROVED LBB DEPENDENCY
M. WOHLMUTH AND M. DOBROWOLSKI
Abstract. We provide a-priori bounds with improved domain dependency for the solution of Stokes equations
and the numerical error of an approximation by conforming finite element methods. The domain dependency appears
primarily in terms of the LBB-constant , and several previous works have shown that degenerates with the aspect
and and improve most
ratio of the domain. We explain the LBB dependency of common a-priori bounds on
of these estimates by avoiding a global inf-sup condition and assuming locally-balanced flow, which is in particular
. In this case, all error bounds on
and
, except for
prove to be
satisfied if
completely independent of .
!
Key words. LBB-constant, inf-sup condition, Stokes equations, a-priori estimates, finite elements
AMS subject classifications. 65N30, 76D07
1. Introduction. Let
equations are given by
"
be a bounded domain of Euclidean
.0/212354
6 '87
in "9+;:,<>=
1 '@A
1 '@?
#%$ , (
& '*),+-
. The Stokes
in "9+
on BC"9D
This system of partial differential equations describes the special case of a stationary, laminar,
and viscous flow in " with velocity 1FEHGJIK #L$ and inner pressure 6FEGJIMK # . Here, 7 models
boundary effects and external forces like gravity. By neglecting any physical parameters (for
instance viscosity) we assume them to be constant in " and set them equal to 1. In our case,
any in- and outflow is described by sources and drains, which are modeled by the function ? .
this paper, we adopt the standard notation for Lebesgue and Sobolev spaces
NPO E " Throughout
I and QSR E " I ; see Section 2.1. QUV T E " I denotes the closure of WYV X E " I in QUT E " I . For
the analysis of the Stokes equations we need the spaces
]
Z '@Q V T E " I $ +;[\' N VO E " I ' 6^K N O E " U
I __ 6cbG 'dAef+
__ `a
g
N
O
equipped with the -norm and QhT -seminorm
i 6 i Oj
'
i6 iO
'
6 O b G +
` ack k
ilmion
'
l
k k
T '
i 4 Cl i
D
The weak formulation of Stokes equations is defined as follows: For 7
Zqp [ , such that
seek EH1 + 6JIK
(1.1a)
z
(1.1b)
Er4s1 + 4 l IF.tE :,<>=
E :,<v=
K N O E " I ,? K [
,
l m6 I E l I u l K Z
+ ' 7J+
+
1 +xw I ' E J? +xw Iyu w K g[ D
Received November 29, 2007. Accepted for publication July 14, 2008. Published online on April 29, 2009.
Recommended by T. Apel.
Department of Computer Science 10 and Erlangen Graduate School in Advanced Optical Technologies (SAOT),
Friedrich-Alexander University Erlangen-Nuremberg, Cauerstr. 6, D-91058 Erlangen, Germany
(wohlmuth@informatik.uni-erlangen.de).
Department of Mathematics, University Wuerzburg, Am Hubland, D-97074 Wuerzburg, Germany
(dobro@mathematik.uni-wuerzburg.de).
173
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174
M. WOHLMUTH AND M. DOBROWOLSKI
The existence of a weak solution is connected to the so-called LBB-constant
defined by
N ' N E" I
E :,<v= l + 6JI
N
‚
j
„

†
ƒ
(1.2)
' €~ <v|,} ‡ˆ n l i 6 i D
{
T
i
lCk i k ‰ l
l Z
5
N
‹
‰
Š
, which follows from :†<>=
for all K
.
We remark that
T
The following results stem from Galdi’s book [7], which
uses
previous
works by Bogovk kŒ
N
skij [2, 3] and Gagliardo [6]. In [7] it is proved that
A for a large class of domains,
including Lipschitz domains, which by a simple variational analysis (see [8]) implies that the
weak solution of (1.1a) and (1.1b) is uniquely determined. Furthermore, it is proved that for
domains " , which are star-shaped with respect to a ball of radius Ž ,
Ntt
‘P’ E " I $ +
Ž”“
where ’ E " I denotes the diameter of "9D This lower bound is very convenient in sphere-like
cases, where ’ E " I and Ž are of similar size. Unfortunately, many common domains in fluid
mechanics, such as large and flat water reservoirs, the atmosphere or tubes and pipelines,
have a huge aspect ratio. The results in [4, 5] show that the LBB-constant is not only allowed
to, but in fact does degenerate with increasing aspect ratio.
N
T HEOREM 1.1 (See [4]). Let " be a fixed domain with LBB-constant and • K #%$
with
Š '–• ‰ •F— ‰ • +
$
T
Š‰t˜P‰ &™D
Let further "›š be the associated stretched domain, defined by
]Fœ
œ
œ
K # $ž ‘ T Ÿ+ DoDŸDo+ $ K "2ef+
• T
• $ “
N
with aspect ratio proportional to '–• . Then the LBB-constant š of " š satisfies
$
N ‰tN ‰  E " I
š
D
p E A†+x I , ¡t¢8# $£ T , and plates
Special
cases
of this theorem are channel domains "('@¡
p
¥
Š
¤
"('(¡ ZsE §A†+ Z I ¢–#C¦D §
¢¨[ be finite-dimensional spaces depending on a discretization
Let
and [
Œ ¢ A . For
parameter ©
the analysis of the standard finite element approximation of the Stokes
Zf§ +x[ § , a discrete LBB-constant is defined by
equations in the spaces
E :†<>= l § + 6 § I
§
N
j
„

¬
ƒ
n
(1.3)
' ~Ÿª«<v|, } ª ‡­ª ª l § i 6 i D
T
ˆ®¯Œ A ,
k
k
The uniform§ stability of the method is equivalent to the existence of a constant
N
°

N
€
¤

®
such that
; see Lemma 4.1. On the other hand, if the finite element method is
§ N § ‰(N . Hence, for a uniformly stable and convergent method
convergent, we have ±v<>²t<>|†}
N
 ® ‰ ±v<>²t§ <>|†} N § ‰(N D
"Mšf'
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
175
This inclusion implies that for stable methods small discrete LBB-constants are only caused
N
by small .
The algebraic aspect of a small LBB-constant is visible. Most of the methods for solving the discrete linear system (Uzawa, CG for the Schur complement, etc.) deteriorate for
domains with high aspect ratios.
But the analytical aspect of a small LBB-constant is not visible in practice. For instance,
p E A†+x I , we usually obtain similar errors for
if we solve the Stokes equations on a channel ¡
the velocity for all in spite of the fact that standard finite element error estimates depend on
N § or N ; see [8].
The object of the paper is to refine the standard error estimates
N by avoiding the
N O global
LBB-condition (1.3). In Section 5, error estimates independent of in the Q T . and . norm
are given for the velocity 1 . From these estimates it follows that the error estimate for the
N
NMO depends
pressure
gradient is also independent of . Only the estimate for the pressure in
N
on which is also confirmed by numerical experience.
N
For error estimates independent of using only the data 7 and ? , refined regularity
results are required. This is carried out in Section 3 for locally balanced flows (see Definition
3.4) which include the common case ?c'‹A¬D For locally balanced flows it can be proved that
N
most quantities in the standard regularity results are independent of , the only exception is
N
O
the -norm of the pressure; see Corollary 3.10. In an example in Section 3.3 it is shown
that, if the condition of a locally balanced flow is violated, then the estimates for the velocity
N
depend on D
2. Technical preliminaries.
Notation. We use the following standard notation for Lebesgue and Sobolev spaces
N O E " 2.1.
I and Q R E " I , together with their related (semi-)norms.
i­lCi O
l O b«G
'
+
a
` afk i k 4 š lCi O
i­lmi O
µ
'
+
R›³ a š,´ ¶
a
R i lCi O
l O
4 š
µ
'
´
D
R›³ a š«·
a
k k
R
Here Q V T E " I denotes the closure of W V X E " I in Q T E " I . Furthermore, we use
Z¸ 'º¹ l K Z E :,<v= l +xw I '(? E w I»u w K [½¼P+¾? K [Y¿rD
k
We assume "–¢À#C$ to be a bounded
domain with diameter ’ E " I and Lebesgue measure
Á E " I . If we omit the domain index by writing i€ÂÃi + i€ÂÄi +  , we always assume " to be
R
R
the considered domain.
k
k
We denote the dual space of Q V T E " I by Q£ T E " I equipped with the negative norm
El I
i i
7 £ T ' ‡ˆ«„ÅMƒ†Æ … 7 l D
ǀÈ
aÊÉ k k T
Â
i¥ÂÄi
Z
2.2. Poincaré’s inequality. Note that
is equivalent to the standard norm
in
T
T
by Poincaré’s inequality.
k k l K Z
L EMMA 2.1 (Poincaré’s inequality). For all
we have
i­lCi ‰tË l
T +
Ë

 Ë is independent of the aspect
k
k
where
only depends on the smallest dimension of " , i.e.,
p
ratio of " and is finite for "('(¡
# , ¡t¢8#L$£ T .
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176
M. WOHLMUTH AND M. DOBROWOLSKI
2.3. The negative norm of 4
6 . The LBB-condition (1.2) also reads
N i6 i ‰ i ›
. 4
6 i ' i 4
6 i
u 6ÌK [P+
£ T
£ T
Z as a functional in terms of
where we understand .›4
6^K
¿
l K Z¯Í.JÎÏE l 6mI
:,<v= + D
According to the Riesz representation theorem, the functional .›4Y6 can be represented by a
Z satisfying
function Ð K
EÑ4 ÐY+ 4 l I ' .›4
6FE l I ' E :,<v= l + 6mI + Ð ' i 4
6 i D
£ T
T
k
k
Hence, the supremum in (1.2) is attained at Ð . This yields
N i 6 i ‰ E :†<>=ÐY+ 6JI ' i 4
6 i ' Ð D
£ T
T
Ð T
k
k
k k
˜ Š +oDŸDoD„ÒÊ+ are
L EMMA 2.2. If "—+ '
disjoint open subsets of " then
Ó
i iO
i iO
´ 4 w £ Tx³ ‰ 4 w £ Tx³ u w K [gD
aÊÔ
a
—· T
Z E "— I  '–Q V T E "— I $ and Ð K Z be the solutions of the problems
Proof. Let ÐM— K
EÑ4 ÐM—„+ 4fÕmI ' E :,<>= Õ +w I
uCÕSK Z E "— I + EH4 ÐY+ 4fÕmI ' E :†<>= Õ +w I»uLÕ^K Z D
a Ô
aÔ
Using (2.1) and extending the functions Л— by A , we obtain
Ó
Ó
Ó
Ó
i4 iO
O
«
b
G
4 Ð 4 ÐM— bG
´
w £ T³ ' ´ ÐM— T­Ö ' ´
w×:,<v=З ' ´
¬
a
Ô
Ê
a
Ô
—· T
—· T k k
— · ` aÊÔ
— · T ` a¬Ô
Ó T
Ó
Š
Š
Š
‰ Ð O 3 ´ Ð — O ' i4 wiO 3 Š ´ i4 wiO
£ T ) —·
£ Tx³ a Ô D
Tx³ a Ô )
) T ) —·
l
k k
T k k
T
2.4. The solvability of the equation :,<v=
'87 .
(2.1)
T HEOREM 2.3 (Bogovskij’s Theorem, [3, 7]). Let
"t'ÀØÚٗ · T " —+;Û ÀŠ +
where each " — is star-shaped with respect to some open ball Ü — with
l Ü
the smallest radius of the balls Ü — . Then, for any 7 K [ there exists K
:,<>=
l
where the constant
Ý
k k
l
'–7J+
‰5Ý i 7 i +
T
obeys the upper bound condition
 Ý ‰( V W ‘ ’ E " I $  C‘ Š 3 ’ E " I +
Ž “
”
Ž “
”
" — , where Ž
Z — ¢\
satisfying
is
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY

177

with constants V ' V E & I and W‹'ÀW E Á E " — II . An explicit expression for W is given in [7].
Proof. See [7, Chapter III.3].
Bogovskij’s Theorem 2.3 bears directly on the divergence operator and the solvability of
the constraint :†<>= 1 '@? , whereas the LBB-condition is related to its adjoint operator, namely
the gradient .›4 , and is essential for the existence of a pressure function 6 . As we show in
the following, these two points of view are mutually equivalent.
Z Zf¤Fß«àŸá E :,<v= I ,
Considering the divergence and gradient operators on the quotient space Þ =
we obtain
â
'–:†<>=  Z Þ Î g[ +
¿ ' .›4  [ ¿ Î Z Þ ¿ +
â
â
where both
â
â
Since
and
¿
are bijectiveâ operators with norms
â
â
and
¿
i i
' oÝ +
i
Š
N D
¿ ' s
i
are adjoint operators, these norms have to be equal,
Š
N ' Ý D
(2.2)
Ý
Equation (2.2) is noteworthy since the domain dependency of
(expressed by its upper
N
bound in Theorem 2.3) transfers to .
By definition, the LBB-condition assures that the gradient .›4 has a closed range ã EH4fI .
According to the closed range theorem, ã E :,<v= I is also closed and given by
Vä
ã E :,<>= I ' ßàoá EÑ4fI ' [g+
ß àoá ÑE 4fI V denotes the annihilator of ß«àŸá EÑ4fI . With these results in mind, one can easily
where
prove the unique solvability of the Stokes problem, as for example done in [7]. We omit these
details and continue with deriving some a-priori bounds.
3. Regularity results.
3.1. Standard regularity results. For ? K [ , according to Theorem 2.3 there exists
1 ¸ K Z with
i i
:,<>= 1 ¸ '@?J+ 1 ¸ T ‰5 Ý ? D
k for given ? K [ can be split into 1 '
Then, the solution 1 of the Stokes equationsk (1.1)
1 V 31 ¸ with 1 V K Z V . By the Riesz representation theorem such a 1 V exists and is uniquely
defined by the solution of
EÑ4s1 V + 4 l I ' E 7J+ l IÚ.tEÑ4s1 ¸ + 4 l I + u l K Z V D
Taken together, we obtain the following estimate for 1 ,
T
1 ‰ 1 V 3 1 ¸ k k
T
T
T
k k ‰t k Ë ik 7 i 3 k ) k Ý i ? i D
Now (2.2) provides the same bound in terms of the LBB-constant,
(3.1)
1
k k
T
‰5Ë i 7 i 3 N ) i ? i D
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178
M. WOHLMUTH AND M. DOBROWOLSKI
The next inequality follows immediately from (1.2), (1.1a), and (3.1). It shows the LBB
dependency of the pressure bound.
i 6 i ‰ Š „ƒ†n … E :,<v= l l + 6JI
N ˆ‡ 
T l
l
‰ N Š „ƒ†n … EH4sk1 + k 4 l ÚI .(E 7J+ I
‡ˆ
T
Š
‰ N E 1 3 Ë i 7 k i k I
T
k k

‰ N ‘  Ë i 7 i 3 NŠ i ? i D
“
 )
We cite, without proof, important results on regularity of higher order, å
found in [7].
O
T HEOREM 3.1 ( å -Regularity). If 7 K Q R £ E " I +ˆ? K Q R £ T E " I and
1. " is a convex polygon (&Ì'À) and å'À) ) or
2. BC" K W R ,
then the Stokes problem is å -regular, i.e., it obeys
i1 i
R
3 i6 i
, which can be
 i i
i i
‰ N\
æ 7 R £ O 3 ? R £ Tç D
R £ T
In the following,i especially
case å¾'q) is important to us since it is used in the
§ i . Allthecommon
a-priori estimates of the solution EH1 + 6JI depend
duality approach on 1c.U1
N
N
on ; therefore, we now want to elucidate the domain dependency of .
3.2. Refined regularity results. We first define a modified 2-regularity that allows us
N
to give a 2-regularity estimate which solely depends on local LBB-constants — , i.e.,
N we
may think of " as an open covering of its parts " — , chosen such that all constants — are
independent of the aspect ratio of " .
D EFINITION 3.2 (Local 2-Regularity). Let ¹F
Þ" —è¼ be an open covering of " , such that "M— ,
Þ" —ÊêÌ" , has diameter ’ E "M— I and is star-shaped with respect to a ball
which is given by "M—é' Þ" —è¼ . The open
of radius Ž9— . Furthermore, let ¹ Õ —ë¼ be a partition of unity with respect to ¹ covering ¹éÞ" — ¼ of " is called locally 2-regular if and only if the Stokes problems
.0/21 — ì
3 4Y6 é— '87€—
:,<v= 1 é— '(?«—
1 — '–A
in "—+
in "—ë+
on Bm"
are regular in the following sense: There exists a constant
and ?«— K Q T E "— I ês[ E "— I the weak solution E1 —„+ 6 — IK Q V T
Q T E "— I and satisfies
—+
 — , such that for all 7— K PN O E " —I$
E "— I $ pcN VO E "— I is in Q O E "— I $ p
i1 iO 3 i6 i
i i
i i
— ³
— T ³ ‰( — E €7 — 3 «? — xT ³ I D
a Ô
a Ô
aÔ
a Ô
p E A†+­í I , í K(î , we can choose
R EMARK 3.3. For instance, for a channel domain ¡
p
˜
Š
˜
Š
t
‰
P
˜
‰
" —é' ¡ Þ E . + 3 I +€A
í , where ¡t¢
¢ ¡›Þ D Then ²sïð†—  — is independent of í .
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
179
Later on, the next definition proves to be a good criterion for an improved LBB dependency.
D EFINITION 3.4 (Locally Balanced Flow). The Stokes problem has locally balanced
flow if and only if there exists a partition ¹é
ñ" —è¼ of " with LBB-constants N — independent of
the aspect ratio of " and ? satisfies
` a¬Ô
? b G 8
' A
u "—ëD
We now discuss a lemma about the class of Stokes problems with locally balanced flow to
emphasize its advantage.
L EMMA 3.5. If the Stokes problem has locally balanced flow, then 1
T can be estimated
as follows:
k k
‰( E i 7 i 3 i ? i I +
LB
T
k
k

N
oË
where LB depends solely on " — and — , used in Definition 3.2, and Poincaré’s constant .
l 1
Proof. Setting '
, w
' 6 in (1.1) results in
1 O ' E 7J+ 1CI%38E ?m+ 6mI D
T
k k
60bG in the following derivation.
The conditions on ? allow us to insert 6 —  ' ò È T
a Ô ÉLó a Ô
E ?J+ 6JI '
6 ? bG '‹´
6 ? b«G '‹´
Eô6c.õ6 — I ? bG
—
—
`a
` a¬Ô
` a¬Ô
‰ ´ i 6s.ž6 — i i ? i D
a Ô a Ô
—
N VO E "— I , we apply the LBB-condition (1.2) for 6õ.S6 — on "— and then use
Since Eô6ž.ö6 — I
K
1
Lemma 2.2 and (1.1a) to obtain
Š i i i i
4
6
N — ?
£ Tx³ a Ô
a Ô
—
T„ù O
‰ ²sïð ‘ N Š i ? iY÷ ´ i 4Y6 i O
—
£ T³ a ÔÑø
—“
—
l
‰5 i ? i „ƒ†n … E :,<v= l + 6JI
‡ˆ
T l
l
‰5 i ? i „ƒ†n … EH4sk1 + k 4 l IÚ.(E 7J+ I
‡ˆ
‰5 i ? i E 1 3 Ë i 7 k i k TI D
T
k k
E ?J+ 6JI ‰ ´
Using the above inequality together with Young’s and Poincaré’s inequality we finally get the
following desired result
1 O ' E 7J+ 1mIL38E ?m+ 6mI ‰t LB ú i 7 i O 3 i ? i ŸO û D
T
k k
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180
M. WOHLMUTH AND M. DOBROWOLSKI
Later in this paper, we will need estimates for dual problems according to Aubin and
Nitsche. These problems, in general, do not have locally balanced flow. For these cases, we
provide the following definitions.

 E 7J+ë? I be defined by
D EFINITION 3.6. Let '
T
T
 ' i i 1 Ti i +
T E 7 k3k ? I
for given Stokes problems with data 7J+ë? and solution 1 + 6 .
Ÿü
ü
D EFINITION 3.7 (Worst Case Regularity Constant E " I ). Let E " I be the supremum

of the constants with respect to all possible Stokes problems in " .
T
oü
R EMARK 3.8. In general, E " I is bounded by þ ý , yet in the case of a simple channel,
the example of Section 3.3 suggests the dependency
 ü E channelI ä ÿ Š D
N
In case of locally 2-regular domains, the following theorem shows that the LBB dependency appears only in terms of 1 .
T " is locally 2-regular according to Definition 3.2, the
T HEOREM 3.9. Assuming
that
k
k
Z
p
weak solution EH1 + 6JIK
[ V of the Stokes problem
.0/21×3 4
6 –
' 7J+;:†<>= 1 '(?J+
p Q T E " I . Furthermore, we obtain
O
with boundary condition 1 '–A on BC" is in Q E " I $
i 1 i 3 6 ‰( E i i 3 i i 3 1 I
O
7
? T
T N
T +

k k .
k k
with independent of the LBB-constant
Õ
Proof. Let ¹ —„¼ be the partition of unity from Definition 3.2. Then, we define 7—+ë?«— as the
60bG :
data of the Stokes problem with solution E1CÕ —+ Eô6s.ž6 — I„Õ — I on "— and 6 —  ' ò È T
a Ô ÉLó a¬Ô
.0/õE1CÕ — I%3 4SE„Eô6c.õ6 — IÕ — I '87 — +
:,<v= E1CÕ — I '(?«—„D
Then 7€—+ë?«— satisfy the bounds
i i ‰
7€—
i i ‰
?— T
PN VO E "— I , i 6c.ž6
Since E6f.õ6 — I›K
as follows:
i 6s.ž6 i
—
‰ NŠ
—
a¬Ô
Š
' N —
i i 3  i1 i
3  i 6s.ž6 — i +
7
Tx³
a Ô
 i 1 a i Ô 3  i ? ai Ô D
T³ a Ô
Tx³ a Ô
i
— can be estimated using a local inf-sup condition on "›—
a¬Ô
E l + 4SE6f.ž6 — I„I
„ƒ¬
l
‡ˆ«Å Ç Æ È
a¬ÔÉ l k k T
E +­7 35/21mI ‰ Š 1
i i û
„ƒ¬
l
N — ú Tx³ 3  Ë 7
D
‡ˆ«Å Ç Æ È
Ô
a¬Ô
a¬Ô
T
k
k
a¬ÔÉ
k k
Now, using Definition 3.2, we have
i 1CÕ i O 3 i E6f.ž6 I„Õ i O ‰  i i O 3 i 1 i O 3 i i O û
— O³
— — T
ú 7
xT ³ ¬a Ô ? T ³ Êa Ô D
a¬Ô
Êa Ô
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
181
˜
Summation over results in
i1 iO 3 6 O ‰
O
´
T
—
k k
‰ ¿
‰ 
i iO
i
ú C1 Õ O ³ 3
a Ô
i iO i iO
ú 7 3 ? T 3
i iO i iO
ú 7 3 ? T 3
E6s.õ6 — „I Õ — i O û
T
i1 iOû
T
O
û
1
T +
k
k
   E Š 3  ~ I O independent of the LBB-constant N and hence, also independent of
with '
¿
the aspect ratio of the domain " .
Clarified by Theorem 3.9 and purged by virtue of Lemma 3.5 we finally obtain the primary result of this section as follows.
C OROLLARY 3.10. If " is locally 2-regular and the Stokes problem has locally balanced
flow, then the estimate
i 1 i 3 6 ‰t E i i 3 i i I
O
7
? T
T
kN k
holds independent of the LBB-constant and the aspect ratio of " .
Proof. This corollary follows immediately from Lemma 3.5 and Theorem 3.9.
R EMARK 3.11. Besides its obvious advantage of estimating 1 and 6 , we obtain an
important application for the case ?×'–A . This allows us to use the dual problem according to
§
N O
Aubin and Nitsche with data 7U' 1f.1 , ?õ' N A , and hence to refine the -approximation
O
of 1 . Unfortunately, a similar approach to the -approximation of 6 would require a dual
§
problem with 7@' A and ?ì' 6^.h6 . Since we cannot assume this dual problem to have
locally balanced flow, the improvement fails at this point.
The following example shows a Stokes problem which does not have locally balanced
flow. Yet even in this case an explicitly calculated bound for 1
T in combination with Theorem 3.9 delivers an improved LBB dependency.
k k
3.3. Example: Free channel flow. We now take a look at an example without locally

N
balanced flow, i.e., with LB depending on . For the sake of simplicity,
we choose the
p
Š
O
E A†+ I E A¬+ I ¢¨# and the subdomain " V ' æ £ O Ç +x . £ O Ç p
"
'
stretched
domain
ç
E A¬+ Š I ¢‚# O according to Figure 3.1. In particular, we are interested in very long domains
with V . Since fluids are almost incompressible, a realistic possibility of a non-vanishing
constraint term ? is in the form of sources ? and drains ? to model inflow and outflow.
£
Obviously, these sources and drains mainly appear at the ends of the channel and thus, are
separated by a distance of about the aspect ratio V . Here, source ? and drain ? £ can be
of arbitrary shape, but we assume ? to vanish outside the subdomain " and ? to vanish
£
outside the subdomain " . Thus we get
£
?s'
and ?
£
?+
in "+
?£ +
?V+
in "
£ +
in " ñ V +
and ? satisfy the relation
` a
œ
? £ bJEG + I '
`a
œ
? £ bJEG + I ' .
`a
?
œ
bJEG + I ' .
` a
?
œ
bÊEHG + I +
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M. WOHLMUTH AND M. DOBROWOLSKI
" V
"
source ?
" £
drain ?
£
V
F IG . 3.1. Stokes problem with maximally separated in- and outflow on a channel domain with aspect ratio
and for long channels.
where the chosen nomenclature suggests predominantly positive values for the source ? and
negative values for the drain ? .
£

Note that the chosen case promises a very large constant LB , because it has a constraint
term ? that vanishes on most parts of the channel domain " . Consequently, the support of
? does not increase together with the integration area and the aspect ratio of " , when the
length of the channel is increased. In fact, we will see that the enlargement of the integration area lets 1
with the aspect ratio, which has to be compensated by an LBB
 , since ? is chosen independent ofœ V . œ
T increase
œ
dependency ofk the
constant
LB
k
1
†
H
E
G
I
1
†
E
I
1
H
E
G
œ
œ
Within this setup, we want to model a force-free flow
+ '
,
+ I A in
V" with zero boundary conditions on both vertical ends ( ' A ' Š ) of the channel. At
both horizontal ends of " V we only require
Ç
1b# '8A†D
`"Ÿ! a
The latter is satisfied since the flow profile does not change along the channel. In order to
obtain zero boundary conditions all over BC" we add domains "$ and "
with appropriate
œ boundaryœ £ conditions on BC"
source ? and drain ? to model the connection between zero
£
and the flow profile at both ends of " V . By virtue of 1FEHG + I ' E1ÊE I +xA I&% we are able
to explicitly calculate the solution of the present Stokes equations in " V . The first Stokes
equation
.0/2123ì4
6 '8A
simplifies to
œ
1 E éI 3ì4 6 8
' A†+
4 6 '8A†+
œ
4 ­6 '8A†+
'( .›4)*'o1¬E IL3 )
where we assumed 1×K W ¦ E " I + 6^K W O E " I . Hence, by virtue of 1ÚE A I ' F1 E Š I '8A
œ
œ œ
1 ' 1 E I ,
' + E . ŠI+
6 ' 6FEGJI ' . ) + G23  +
.›4
'
, we have
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
183

for any desired flow rate + and constant pressure . That is, we obtain a quadratic flow profile
1 independent of G and a pressure 6 depending linearly on G . Therefore 1 Ç is proportional
ÿ
T³
to V :
k k a
By this result, we conclude
Š
 bG ä V ä s
1 O Ç ‰
N
D
Tx³ a
Ç
`a
k k
Š
1 O ä ä NsD
T
k k
i i i i
N
Since 7 + ?
T are independent of , we obtainŠ
 ä ÿ D
N
T
Hence, even in this disadvantageous case an improved 2-regularity result holds by virtue of
Theorem 3.9:
i1 i 3 6 ‰ ÿ  Ei i 3 i i I
O
? T +
N 7
T
k k
where the LBB dependency reduces from þ T to - þ T .
Having established all these results, we now concentrate on a discrete approximation of
Stokes equations by a finite element method and its associated a-priori error bounds.
4. FEM setup. In this section, we are concerned with the approximation of weak Stokes
equations (1.1) by a conforming finite element method (FEM), i.e., we would like to deter§ § Z § p [ § of
mine the solution EH1 + 6 IK
EÑ4s1 § + 4 l § ™I .tE :,<>= l § + 6 § I ' E 7J+ l § I u l § K Z § +
E :,<v= 1 § +w § I ' E ?m+w § I u w § K [ § +
(4.1b)
Z § ¢ Z +[ § ¢\[ . In the following, all constants including
with finite dimensional spaces

the generic constant are assumed to be independent of © .
(4.1a)
Similar to the continuous problem, a discrete LBB-condition ensures the solvability of
Z § +[ § I
the constraint (4.1b) and the existence of a discrete solution of the FEM. Therefore E
is said to be uniformly stable, if there exists a constant å , such that (see [8])
E :,<v= l § +xw § I
§
t
‰
N
j

†
ƒ
(4.2)
A/.5å
' 0誫<v|, } ª ‡ ª  n ª l § i w § i D
T
k
k
The following lemma (see [8]) provides a criterion for the existence of such a constant å
,
although we only need its statement in one direction, namely to ensure the discrete LBB§ K329E Z + Z § I ,
condition. For § this purpose, we consider a linear approximation operator
1
§
Z
Z
Z
Z
2 E +
I denotes the space of linear operators from to . This operator 1 § K
where §
Z
Z
2 E +
I is used in the following lemma and will be an essential part in the derivation of the
a-priori estimates in Section 5. Its properties are specified in (5.2a) and (5.2b) in detail.
L EMMA 4.1 (Discrete LBB Condition). Assume the inf-sup condition holds for the
continuous Stokes problem (1.1). Then the following two statements are equivalent:
i) The condition (4.2) holds.
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184
M. WOHLMUTH AND M. DOBROWOLSKI
§ K29E Z + s
Z § I and a constant  ® satisfying
E :,<v= E l . 1 § l I +xw § I @
' A¬+
l ‰5 ® l
§
1
T
T +
l K Z § K §
k k
k k
for all
,w
[ .
§
§
˜
˜
˜
5
I
4
I
K
Proof. To prove
, we have, for any w
[ ,
l
l
l §
§
§
§
§
„ƒ†n … E :,<v= l § +xw I  „ƒ¬n … E :†<>=61 § l +w I ' ƒ†n … E :,<v= § l +w I
‡­ª€ ª
‡ˆ
‡ˆ
1
1
T
T
T
l
§
N
Š
k k
k
k
k
k
E
I
i
i
 „ƒ¬n …  ® :,<v= l +xw   ® w § D
‡ˆ
T
˜
˜
˜
k k
I can be found in [7].
The proof of I (
ii) There exists an operator 1
With these preparations we can now approach the a-priori estimates of the approximation
errors of the FEM.
5. A-priori error bounds. In this section, we investigate the convergence of the discrete
FEM solution. Hereby, the error relations
EÑ4SE1ž.U1 § I + 4 l § I ' E :,<v= l § + s
6 .ž6 § »
I u l
E :,<v= E1ž.U1 § I +w § I 8
u w
' A
(5.1a)
(5.1b)
§ K Z §+
§ K [ §+
which we obtain by subtracting the FEM formulation (4.1)§ from the
§ Stokes problem (1.1), are
an important tool to estimate error bounds for both 1s.U1 , 6s.ž6 and their derivatives. The
derived results are independent of the chosen conforming FEM as long as it is based on a uniformly regular discretization. For example, the latter is required to have approximation oper§ K79
2 E Z + Z § I was already used in Lemma
ators with the desired properties. The operator 1
4.1, and a similar one is needed for the pressure
spaces
[ § and [ § . We§ therefore assume
§
§
Z
Z
K3
2 E +
I +8 K39
2 E [g+[ I , and í max ”Š ,
that there exist approximation operators 1
which comply with the requirements
(5.2a)
E :†<>= E l .
(5.2b)
(5.2c)
l Z
for all K
,w K [
§ l I +w § I 8
' A
l . §l (
‰  © :
1
k w . 8 § w k 9 ‰( ©:
k
k9
1
l
u w§ K [ §+
kw k
k k
:
:
9
+ L'–A¬+ Š +
+5;L'–A¬+ Š +
u ís'@A¬+oDoDŸD+xí
max 5;
u ís'@A¬+oDoDŸD+xí
max
9
.
R EMARK 5.1. Note that Condition (5.2a), together with error relation (5.1b), involves
(5.3)
E :,<>= E 1 § 1c.h1 § I +xw § I '@A
u w§ K [ § D
The existence of such operators is discussed in [9], including an explicit expression for
these operators in case of low-degree finite elements. For example, in case of the MiniElement [1] one can use the approximation operator by Scott-Zhang [10] on each macro
element to obtain the properties (5.2b)-(5.2c) and locally add appropriate constants to fulfill
condition (5.2a).
In the following, we implicitly assume that " , and consequently 1 + 6 , provide sufficient
regularity. Especially, " is assumed to be a Lipschitz domain. This guarantees the existence
of a locally 2-regular open covering of " .
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
5.1. Approximation of 4c1 and 1 . The above setup, consisting of the refined regularity
estimates and appropriate approximation operators, allows us to state the following error
bounds for our discrete solutions.
T HEOREM 5.2. Let the conditions of (5.2) be fulfilled, then the discrete solution satisfies
1c.h1 §
‰( © : E 1
3 6 I +
T
: T
:
ki 1f.U1 § k i ‰( © 1ck .hk 1 § 3 k  k © : T 6 ‰( ©
T
 k independent
N :
k
In both cases the constants are
of k .k
T æ 1 3 6 ç D
: T
:
k k
k k
l§
'<1 § 1c.h1 § , and we have
Proof. We start by employing error relations (5.1) with
1c.h1 § O ' EÑ4SE1f.U1 § I + 4SEH1c. 1 § 1mII%3–EÑ4SE1ž.U1 § I + 4SE 1 § 1c.h1 § I„I
T
k
k ' EÑ4SE1f.U1 § I + 4SEH1c. 1 § 1mII%3–E :,<v= E 1 § 1c.h1 § I + 6JI D
:
Using (5.1), approximation properties (5.2b) and (5.2c), and finally Young’s inequality, we
obtain
1c.h1 § O ' ÑE 4SE1f.U1
T
k
k ‰ c
1 . 1 §1
‰ k Š 1c.h1 §
)
k
§ I + 4SEH1c.
1c.h1 §
T
kO k  O
3
T © :
k
§ m1 II%3–E :,<v= E 1 § 1c.h1 § I + 6s. 8 § 6mI
38E 1c. 1 § 1 3 1f.U1 § I i 6c. 8 § 6 i
T
T
T
k O k
k k
k
O
û
ú 1 : T 3 6 : +
k k
k k N O
which proves the first part of the theorem. To approach the -estimate and hence
prove the
Z p [ toaccording
second part, we define the dual problem with associated solution E ¡›+ ÕmIMK
1
to Aubin and Nitsche:
EH4 l + 4 ¡ IF.(E :†<>= l + ÕmI ' E1f.h1 § + l I
E :,<v=¡›+w I –
' A
(5.4b)
l 1f.U1 §
,
Here, we start with (5.4a) and '
i 1c.h1 § i O EH4SEH1f.h1 § I 4 FI .(E E1f.U1
'
+ ¡
:,<v=
(5.4a)
u l K Z +
u w K [PD
§ I + ÕmI +
and then employ error relation (5.1a) for the dual problem (5.4) and error relation (5.1b) for
the original Stokes problem (4.1) to yield
i 1c.h1 § i O HE 4SEH1f.h1 § I 4SE . § I„I
'
+ ¡ 1 ¡
§
38E :,<v=61 ¡›+ 6s.õ6 § ÚI .tE :,<v= E1c.h1 § I + Õc.
Using (5.2a) and (5.4b) results in
E :,<v=1 § ¡›+ 6s.ž6 § I ' E †: <>= E 1 § ¡ . ¡ I + 6s.ž6
' E :†<>= E 1 § ¡ . ¡ I + 6s. 8
Taken together, we obtain
i 1c.h1 § i O ‰ f
1 .U1
‰ k 1f.U1
k
§
T
§k
k
T
E ¡ . 1 § ¡ 3 i Õf. 8 § Õ i I%3
T
 ©k E ¡ O 3 k Õ éI 3  © : T ¡ O
T
k k k k
k k
¡ .
6k +
:
k k
8
§ ÕmI D
§I
§ 6mI D
1
§ ¡ i 6c.
T
k
8
§6 i
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M. WOHLMUTH AND M. DOBROWOLSKI
which finally reveals
i 1c.h1 § i O (
‰  i 1c.h1 § i æ © c
1 .h1 § 3 © : T 6 +
T
: ç
k
k
k
k
due to (5.2) and Corollary 3.10.
§ was obtained solely by using the properties of
As we have seen, the result for 1c.h1
i
§ i
T
the previously defined approximationk operators.
k However, the estimate of 1f.U1 requires
an auxiliary problem with homogeneous constraint ?½' A . Therefore we can eliminate the
N
dependency on i for arbitrary
? as soon as " is 2-regular. As it will become clear below, a
§ i would
similar result for 6s.ž6
require Corollary 3.10 without any assumptions on ? .
6
5.2. Approximation of and 4
6 . Firstly, we look for more promising norms to measure the pressure error with improved LBB dependency. Secondly, we use these results to
NgO
estimate the -norm of the pressure error.
Since the LBB-condition suggests
i 6s.õ6 § i ‰ Š i 4SE6s.õ6 § I i
N
£ T +
the negative norm seems to be a reasonable candidate for an LBB-free error bound. Indeed,
§ and obtain the following lemma for
we find a result comparable to the one for 1c.U1
N
O
T
V E" I.
arbitrary ? K
k
k
L EMMA 5.3. The approximation error of the pressure gradient 4
6 in its negative norm
can be estimated by
i 4SEô6s.ž6 § I i
‰t © : æ 6 3 1
D
£ T
:
: T ç
k k k k
Proof. Just by the definition of the negative norm, we obtain
l
§
„ ƒ¬n … E :,<v= +l 6s.õ6 I
'
£ T ‡ˆ
lT
l
l
‰ „ƒ¬n … ‘ E :†<>= k E k . 1 l § I + 6c.õ6 § I 3 E :,<v=1 § l + 6s.ž6 § I
‡ˆ
“
T
T
i l . § lCik 6s
l
§
§
§
k . 8 6
k k .õ6 II
‰ „ƒ¬n … ‘
1
3  ® E 1 + 4S§ Eô6c
l
l
T
+
‡ˆ
1
“
kT
k
T
k k
k k
where the last line follows from (5.2b) and was already used for Lemma 4.1. The last steps
i 4öE6s.õ6 § I i
follow by error relation (5.1a), approximation properties (5.2b) and (5.2c), and Theorem 5.2:
l
‰t © : 6 3 ƒ†n… EÑ4 §
‡xª« ª
£ T
:
k
k
‰t © : E 6 3 1
I D
:
: T
§
k k k k
Furthermore we must control the approximation 6s.ž6
§
therefore have to assume an inverse inequality for [ : k
i 4 § i ‰5 £ i 4 § i
(5.5)
w
© T w £ T D
i 4SE6f.ž6 § I i
+ 4Sl § EH1c.U1 § I„I
T
k k
k
T
of the pressure gradient. We
R EMARK 5.4. Equation (5.5) holds for all finite elements, i.e piecewise polynomial
§
functions on elements = of diameter > E © I . Denoting the mean value of w over = by w*? , we
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ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
187
obtain from the inverse relation, the LBB-condition on each = and Lemma 2.2
i4 § iO
w
' ´
iö
4 E w § . w'? I i ?O ‰5 © £ O ´
?
‰5 © £ O ´

Clearly, the constant
?
?
i § . iO
w
'w ? ?
i4 § iO
iO
Oi
w £ T³ ? ‰( © £ 4 w § £ T D
in this estimate does not depend on the LBB-constant of " .
The above preparations now lead to the desired estimate of the pressure gradient as follows.
T HEOREM 5.5. The error of the pressure gradient can be estimated by
6c.õ6 §
‰t © £ T æ 1
í ÀŠ D
:
3 6 ç +
: T
:
k
k
k k
k k
§
Proof. We insert the approximation operator 8 and use its property (5.2c) together with
T
the above inverse inequality (5.5) to obtain
3 iS
4 E 8 § 6s.ž6 § I i
T
i
i
k
T k 3  © £ T 4SE 8 § 6c.õ6 § I £ T
:
k 
E :,<v= l +@8 § 6s.ž6 § I

†
ƒ
3
l
T
© £ T ‡ˆ n
:
k
§ k l k +@T 8 § 6s.ž6 § I
E


,
:
v
<
=
6
1
3 © £ T ƒ†n
l
' ©: £ T
‡ˆ
:
T
k
k k E :,<v=61 § l + 6c.ž6 § I
i
i
‰( © : £ T

§
‘
3 © £ T 6. 8 6 3 ƒ†n
l
‡ˆ
“
:
T
k
l
§
§
‰( © £ T
= 1 l + 6s.ž6 I D k k
3  © £ T ƒ†n … E :,<v6
:
ˆ
‡

:
T
k
k k and Theorem 5.2 yields
Again, employing error relations (5.1), Inequality (5.2c),
§l
§
6s.ž6 § ‰( © : £ T 6 3  © £ T ƒ†n … EÑ4 1 + 4Sl EH1c.U1 I„I
‡ˆ
T
:
T
k
k ‰(
k k
k k
© :£ TE 6 3 1 TID
i k : .ž6 k § i k :
N O
Finally, we consider the -error k 6s
. For this purpose, we define the dual problem
6s.õ6 §
‰ s
6 .
T
k ‰( k © : £
' © :£
8
§6
6
k
6
k
6
k
6
k
6
k
EH4 ¡›+ 4fÕmI%38E :,<v= Õ +w I '8A
uLÕÌK Z +
E :,<v=g¡›+BA I ' E6c.õ6 § +CA I
u A K [g+
(5.6b)
Z p [ . By testing the second equation with 6c.^6 §
with solution E ¡›+w IK
N O
(5.6a)
error relations (5.1), we obtain for its
i 6f.ž6 § i O ‰ E E
†: <>= ¡
‰ ¡ . 1
k E :,<v=
.(
-bound
§ ¡ I + 6s.ž6 § Ié3–E :,<>=61 § ¡›+ 6s.ž6
§ ¡ i 4SEô6s.ž6 § I i 38Er4SE1c.h1 § I
£ T
T
E1f.Uk 1 § I +D8 § w I%38EH4SEH1f.h1 § I + .›4 ¡
.
1
§I
+4 1 §
3 4 ¡
and by virtue of
¡ I
I +
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M. WOHLMUTH AND M. DOBROWOLSKI
where the last line is equal to 0. By reordering these terms and using (5.4a), this results in
i 6s.ž6 § i O ‰
 i 4 E6c.õ6 § I i
¡ . 1 §¡ T ö
38
4 E 1 § ¡ . ¡ £ „I T I%38E :†<>= EH1c.U1 § I +w .
k EH4öE1c.hk 1 § I + S
8
§wI+
and finally yields by Lemma 5.3, approximation properties (5.2b), (5.2c), and by Young’s
inequality
i 6s.ž6 § i O t
‰ © O ¡
‰t O © k O
T
‰t O © O
T
O 3 
O
©
k 6s.õ6 §
k 6 O 3
ú T
k k
O
O
O
w T 3  1c.U1 § T
O k 3 k  O 1k O
k
O
©
T
k1 Oû k k
O +
k k
where the last inequalities follow from Theorems 5.2 and 5.5.
N O
The problem of the -error bound is, that the dual problem requires 6c.õ6
term which does not have locally balanced flow. Thus, we only obtain
§
as a source
Š
 t
‰ ü ‰ c
N
D
T
But for instance, if we consider the example of Section 3.3, then 6 is linear and is exactly
§
approximated in the free flow part " V of the domain " . Hence, 6Ì.h6 corresponds to the
source terms ? and ? which we assumed in the example. In this case we obtain the partial
£
improvement
 ä ÿŠ D
N
T
N O
Since in general, we can not improve the first i order § i -error of the pressure, we do not
want to consider any higher order estimates of 6s.ž6
. To obtain higher order estimates
comparable to those we provided for the other errors, an immediate generalization of the
§ . The required estimates
above proof would require estimates for Ð + w , and 6s.õ6
: T
:
:
3.10.
for Ð + w can be obtained by a generalization
k k
k k of Corollary
k
k
T
:
:
k Finally,
k  k ifk the Stokes problem has locally-balanced flow or even satisfies ?'8A ,i thei righti i
hand side E 1 O 3 6 I of all our first order error bounds can be estimated by 7 + ?
T
T
without any further
k k LBB
k k dependency.
Acknowledgments. The first author gratefully acknowledges funding of the Erlangen
Graduate School in Advanced Optical Technologies (SAOT) by the German National Science
Foundation (DFG) in the framework of the excellence initiative.
REFERENCES
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ETNA
Kent State University
etna@mcs.kent.edu
ANALYSIS OF STOKES EQUATIONS WITH IMPROVED LBB DEPENDENCY
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