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Electronic Transactions on Numerical Analysis.
Volume 20, pp. 154-163, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
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CONVERGENCE ANALYSIS OF THE ROTATED
ELEMENT ON
ANISOTROPIC RECTANGULAR MESHES SHIPENG MAO AND SHAOCHUN CHEN
Abstract. The main aim of this paper is to study the convergence of the well-known nonconforming rotated
element for the second order elliptic problems on anisotropic rectangular meshes, i.e., the meshes considered
in our work do not satisfy the regular assumption. Lastly, a numerical test is carried out, which coincides with our
theoretical analysis.
Key words. anisotropic, interpolation error, nonconforming, the rotated
element
AMS subject classifications. 65N30, 65N15
1. Introduction. The regular assumption (cf. [8], [13]) of finite element meshes is a
basic condition in the convergence analysis of finite element methods (FEMs), whereas some
early papers have been written to prove error estimates under more general conditions (cf.
[7], [19]). Recently, much attention is paid to FEMs on anisotropic meshes. In particular,
for rectangular meshes we refer to Acosta [1], [2], Apel [3], [4], [5], [6], Chen [11], [12],
Duran [16], [17], Shenk [24] and references therein. The studies mainly concentrated on
some Lagrange type elements (conforming elements). But nonconforming methods are
hardly treated, as far as we know, there are few papers on the nonconforming elements under
anisotropic meshes. On the other hand, most of the former works are concentrated on the
estimates of the interpolation error under anisotropic meshes, in particular, the readers are
refer to [4] and [11] for some techniques of the anisotropic interpolation error estimates.
However, some elements do not
satisfy the anisotropic interpolation properties. A case in
element
point is the famous rotated
of Rannacher
and Turek [23]. In this paper, we will
element
show that the interpolation error of the rotated
is not convergent, while it can be
applied to anisotropic rectangular meshes.
nonconforming elThe goal of this paper is to obtain the error estimates of the rotated
ement on anisotropic rectangular meshes. Our estimates improve the previous known results
in several aspects: Firstly, the anisotropic
approximation error is obtained in a different way.
element
The interpolation error of the rotated
is not convergent under anisotropic rectangular meshes. We overcome this difficulty by constructing another operator instead of the interpolation operator. Then we come to the interesting conclusion that the
interpolation error may say nothing about the convergence of some FEMs. Secondly, our results improve the previous consistency error estimate of
the
rotated
element. In section 3,
element under anisotropic
we mainly study the consistency error of the rotated
rectangular
meshes. Under moderate smoothness of the solution, accuracy with !" and !"#$ order
of the consistency error are both obtained by the trick of element cancellation. Lastly, the
techniques developed in our analysis can give some hints to other nonconforming elements
under anisotropic rectangular meshes.
Before the end of this section, we will recall some notations and terminology (or refer to
[8], [13]). Let &%(')%* denote the usual +, -inner product and -/.0-)132 452 6 (resp. 7 .7 132 452 6 ) be the
8
Received June 28, 2004. Accepted for publication June 19, 2005. Recommended by R. Lehoucq. The research
is supported by NSF of China (No.10471133, 10590353) and the project of Creative Engineering of Henan Province
of China.
Department of Mathematics, Zhengzhou University, 450052, China.
Current address: Institute
of Computational Mathematics, Chinese Academy of Science, P.O. Box 2719, Beijing, 100080, China.
(shipengmao@yahoo.com.cn).
Department of Mathematics, Zhengzhou University, 450052, China. (shchchen@zzu.edu.cn).
154
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CONVERGENCE ANALYSIS OF THE ROTATED
usual norm (resp. semi-norm) for the Sobolev space 9
1
by .
ELEMENT
155
1:2 4 . When ;=<?>@' denote 9 :1 2 2. Nonconforming rotated
element and its approximation error estimate on
anisotropic rectangular meshes. For simplicity, assume that is a polygon with boundaries parallel to the axes and let A, be a partition of by rectangular meshes which need
EFA , denote the barycenter
C by
not satisfy the regular assumption. BDC
of element
HGI':JIK/' the length
of edges parallel to x-axis and y-axis by >"LI 'M>"NI
respectively,
"OIF<QPRTSNU5"OI 'V"NI 'V"X<YI\PZ[R5]@S ^ "NI . Assume that `
_C <bacd'edgfhXacd')d)f is the refer-
5W
j
< &d'ed$g' _i <j:d') dk/' _ iml <n&d')d$g' _imo <j:d'ed$/'
_i l ' _ p l < _i l _ i o ' _ p o < _i o _i . Then there exists an unique
G=<tG IXu " I /v '
JZ<tJ IXu " I gwLx
element.
To begin with, we will introduce the finite element space of the rotated
y
Set
z <|{3; Ud' v ' ' v '
im}
w
w W
and BLqn~€LC' for any ‚Eƒ „C…/' we define
†ˆ‡ „‰< d ‡ m‹‰{
x
7 q‚7mŠ
Then the finite element space isy defined as
<ŒU$‚Eƒ+ )7 KŽq I E z ': y is continuous
†y ‡ &%*/' †ˆ‡ H@<‘’'&BLq“E”D
regarding
y
y
Wx
•
–
We will define two interpolations and on C— , C— respectively,
yy
• ˜< _ u _ l u ™ _ l:o u _ o u _ l _ o v u _ l3o _ > w
u   u  v>
&

y y y y
y y u›y š _ y y_ l œ y _ l:o _ y o y w y /' y
– ˜<  y u  y u ™  y l u  y o u  u  ™ u  l  o v u   ™ u  l u  o
w
 u   v

™ l o w'
y u
y
y y y
&
< y ¡ _ ¢¤£ ¡$¥ _ ¢ £ ‰‹ _{'   < D i  /'ˆ¦§<¨d'V>’' š ' ™ x Let • L7 I©< • I and – 7 Iª< – I
where T„žc(Ÿ
y
rotated and bilinear element on C , respectively, where • I <
be the interpolations
• Žq˜« ' – I < – Žq˜of« the
I
I .
For the convenience of simplicity,
we consider the following Poisson problem:
s
­¬=.®<?¯'
in \'
(2.1)
.°<|O'
on D
x
s is:
Then the weak form of (2.1)
Find .Eƒ /'
such that
(2.2)
i H.±':@²<?¯
H@/' B˜®E° g'
ence element, the four vertices are: _ i
and its 4 sides are _ p < _ i _ i ' _ p < _ i
ass
_C rC defined
mapping q I b
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156
SHIPENG MAO AND SHAOCHUN CHEN
where
i H.±':@²< Š 6³ . ³ m ‹GN‹JL'´¯H@< Š 6 ¯m‹GN‹J x
The approximation of (2.2)s reads as:
Find . E '
such that
(2.3)
i H. ': <|¯„ g'µB E with
i „. ': ²<·I\¶ []@^ Š I¸³ . ³  ‹GN‹J x
We will define the broken norm as
-%m- <º¹»¶
I\[]@^
It is obvious that -%‰- is a norm on .
¾
7k%‰7 2 IX¼—½
x
•
The following theorem shows that the interpolation is unstable or even divergent in
the energy norm sense under anisotropic rectangular meshes.
T HEOREM 2.1. (cf.[22]) Let be a rectangular and a uniform mesh division in each
direction with the diameter " and "¿ respectively, and dKÀ eÁ ÀXÂ . we have
(2.4)
-/.Z • .0-)˜Ãt"Ä:d u "LÅT" ¿ e7 .07 2 6 x
It’s seen from Theorem 2.1 that the aspect ratio "LÅT" ¿ appears in the right hand side of
(2.4). The following counterexample shows that this is the true case (cf. [5]).
Taking an element CÆ<Ça¤È"É@'M"OÉTf,hta¤È"NÊm'V"NÊ$fˍ"OɈÌÍ"NÊT/':.?<QGN , then by a direct
calculation, we have
y
y
•ÏÎ .=< > " É
I
š
•
- H.‚  J I\.D - 2 I?<?" Ê« "OÉm"OÊT
d v
> " É y
¾ -  .‚

½
w g'
y y
• Î .D Î
I - 2 I < Ð > ",É ¾Ñ " Ê« ¾ '
½
w
š
7 .7 2 I <j Š I > ‹GN‹J’ ¾ < ™ " É ¾ " Ê ¾ x
½ ½
½
So
Ð
žcÔ «NÕ#Ö Ô - 2 I
•
N
Ó
7
‚
.
\
I
.
7
I
Ê
2
< × š "O" ÉÊ ÙØ x
(2.5)
Ò
7 .7 2 I
7 Ó .07 2 I
element is convergent under
From the above discussion one may ask if the rotated
anisotropic rectangular meshes, the answer is affirmative. In fact, from
the numerical test
element
and theoretical analysis in this paper, one can see that the rotated
preserves the
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CONVERGENCE ANALYSIS OF THE ROTATED
ELEMENT
157
optimal order of convergence even if the interpolation error is divergent, which is a very
interesting result.
In order to obtain the anisotropic approximation error estimate, we construct the operator
with I < • I – I ': 7 I <| I '&BLCbEƒA .
yy
y y holds: y
y
L EMMA 2.2. y For y the operator
, there
(2.6)
- Ú®Û  ’ )- 2 I Î ÃtZ7 Ú=Û D7 2 I Î Ü' B ®Eƒ C—/'ˆ7 Ý7m<“d x
Proof.
y y calculation
y y gives
y A simple
y y y y
y y y y
y
 u  u  u 
 u  u 
v
u
u





l
o
l
o
™ l o Lw x
™
u
™
u
 <
For the case ݅<j:d'3m ,
y yy
y y
y y
y y
y
ß qZ Ú Û @
Ú Û < d
Ú Û  v ')d$:‹ v u Ú Û D v ')d$3&‹ v˜<‘
(2.7)
x
7 CÞ7 Š « y
It can be verified that
à
y
y
y
y Z- á - 2 I Î y '&B á Eƒ y C—/'
7 qZ á )7‰Ãt
qZ á ²<‘’'#B á E z Câ x
y y y y lemma yields
y y
y y
y y
Employing the Bramble-Hilbert
Ú- Û  ‰e- Î <n- Ú Û ¸qZ Ú Û @e- Î ÃtZ7 Ú Û D7 Î
2I x
2I
2I
Similarly, we can prove (2.6) for the case ݅<j„O')d$ .
Assume . and .D to be the unique solution of (2.1) and (2.3) respectively, .ˆE…⍠x
Then the second Strang’s Lemma (cf. [8], [13]) gives
(2.8)
. ': )7
-/.Z….DN-eÃã>åè ^Täcæ@[mç ée^ -/.Z…-) u è ^[mê3é$ë’^kì í/î 7 i „.®g-
x
-
gï
Now, we are in a position to bound the first term on the right hand of (2.8) firstly.
L EMMA 2.3. Under the above assumptions, we have
(2.9)
è ^Täcæ’[mç é$^ -g.Z—-)Ãt"±7 .07 2 6 x
Proof. By lemma 2.2, we have
è ^Tä(æ@[mç é$^ -g.Z—L-e˜ÃŒ-g.Z—#.-e <º¹ I\¶ [ ]@^
¾
7 .®—DI­.7 2 I\¼—½
¾
Ú
Û
<»ðñò¶ ¡ ¶ ¡ ó - „.‚— I D. )- 2 I­ôõ ½
I\[]@^ Û
y y yy
¾
Û
Ú
Û
Î
<»ðñ ¶ ¡ ¶ ¡ ó "#I« "OI "OI )- .‚ .D) - 2 I ôõ ½
I\[]@^ Û
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SHIPENG MAO AND SHAOCHUN CHEN
y y
¾
Û
Ú
®
Û
Î
ÃXQðñ ¶ ¡ ¶ ¡ ó #" I«  "OI "OI )7 .±7 2 I ôõ ½
I\[]@^ Û
¾
ö Ú ÛŸö
ÃX ðñ ¶ ¡ ¶ ¡ ó ¡ ö¶ ¡ ó " I .0- 2 I­ôõ ½
I\[]@^ Û
ÃX"±7 .7 2 6 x
R EMARK 2.1. Recently, Cai, Douglas and Ye [9], [15] have proposed some new nonconforming midpoint-oriented four-node elements. Note that the associate interpolations have
the same properties as the rotated
element.
R
EMARK
2.2.
In
order
to
overcome
the deficiencyof the interpolation of the rotated
y
K element,
reference
[5]
proposed
a
modified
rotated
element with the shape function
z <÷{3; Ud' v ' ' v (cf. [6], [20]). However,
space
the
dissymmetry of the shape space
im}
w W
will influence the element geometrical inflexibility (cf. [14]).
3. Consistency error estimate on anisotropic rectangular meshes. In this section, we
will turn to the second term on the right hand of (2.8), i.e., the consistency error. The standard
technique of the consistency error estimate (cf. [8], [13]) is invalid under anisotropic meshes,
then we develop a new
one for the estimate of the consistency error.
For any =E° C— , we define
z  ˜< d
™
7 p  7Š ¤¢ £ ‰‹‰{'©¦0<Œd'M>@' š ' x
Then by Green’s formula we have
(3.1)
¶
i H.‚…. ': < i H.±': X¯'3 <ø¶
I\[]@^ ¢c£Ëù
<ú¶ a Š „ z  g .J
I\[]@^ ¢
u Š „½ z l mg .J z
¢¤û
u Š ¾ H z V ) .G z
¢
Š H z o  ) .G z
¢ýü
<ø¶ a • u • l u • u
I\[]@^
.  ‹‰{

Š
Ó I ¢ £ }
z N.J :‹ G
.
l NJ :‹G
N.
M NG &‹J
N.
o NG &‹J
• f
o x
We will show that the conventional technique of nonconforming error
will be• forestimate
come invalid under the consideration of anisotropic meshes. Take
example, in the
conventional method, by the trace theorem and the Bramble-Hilbert lemma, we will estimate
it as follows:
7 • 7 <©þþ Š H z  mg N.J z  .J & ‹Gþþ
þþ ¢
þþ
½
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CONVERGENCE ANALYSIS OF THE ROTATED
159
ELEMENT
Ó-/K z  N- 2 ¢ y .J z Ny .J y2 y
y
y
¢
z
Î
¾

.
m
½
ÿ
ÿ
<?"OI " I« - K ÿÿ y - 2 ¢  ÿÿ z ½  . 2 Î
½ y
w
w ¢
Î
¾

.
ÿÿ ½
ÿ
½
Î
Ãt"OI " I« 7 L7 2 I þ  þ 2 I ÿÿ
ÿ
þ
þ
½
þ w þ
¾
<? "O" II ðñ ¡ ¶ ¡ ó " I Û - Ú Û L- 2 I ôõ ½ þ .J þ 2 I x
þþ þþ
Û
When the regular assumption is satisfied, which yields Ö ¾ ÃX , then we can get
Ö
7 • 7@ÃX"OIZ7 .7 2 IZ7 N7 2 I x ½
(3.2)
However, we do not have the regular assumption under anisotropic rectangular meshes.
On the contrary, we may have Ö ¾ ÙØ when "NInǐ , and the desire convergence result
Ö ½ Thus it is more difficult for us to estimate nonconformof (3.2) can not be obtained as usual.
ing error on anisotropic meshes. Now, let us turn to (3.1) again.
è^
From the construction of the element we know that Ó É is independent of J , hence
Ó
H z #HGÄ'3JI?¸"NI É Ÿ
<t HG ':J I ¸" I ± >"Nd I Š É Ö Ö  '3J I ˆ" I &‹ Ö « Ö ½
Ÿ
É
< >"Od I Š É Ö Ö a  „GÄ'3J I ˆ" I ½ … ':J I ¸" I Üfk‹ Ö « Ÿ Ö ½ É
É
(3.3)
< >"Od I Š É Ö Ö ½ Š   5':J I ˆ" I &‹ —‹ Ö « Ÿ Ö ½ É
É
< >"Od I Š É Ö Ö ½ Š  5':J Iãu " I &‹ —‹ Ö « Ö ½
z
<j„ l  D„½GÄ':J Iãu " I <ß á „G x
It is easy to see that
É Ö Ÿ Ö É Ö Ÿ Ö Ê Ö Ÿ Ö ¾  "NI Š É Ö « Ö ½ Š É Ö « Ö ½ Š Ê Ö « Ö ¾ þþþ G
 HGÄ'3J’½ 5þ ‹GN‹J°Ã ½ d >
" I þ " I
(3.4)
þ
þþ
>"NI Š I þþþ G
þ
i) Suppose .”E”â , we modified (3.1) slightly,
z N.
i ’„.®….L’':mV<ø¶ a Š „  J ‹G u Š H
I\[]@^ ¢
¢¤û
z
z

.
u Š ¾ H ½ V  G ‹J Š „ o  (3.5)
¢
¢ü
<ø¶ a u l u u o f x
I\[]@^
7 á „G)7Oà ™ N" I
à >"Nd I
d
„GÄ':J’ þþ ‹GN‹J‰‹
þþ
7 7 2I x
z l m N.J ‹ G
NN.G ‹J
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SHIPENG MAO AND SHAOCHUN CHEN
u , we get
l
Ÿ
É
Ö Ö á „G N. HG ':J IXu " I . „GÄ':J I ˆ" I ‹G
u l <
J
Š É Ö« Ö ½
J
¾
É Ÿ
Ê Ÿ
< Š É Ö Ö ½ á „G Š Ê Ö Ö ¾ J . ‹JK‹G
(3.6)
Ö « Ÿ Ö ½
Ö « Ö
É
ß<
Ö Ö ½ á „GZHGL&‹G
x
ŠÉÖ «Ö ½
A combination of (3.4) and
½ (3.6) yields
(3.7)
u l ÃX" I 7 .07 2 I 7  7 2 I x
Substituting (3.3) into By the same argument, we can prove
(3.8)
u o ÃX"NI 7 .07 2 IZ7 7 2 I x
Then we have
i O„.‚….LO':mÃX"±7 .7 2 6 -/-) x
Ô
ii) Suppose .Eƒ l , we consider (3.1) again. Set < Ó Ê ' there hold
Ó
É Ö Ÿ Ö É 
z
d
(3.10) LHG ':J I ¸" I 0
˜< >" I Š É ½ Š  5':J I ¸" I :‹â‹ '
Ö²« Ö
and
½
É
Ÿ
É
z
d
Ö
Ö

(3.11) „GÄ':J IXu " I l < >"OI Š É ½ Š  5':J Iãu " I &‹ ‹ x
Ö « Ö • •
u l , we get
Substituting the above two formulas and (3.3) into
½
ÉÖ Ÿ Ö
É Ö Ÿ Ö É N
•
•
d
á
u l < >" I Š É „G Š É (3.12)
 5'3JI u "NI Ö« Ö ½
Ö« Ö ½ Š
N  5'3J I ˆ" ½ I ‹ ‹ ‹G ½
É Ÿ
É Ÿ
É Ê Ÿ ¾
< >" d I Š É Ö Ö á HG­¹ Š É Ö Ö ¹ Š Š Ê Ö Ö ¾  L5/ J 5':J’&‹ 5‹J ¼ ‹ ¼ ‹G
Ö« Ÿ Ö ½
Ö« Ö ½
Ö²« Ö
É
ß< d Ö Ö ½ á HGZHGL&‹G ½
x
>" I Š É Ö « Ö ½
Obviously,
½
7 ZHGL)7cÃX>"OI Š I þþ GN) J „GÄ'3J@ þþ ‹GN‹J=Ãt>"NI x >
"OI "OI 7 D7 2 I
þ
þ
(3.13)
à ™ " I " I þ " I 7 .7 l 2 I þ x
(3.9)
Substituting (3.4) and (3.13) into (3.12) gives
(3.14)
u Ãã" 7 .07 2 I 7  7 2 I
I l
l
x
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CONVERGENCE ANALYSIS OF THE ROTATED
Similarly,
(3.15)
161
ELEMENT
u o ÃX" I 7 .7 l 2 I˜7 L7 2 I x
Thus we have obtained
(3.16)
i „.‚…. ': ÃX" 7 .07 l 2 6-/ - x
Then we can get the following anisotropic error estimate of the nonconforming rotated
element.
T HEOREM 3.1. Suppose .|EXâ/' under anisotropic rectangular meshes, we have
the following error estimates
g- .‚…. - ÃX" 7 .7 2 6²' -/.Z…. - 2 6…Ãã" 7 .07 2 6
If further assume .E° l g' the consistency error is of one higher order, i.e.,
7 i NH.®—.D’'3e7 Ãt" 7 .07 2 6
(3.18)
l x
è ^[mê:é$ëO^eì í/î /ï
-/ - (3.17)
Proof. (3.18) is a direct consequence of (3.16). Substituting (2.9) and (3.9) into (2.8), we
can obtain the first identity of (3.17). By the usual dual argument as standard finite element
theory (cf. [8], [13]), we will get the second identity of (3.17). Then the proof is completed.
R EMARK 3.1. The results of (3.17) can be applied to the nonconforming elements discussed in [5], [6], [9], [15], [18], [21], [23]. However, the result of (3.18) does not stand for
the midpoint-oriented elements proposed by [23].
\
4. Numerical experiment. In order to investigate the numerical behavior of the rotated
element under anisotropic rectangular meshes, we consider the second order problem
(2.1) with ¯HG ':J’K< ² ê DG ê ä(æ DJ’ u ê DJ’ ê äcæÄDG3Eã+ , and n<Qa ’')d)f±h
a O')d)f . It can be verified that the exact solution of problem (2.1) is .±„GÄ'3J@²< ê äcæ#DG ê äcæDJ’ .
The meshes on can be obtained in the following way, the edges of parallel to Gˆ
&%
i GO¦&{ (J' i GO¦&{ resp.)&% are %divided into' } segments with } u d points :d­"!$#T{m ' 33ÅT>’':¦§<
’'ed' x(× x(x ' 'e× &d u {e¦ } ' 33Å>@':¦< u d' x(xcx ' } (m resp.). The mesh obtained in this way
for d hˆ d is illustrated in Fig' 4.1. Note
that the large aspect ratio meshes can be obtained
by adjusting the ratio value of ( ( refer to Table 4.1 for the aspect ratio I\
PZ[R5]@S ^ ) Ö ).
Ö
In Table 4.1, we list the numerical results of the rotated
nonconforming element,
•
Ô ^Ô ^
Ô Ô^ ^
here *e¦<,+ «.- + and *5"ã</+ « + , where . denotes the interpolation of the exact
solution. From the results of this table, we can see that the optimal energy norm error between
. and . is obtained under large aspect ratio meshes (strongly anisotropic meshes), while the
interpolation error is divergent in such case. These results show that the constant at the
right hand side of the interpolation error is dependent on the aspect ratio, while that of the
error of . and . is independent of the aspect ratio, which coincides with our theoretical
analysis.
nonconforming element, we also have
For the sake of a comparison with the rotated
y space isy {3; im} UdÎ ' v ' '
y y in [18].
y yThe shape
computed the five-node element proposedy by Han
v T ' , and the degrees of freedom are U  '  '  '  ' 10
w W
l l:o o W
w
32 ¡ è Î 46¡57478
¥
on C , where 10<
Ö I
.
The results are listed in Table 4.2, from which we can see that both the interpolation error and
finite element error are of optimal order on anisotropic rectangular meshes.
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SHIPENG MAO AND SHAOCHUN CHEN
F IG . 4.1. The anisotropic rectangular mesh for the case 9:<;=:?>A@
TABLE 4.1
Numerical results of the rotated
;CBD9
P6Q
PAR
SWYUX[T6Z V ]
^\
EFBE
ÖÖ
1.92
1.84
1.41
EGBH
1.98
3.49
18.58
EFBIJE
2.04
15.24
293.69
,
EFBK>AEJH
2.04
68.80
4695.56
element
EFBL>AE
2.04
303.67
75125.35
EFBM>ANJE7O
2.04
633.91
300500.71
5. Conclusions.
This paper has been studied the convergence properties of the famous
nonconforming
element under anisotropic rectangular meshes. The interpolation
rotated
error in the discrete norm is not convergent on anisotropic
meshes. But this
element. rectangular
From this paper we can see
does not influence the convergence of the rotated
that the interpolation error does not indicate the convergence of some elements. What’s more,
the techniques developed in this paper can be applied to other nonconforming elements.
The higher order of the consistency error estimate obtained in this paper is of interest in
the superconvergence analysis of nonconforming elements (cf. [10], [25]). In fact, combined
•
the result of this paper with i H. . '3 Ãt"Om7 .07 l 2 6-g - (If we can prove this!) yields
•
a superclose result - .°Þ. - Ãn"O‰7 .7 l 2 6 . Some superconvergence results of the fivenode element proposed in [18] can be obtained in
this way (will be addressed elsewhere).
element
However, the superconvergence of the rotated
is still an open problem, which
will be a future work of us.
Acknowledgments. The authors would like to express their sincere thanks to the anonymous referees for many helpful suggestions and corrections of the English and typesetting
mistakes.
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ETNA
Kent State University
etna@mcs.kent.edu
CONVERGENCE ANALYSIS OF THE ROTATED
ELEMENT
163
TABLE 4.2
Numerical results of the five-node element
;CBD9
PQ
PR
SWYUXT6Z V ]
^\
EFBE
ÖÖ
1.49
1.39
1.41
EFBH
1.46
1.37
18.58
EGBIJE
1.48
1.39
293.69
EFBM>AEJH
1.48
1.40
4695.56
EGBL>AE
1.48
1.40
75125.35
EFBK>ANJE7O
1.48
1.40
300500.71
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