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A discrete Korn's inequality in two and three dimensions
Applied Mathematics Letters
Applied Mathematics Letters 13 (2000) 99-102
PERGAMON
www.elsevier.nl/locate/aml
A Discrete in Two and
Korn's Inequality Three Dimensions
XUEJUN XU Institute of Computational Mathematics, Chinese Academy of Sciences P. O. Box 2719, Beijing 100080, P.R. China
(Received May 1998; accepted April 1999) A b s t r a c t - - I n this paper, we present a simple and general proof for Korn's inequality for nonconforming elements, like Wilson's Element and Carey's Element. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords--Korn's
inequality, Wilson's Element, Carey's Element.
1. I N T R O D U C T I O N Let Q C R d (d = 2,3) be a b o u n d e d domain with b o u n d a r y 0 ~ . where v = ( V l , . . . , V d ) , we set
1
For any v c V ~ ( H ~ ( ~ ) ) a ~
j<_d.
(1.1)
d
Moreover, for any v E V, the expressions ( E ~ = I Ivi]i) 1/2 and (~-~d 1 tlv~[ll) 1/2
llvll~ <_c
c~o(v \i,j=l
,
Vv e (g~(a))"
(1.2)
/
It is known t h a t K o r n ' s inequality is very i m p o r t a n t for the analysis of elasticity p r o b l e m (see [1]). If we use conforming finite element space Sh to a p p r o x i m a t e space V, because Sh C V, K o r n ' s inequality is trivial. B u t if we choose nonconforming finite element space Vh to a p p r o x i m a t e space V, as pointed out in [2-4] it is not straightforward. In this direction, Lesaint [2] proved t h a t K o r n ' s inequality held for rectangular Wilson's Element in two dimensions. In [3], Zhang gave a proof of K o r n ' s inequality for quadrilateral Wilson's Element. In this paper, we will present simple and general proof for the discrete Korn's inequality which is easily extended to three The author would like to thank a referee for his valuable comments. 0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00217-7
Typeset by AA/b%rPFjX
100
X. X u
elasticity problems and Carey's Element. Carey's Element was developed in [5], convergence results were analyzed in [6]. Recently the superconvergences of the element were obtained in [7]. This paper is organized as follows. In the next section, under an assumption, we prove that Korn's inequality is valid. In Section 3, we show that Wilson's and Carey's Elements satisfy the assumption. 2. A N
ABSTRACT
RESULT
Assume that there exists a conforming counterpart space Uh C V for the space Vh. Moreover, we have the following. ASSUMPTION (H). There exists an operator Ih : Vh ~ Uh such that
llv - &Vllh <_ ch
~ (Ivl~,K) ~/2, KEFt,
Ivl~,~. Based on this simple assumption, we can prove that the following discrete Korn's inequality. THEOREM 2.1. For any v E Vh, we have d
Ilvll~ _
~
II
IgEFI, i , j = l
PROOF. Because Ihv E (H~(f~)) d, using Korn's inequality (1.2), we have d
(2.1) i,j=l
Moreover, it is straightforward to check that 02vm - ~ ~m/(v) + ~zt ~ s ( v ) _ 0@m ~st(v), OxsOxt where 1 _< s, t _< d. Then using inverse inequality, we obtain
m = 1,...,d,
(2.2)
d
Iv~12,K ___Ch-1
Ile~j(v)ll0,K,
Z
/C c rk.
i,j=l
So, by
(2.1),(2.3), we have d
< c ~
II¢~j(&v)llo2 + IIv -
Zhvll~,
i,j=l d
~
~
d
He~j(v)ll~+ ~
KcFj, i,j~l
~
I[e~j(v-Ihv)lJ~+Hv-Ihv]l~
KEFt, i , j = l
d _<
c ~
~
Ilc,j(v)ll~ + Ilv -/hVH 2h
KEFh i , j = l d
_< c
E
E If (v)rro + h2 E
KEFI, i , j = l
KCFk
d
<
c
iij(V)No KEFh i,j=l
which completes the proof.
ivj ,
(2.2)
Korn's Inequality
3.
101
APPLICATIONS
In the following, we will apply the general theorem developed in Section 2 to Wilson's Element and Carey's Element.
3.1. W i l s o n ' s E l e m e n t For simplicity, we only consider two dimension case. Define Wilson's Nonconformintg Element Space as follow. First, on the reference element [( = [-1, 1] x [-1, 1], the shape function of Wilson's Element defined by the values of/3 at four vertices of [¢ and the value of is quadratic polynomial and ~
on/~'. We have ]) =
(1
+
{)(1
+
r/)
4
(1 + ~)(1
(1
Pl +
-f])P4
-
{)(1
r/)
+
4
P2 +
(1
-
~)(1
+
r/)
4
1 ({2 _ 1) 021fi + 1 (q2 _ 1) ~)2/5
--
P3 (3.1)
,
where P i is the value of/3 at the vertex A,. Then, for a rectangle K E Fh, using as affine transformation, we can also define a Wilson's Elen:ent. W'e denote Wilson's Finite Element Space on Fh by Vh. It is known that the Wilson's Element is not continuous on interelement boundaries, so it is noconforming for elasticity problem. For this element, we choose Uh as bilinear conforming finite element space on Fh and Ih as bilinear interpolation. LEMMA 3.1. Assupmtion (H) is valid for Wilson's Element. PROOF. By standard interpolation estimate [1], we have
I~ - ±,,-~'1~,,~ <- ct~l~l~,~ here IK = Ibis,-
Then summing up all K G Fh gives Assumption (H).
|
1REMARK 3.2. Similarly, it is not difficult to see that Assumption (H) also holds tbr three Wilson's brick [1] and quadrilateral Wilson's Element [6].
3.2. Carey's E l e m e n t Given a triangle K G Fh with vertices pi = (y~, zi), 1 < i < 3, we denote by Ai the coordinates relative to vertices Pi, by E the area of K, and put / ~t]l :
:'~ + ,/~ = l2~3 ,
z2
-
z3,
~ + ~
02
=
IL,
:
z3
-
Zl,
'tl3
~ + ,~ : z212'~
:
Zl
--
z2,
& + ~h + I'~,
:
l ~'
Carey's Element is defined as follows [5,6]: on each triangle K, the shape function has the form tt = 'ttlAI + U2A2 4- U3A3 4- tO,
0 -- AtA2 4- AzAs 4- AaA:.
(3.2)
T h e function ~t has four parameters, namely, the function values zt{ at the vertices of the triangle K
and the parameter t, using the Laplacian of u on K t
=
4E - 75- ( A ~ ) .
(3.3)
For Carey's Element, we choose Uh as piecewise linear conforming element space on Fh and Ih as standard interpolate operator. Then similar to Lemn:a 3.1, we can see that Assumption (H) is valid for Carey's Element.
102
X. Xu
REFERENCES 1. P.G. Ciarlet, The Finite Element Method for Elliptic Problem, North-Holland, Amsterdam, (1978). 2. P. Lesiant, On the convergence of Wilson's nonconformaing element for solving the elastic problem, Comput. Methods Appl. Meth. Engrg. 12, 1-16, (1976). 3. Z. Zhang, Analysis of some equadrilateral nonconforming elements for incompressible elasticity, S I A M J. Numer. Anal. 34, 460-663, (1997). 4. P.S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Com. 57, 529-550, (1991). 5, G.F. Carey, An analysis of finite element equations and mesh subdivisions, Comput. Meths. Appl. Mech. Engrg 9, 165-179, (1976). 6. Z. Shi, Convergence properties of two nonconforming finite element, Comput Methods. Appl. Mech. Engrg. 48, 123-139, (1985). 7. L. Zhang, Superconvergent recoveries of Carey's nonconforming element approximation, Comm. Numer. Methods. Engrg. (to appear). 8. Z. Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44, 349-361, (1984).
Applied Mathematics Letters 13 (2000) 99-102
PERGAMON
www.elsevier.nl/locate/aml
A Discrete in Two and
Korn's Inequality Three Dimensions
XUEJUN XU Institute of Computational Mathematics, Chinese Academy of Sciences P. O. Box 2719, Beijing 100080, P.R. China
(Received May 1998; accepted April 1999) A b s t r a c t - - I n this paper, we present a simple and general proof for Korn's inequality for nonconforming elements, like Wilson's Element and Carey's Element. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords--Korn's
inequality, Wilson's Element, Carey's Element.
1. I N T R O D U C T I O N Let Q C R d (d = 2,3) be a b o u n d e d domain with b o u n d a r y 0 ~ . where v = ( V l , . . . , V d ) , we set
1
For any v c V ~ ( H ~ ( ~ ) ) a ~
j<_d.
(1.1)
d
Moreover, for any v E V, the expressions ( E ~ = I Ivi]i) 1/2 and (~-~d 1 tlv~[ll) 1/2
llvll~ <_c
c~o(v \i,j=l
,
Vv e (g~(a))"
(1.2)
/
It is known t h a t K o r n ' s inequality is very i m p o r t a n t for the analysis of elasticity p r o b l e m (see [1]). If we use conforming finite element space Sh to a p p r o x i m a t e space V, because Sh C V, K o r n ' s inequality is trivial. B u t if we choose nonconforming finite element space Vh to a p p r o x i m a t e space V, as pointed out in [2-4] it is not straightforward. In this direction, Lesaint [2] proved t h a t K o r n ' s inequality held for rectangular Wilson's Element in two dimensions. In [3], Zhang gave a proof of K o r n ' s inequality for quadrilateral Wilson's Element. In this paper, we will present simple and general proof for the discrete Korn's inequality which is easily extended to three The author would like to thank a referee for his valuable comments. 0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00217-7
Typeset by AA/b%rPFjX
100
X. X u
elasticity problems and Carey's Element. Carey's Element was developed in [5], convergence results were analyzed in [6]. Recently the superconvergences of the element were obtained in [7]. This paper is organized as follows. In the next section, under an assumption, we prove that Korn's inequality is valid. In Section 3, we show that Wilson's and Carey's Elements satisfy the assumption. 2. A N
ABSTRACT
RESULT
Assume that there exists a conforming counterpart space Uh C V for the space Vh. Moreover, we have the following. ASSUMPTION (H). There exists an operator Ih : Vh ~ Uh such that
llv - &Vllh <_ ch
~ (Ivl~,K) ~/2, KEFt,
Ivl~,~. Based on this simple assumption, we can prove that the following discrete Korn's inequality. THEOREM 2.1. For any v E Vh, we have d
Ilvll~ _
~
II
IgEFI, i , j = l
PROOF. Because Ihv E (H~(f~)) d, using Korn's inequality (1.2), we have d
(2.1) i,j=l
Moreover, it is straightforward to check that 02vm - ~ ~m/(v) + ~zt ~ s ( v ) _ 0@m ~st(v), OxsOxt where 1 _< s, t _< d. Then using inverse inequality, we obtain
m = 1,...,d,
(2.2)
d
Iv~12,K ___Ch-1
Ile~j(v)ll0,K,
Z
/C c rk.
i,j=l
So, by
(2.1),(2.3), we have d
< c ~
II¢~j(&v)llo2 + IIv -
Zhvll~,
i,j=l d
~
~
d
He~j(v)ll~+ ~
KcFj, i,j~l
~
I[e~j(v-Ihv)lJ~+Hv-Ihv]l~
KEFt, i , j = l
d _<
c ~
~
Ilc,j(v)ll~ + Ilv -/hVH 2h
KEFh i , j = l d
_< c
E
E If (v)rro + h2 E
KEFI, i , j = l
KCFk
d
<
c
iij(V)No KEFh i,j=l
which completes the proof.
ivj ,
(2.2)
Korn's Inequality
3.
101
APPLICATIONS
In the following, we will apply the general theorem developed in Section 2 to Wilson's Element and Carey's Element.
3.1. W i l s o n ' s E l e m e n t For simplicity, we only consider two dimension case. Define Wilson's Nonconformintg Element Space as follow. First, on the reference element [( = [-1, 1] x [-1, 1], the shape function of Wilson's Element defined by the values of/3 at four vertices of [¢ and the value of is quadratic polynomial and ~
on/~'. We have ]) =
(1
+
{)(1
+
r/)
4
(1 + ~)(1
(1
Pl +
-f])P4
-
{)(1
r/)
+
4
P2 +
(1
-
~)(1
+
r/)
4
1 ({2 _ 1) 021fi + 1 (q2 _ 1) ~)2/5
--
P3 (3.1)
,
where P i is the value of/3 at the vertex A,. Then, for a rectangle K E Fh, using as affine transformation, we can also define a Wilson's Elen:ent. W'e denote Wilson's Finite Element Space on Fh by Vh. It is known that the Wilson's Element is not continuous on interelement boundaries, so it is noconforming for elasticity problem. For this element, we choose Uh as bilinear conforming finite element space on Fh and Ih as bilinear interpolation. LEMMA 3.1. Assupmtion (H) is valid for Wilson's Element. PROOF. By standard interpolation estimate [1], we have
I~ - ±,,-~'1~,,~ <- ct~l~l~,~ here IK = Ibis,-
Then summing up all K G Fh gives Assumption (H).
|
1REMARK 3.2. Similarly, it is not difficult to see that Assumption (H) also holds tbr three Wilson's brick [1] and quadrilateral Wilson's Element [6].
3.2. Carey's E l e m e n t Given a triangle K G Fh with vertices pi = (y~, zi), 1 < i < 3, we denote by Ai the coordinates relative to vertices Pi, by E the area of K, and put / ~t]l :
:'~ + ,/~ = l2~3 ,
z2
-
z3,
~ + ~
02
=
IL,
:
z3
-
Zl,
'tl3
~ + ,~ : z212'~
:
Zl
--
z2,
& + ~h + I'~,
:
l ~'
Carey's Element is defined as follows [5,6]: on each triangle K, the shape function has the form tt = 'ttlAI + U2A2 4- U3A3 4- tO,
0 -- AtA2 4- AzAs 4- AaA:.
(3.2)
T h e function ~t has four parameters, namely, the function values zt{ at the vertices of the triangle K
and the parameter t, using the Laplacian of u on K t
=
4E - 75- ( A ~ ) .
(3.3)
For Carey's Element, we choose Uh as piecewise linear conforming element space on Fh and Ih as standard interpolate operator. Then similar to Lemn:a 3.1, we can see that Assumption (H) is valid for Carey's Element.
102
X. Xu
REFERENCES 1. P.G. Ciarlet, The Finite Element Method for Elliptic Problem, North-Holland, Amsterdam, (1978). 2. P. Lesiant, On the convergence of Wilson's nonconformaing element for solving the elastic problem, Comput. Methods Appl. Meth. Engrg. 12, 1-16, (1976). 3. Z. Zhang, Analysis of some equadrilateral nonconforming elements for incompressible elasticity, S I A M J. Numer. Anal. 34, 460-663, (1997). 4. P.S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Com. 57, 529-550, (1991). 5, G.F. Carey, An analysis of finite element equations and mesh subdivisions, Comput. Meths. Appl. Mech. Engrg 9, 165-179, (1976). 6. Z. Shi, Convergence properties of two nonconforming finite element, Comput Methods. Appl. Mech. Engrg. 48, 123-139, (1985). 7. L. Zhang, Superconvergent recoveries of Carey's nonconforming element approximation, Comm. Numer. Methods. Engrg. (to appear). 8. Z. Shi, A convergence condition for the quadrilateral Wilson element, Numer. Math. 44, 349-361, (1984).