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MATHEMATICS OF COMPUTATION Volume 66, Number 220, October 1997, Pages 1375–1388 S 0025-5718(97)00903-4
EXTENSION THEOREMS FOR PLATE ELEMENTS WITH APPLICATIONS JINSHENG GU AND XIANCHENG HU
Abstract. Extension theorems for plate elements are established. Their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented. Numerical results support our theory.
1. Introduction Consider the plate bending problem with the clamped boundary conditions ( 2 in Ω, 4 u=f (1.1) ∂u =0 on ∂Ω, u= ∂ν where Ω ⊂ 0. By Taylor’s formula and the fact that v|ej and v|ej−1 agree at the two endpoints of ∂ej ∩ ∂ej−1 , it is easy to obtain i ∂(v|ej ) ∂(v|ej−1 ) hh 2 (e ) + |v|W 2 (e (p1 ) − (p1 )| ≤ |v|W∞ , | j ∞ j−1 ) ∂s ∂s 2 ∂(v|e )
where s is the arc length along ∂ej ∩ ∂ej−1 . Similarly since ∂ν j and agree at the midpoint of ∂ej ∩ ∂ej−1 , we get i ∂(v|ej ) ∂(v|ej−1 ) hh 2 (e ) + |v|W 2 (e (p1 ) − (p1 )| ≤ |v|W∞ . | ) j ∞ j−1 ∂ν ∂ν 2 Therefore, we have i h 2 (e ) + |v|W 2 (e . |∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 )| ≤ ch |v|W∞ j ∞ j−1 )
∂(v|ej−1 ) ∂ν
Let p01 ∈ ∂e1 ∩ (∂Ωk \Γ) be another endpoint of the edge ∂e1 ∩ (∂Ωk \Γ). Since ∂(v|e1 ) v(p1 ) = v(p01 ) = 0, there exists a point q ∈ ∂e1 ∩ (∂Ωk \Γ), s.t. ∂s (q) = 0. Obviously, Then
∂(v|e1 ) ∂ν (m1 )
= 0, where m1 is the midpoint of the edge ∂e1 ∩ (∂Ωk \Γ).
∂(v|e1 ) ∂(v|e1 ) ∂(v|e1 ) 2 (e ) , (p1 )| = | (p1 ) − (q)| ≤ h|v|W∞ 1 ∂s ∂s ∂s ∂(v|e1 ) ∂(v|e1 ) ∂(v|e1 ) 2 (e ) . (p1 )| = | (p1 ) − (m1 )| ≤ h|v|W∞ | 1 ∂ν ∂ν ∂ν 2 (e ) . So |∂α (v|e1 )(p1 )| ≤ ch|v|W∞ 1 |
|∂α (w − w)(p ˜ 1 )|
= |∂α (v|e )(p1 )| J h i X ∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 ) + ∂α (v|e1 )(p1 )| =| j=2
≤
J X
|∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 )| + |∂α (v|e1 )(p1 )|
j=2
≤ ch
J X j=1
2 (e ) ≤ c |v|W∞ j
J X
|v|2,ej ≤ c
j=1
X
|v|2,e0 ,
e0
where e0 ⊂ Ωk s.t. ∂e0 ∩ ∂e 6= ∅. If p1 6∈ ∂Ωk \Γ, then by the same argument as above, we can easily obtain X |∂α (w − w)(p ˜ 1 )| ≤ c |v|2,e0 . e0
Therefore, we have kv − Ihk vkL2 (e) ≤ ch2
X
|v|2,e0 .
e0
Summing up the square of the last inequality over all the elements e ⊂ Ωk , we eventually get (2.6) by the quasi–uniformness of the mesh Ωh . I J In what follows, pi i=1 denotes the set of the vertices on Γ and mj j=1 the set of the edge midpoints on Γ. Let νk (k = 1, 2) be the unit outward normal vector
1382
JINSHENG GU AND XIANCHENG HU
of Ωk . r0 : Vh →
EXTENSION THEOREMS FOR PLATE ELEMENTS WITH APPLICATIONS JINSHENG GU AND XIANCHENG HU
Abstract. Extension theorems for plate elements are established. Their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented. Numerical results support our theory.
1. Introduction Consider the plate bending problem with the clamped boundary conditions ( 2 in Ω, 4 u=f (1.1) ∂u =0 on ∂Ω, u= ∂ν where Ω ⊂ 0. By Taylor’s formula and the fact that v|ej and v|ej−1 agree at the two endpoints of ∂ej ∩ ∂ej−1 , it is easy to obtain i ∂(v|ej ) ∂(v|ej−1 ) hh 2 (e ) + |v|W 2 (e (p1 ) − (p1 )| ≤ |v|W∞ , | j ∞ j−1 ) ∂s ∂s 2 ∂(v|e )
where s is the arc length along ∂ej ∩ ∂ej−1 . Similarly since ∂ν j and agree at the midpoint of ∂ej ∩ ∂ej−1 , we get i ∂(v|ej ) ∂(v|ej−1 ) hh 2 (e ) + |v|W 2 (e (p1 ) − (p1 )| ≤ |v|W∞ . | ) j ∞ j−1 ∂ν ∂ν 2 Therefore, we have i h 2 (e ) + |v|W 2 (e . |∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 )| ≤ ch |v|W∞ j ∞ j−1 )
∂(v|ej−1 ) ∂ν
Let p01 ∈ ∂e1 ∩ (∂Ωk \Γ) be another endpoint of the edge ∂e1 ∩ (∂Ωk \Γ). Since ∂(v|e1 ) v(p1 ) = v(p01 ) = 0, there exists a point q ∈ ∂e1 ∩ (∂Ωk \Γ), s.t. ∂s (q) = 0. Obviously, Then
∂(v|e1 ) ∂ν (m1 )
= 0, where m1 is the midpoint of the edge ∂e1 ∩ (∂Ωk \Γ).
∂(v|e1 ) ∂(v|e1 ) ∂(v|e1 ) 2 (e ) , (p1 )| = | (p1 ) − (q)| ≤ h|v|W∞ 1 ∂s ∂s ∂s ∂(v|e1 ) ∂(v|e1 ) ∂(v|e1 ) 2 (e ) . (p1 )| = | (p1 ) − (m1 )| ≤ h|v|W∞ | 1 ∂ν ∂ν ∂ν 2 (e ) . So |∂α (v|e1 )(p1 )| ≤ ch|v|W∞ 1 |
|∂α (w − w)(p ˜ 1 )|
= |∂α (v|e )(p1 )| J h i X ∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 ) + ∂α (v|e1 )(p1 )| =| j=2
≤
J X
|∂α (v|ej )(p1 ) − ∂α (v|ej−1 )(p1 )| + |∂α (v|e1 )(p1 )|
j=2
≤ ch
J X j=1
2 (e ) ≤ c |v|W∞ j
J X
|v|2,ej ≤ c
j=1
X
|v|2,e0 ,
e0
where e0 ⊂ Ωk s.t. ∂e0 ∩ ∂e 6= ∅. If p1 6∈ ∂Ωk \Γ, then by the same argument as above, we can easily obtain X |∂α (w − w)(p ˜ 1 )| ≤ c |v|2,e0 . e0
Therefore, we have kv − Ihk vkL2 (e) ≤ ch2
X
|v|2,e0 .
e0
Summing up the square of the last inequality over all the elements e ⊂ Ωk , we eventually get (2.6) by the quasi–uniformness of the mesh Ωh . I J In what follows, pi i=1 denotes the set of the vertices on Γ and mj j=1 the set of the edge midpoints on Γ. Let νk (k = 1, 2) be the unit outward normal vector
1382
JINSHENG GU AND XIANCHENG HU
of Ωk . r0 : Vh →
