with digital signature within the project DML-CZ: The Czech Digital · Mathematics ... The application of the extrapolation method to eigenvalue problems was first.
Applications of Mathematics
Shanghui Jia; Hehu Xie; Xiaobo Yin; Shaoqin Gao Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods Applications of Mathematics, Vol. 54 (2009), No. 1, 1--15
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54 (2009)
APPLICATIONS OF MATHEMATICS
No. 1, 1–15
APPROXIMATION AND EIGENVALUE EXTRAPOLATION OF STOKES EIGENVALUE PROBLEM BY NONCONFORMING FINITE ELEMENT METHODS* Shanghui Jia, Beijing, Hehu Xie, Beijing, Xiaobo Yin, Beijing, Shaoqin Gao, Baoding (Received February 5, 2007)
Abstract. In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions rot for two nonconforming finite elements, Qrot 1 and EQ1 . Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. Keywords: Stokes eigenvalue problem, stream function-vorticity-pressure method, asymptotic expansion, extrapolation, a posteriori error estimates, nonconforming finite element methods MSC 2010 : 65N30, 65N25, 35Q30
1. Introduction There are various approximation methods for solving the Stokes problem, see Bercovier and Pironneau [3], Brezzi et al. [4], Chen and Lin [5], Girault and Raviart [9], Glowinski and Pironneau [10], Han [11], Křížek [13], Mercier et al. [20], Rannacher and Turek [21], Wang and Ye [23], Ye [25], Zhou and Li [26], and references cited therein. rot In this article we will study two nonconforming finite elements, Qrot 1 and EQ1 , for the eigenvalue approximations of the Stokes eigenvalue problems. For simplicity * This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.
1
we consider the model eigenvalue problem stated −∆u + grad p = λu div u = 0 (1) u=0
as follows: in Ω, in Ω, on ∂Ω,
where Ω is a rectangular domain in R2 . For simplicity, we take Ω = (0, 1) × (0, 1). It is well known that the extrapolation method is an efficient procedure for improving the accuracy of approximation of many problems in numerical analysis. The effectiveness of this technique relies heavily on the existence of an asymptotic expansion for the error. This technique has been well demonstrated in its application to the finite element methods [17], [18] and [22]. The application of the extrapolation method to eigenvalue problems was first proposed by Q. Lin and T. Lü [16], and was analyzed in [14], [15], [17], [19], and [22]. In [5], the bilinear finite element has been analyzed for the Stokes eigenvalue problem and the error asymptotic expansion and the extrapolation formula were given. This paper is organized in the following way. In Section 2 we present asymptotic rot expansions for nonconforming finite elements, Qrot 1 and EQ1 . The analysis for the eigenvalue problem is given in Section 3, and in Section 4 we derive the expansions of the eigenvalue error, using some integral identities and the Bramble-Hilbert lemma in our analysis. Section 5 is devoted to extrapolation and an a posteriori error estimate for eigenvalue approximations. And finally, in Section 6 numerical experiments are reported. Throughout this article we shall use the standard notation as in Chen [7] and Ciarlet [8], for example, the notation of the Sobolev space, product, norms, seminorms, and discretized norms. The main technique we use is the eigenvalue error expansion technique first proposed by Lin and Lü [16], and the Bramble-Hilbert lemma [15]. rot 2. Qrot 1 and EQ1 elements
Let Th = {e} be a rectangular partition over Ω, where e = [xe − he , xe + he ] × [ye − ke , ye + ke ], h = max{he , ke }. Moreover, Th is regular, i.e. e
C0 h2 6 meas(e) 6 C1 h2 where Ci > 0 (i = 0, 1). 2
∀ e ∈ Th ,
The Qrot 1 finite element space Vh is defined as follows: Vh := {v ∈ L2 (Ω) ; v|e ∈ span{1, x, y, x2 − y 2 }, ∀ e ∈ Th } with the interpolation uI ∈ Vh which is defined by the edge conditions Z Z uI ds, i = 1, 2, 3, 4, u ds = li
li
where li (i = 1, 2, 3, 4) are the four edges of e. In addition, set V0h = {v ∈ Vh ; v|∂Ω = 0}. The EQrot 1 finite element space Wh is defined as follows: Wh := {v ∈ L2 (Ω) ; v|e ∈ span{1, x, y, x2 , y 2 }, ∀ e ∈ Th } with the interpolation uI ∈ Wh which is defined by the edge-surface conditions Z Z uI ds, i = 1, 2, 3, 4, u ds = Zli Z li uI dx dy, u dx dy = e
e
where li (i = 1, 2, 3, 4) are the four edges of e. In addition, set W0h = {v ∈ Wh ; v|∂Ω = 0}. It is obvious that Vh 6⊆ H 1 (Ω) and Wh 6⊆ H 1 (Ω). We have the following integral expansions (see [15]): Lemma 2.1. For all v ∈ Vh or Wh and u ∈ H 5 (Ω) we have Z X k2 Z X Z ∂u 4k 4 e v ds = (2) uxxy vy dx dy − e uxxyy vyy dx dy ∂n 3 e 45 e e∈Th e∈Th ∂e Z Z h2 4h4 + e uyyx vx dx dy − e uyyxxvxx dx dy 3 e 45 e + O(h4 )|u|5 |v|1,h
and furthermore, if Th is uniform and ∂u/∂n|∂Ω = 0 or v|∂Ω = 0, we have Z X Z ∂u h2 + k 2 (3) v ds = − uxxyy v dx dy + O(h4 )kuk5 kvk2,h , 3 Ω ∂e ∂n e∈Th
where h and k are the mesh step sizes in x- and y-directions, respectively.
3
Lemma 2.2. If v ∈ Vh , Th is a square mesh and uI ∈ Vh then X Z
(4)
e∈Th
(5)
h2 (u − uI )v dx dy = − 6 Ω
Z
∇(u − uI )∇v dx dy = 0,
e
Z
(uxx + uyy )v dx dy + O(h4 )kuk4 kvk1,h .
Ω
Lemma 2.3. For all v ∈ Wh , uI ∈ Wh we have X Z
(6)
e∈Th
(7)
Z
∇(u − uI )∇v dx dy = 0,
e
(u − uI )v dx dy = O(h4 )kuk4 kvk1,h .
Ω
3. Approximation of the Stokes eigenvalue problem In this section we consider a stream function-vorticity-pressure method to solve the eigenvalue problem (1). We introduce the stream function ψ for the velocity (u = curl ψ = (∂2 ψ, −∂1 ψ)), based on the identities ([4], [8]) curl(curl u) = − ∆u + grad(div u), curl(curl ψ) = − ∆ψ where curl u = −∂2 u1 +∂1 u2 . Problem (1) can be expressed as the following buckling plate problem: Find λ, ψ satisfying −∆2 ψ = λ∆ψ ∂ψ ψ = =0 ∂n
(8)
in Ω, on ∂Ω,
where n is the outward unit normal. Then we can obtain the following weak mixed formulation for (8) which seeks λ ∈ R, (ψ, ω) ∈ H01 (Ω) × H 1 (Ω) satisfying s(ψ, ψ) = 1 and (9) 4
(
a(ω, θ) + b(θ, ψ) = 0 ∀ θ ∈ H 1 (Ω), b(ω, ϕ) = −λs(ψ, ϕ)
∀ ϕ ∈ H01 (Ω),
and find p ∈ H 1 (Ω) such that (grad p, grad q) = λ(u − curl ω, grad q) ∀ q ∈ H 1 (Ω), Z (10) p dx = 0 Ω
with u = curl ψ, where ω = −∆ψ and Z a(ω, θ) = ωθ ∀ ω, θ ∈ H 1 (Ω), Ω Z b(ω, ϕ) = − curl ω curl ϕ dx dy ∀ ω ∈ H 1 (Ω), ϕ ∈ H01 (Ω), Z Ω curl ψ curl ϕ dx dy ∀ ψ ∈ H01 (ω), ϕ ∈ H01 (Ω). s(ψ, ϕ) = Ω
Problem (9) has an eigenvalue sequence {λj } ([1], [2]): 0 < λ1 6 λ2 6 . . . 6 λk 6 . . . ,
lim λk = ∞,
k→∞
and the associated eigenfunctions (ψ1 , ω1 ), (ψ2 , ω2 ), . . . , (ψk , ωk ), . . . , where (curl ψi , curl ψj ) = δij , ωk = −∆ψk . The finite element approximation of (9) is to seek λh ∈ R, (ψh , ωh ) ∈ V0h × Vh or W0h × Wh such that sh (ψh , ψh ) = 1 and ( ah (ωh , θ) + bh (θ, ψh ) = 0 ∀ θ ∈ Vh or Wh , (11) bh (ωh , ϕ) = −λh sh (ψh , ϕ) ∀ ϕ ∈ V0h or W0h , and find ph ∈ Vh or Wh such that (grad ph , grad qh )h = λ(uh − curl ωh , grad qh )h ∀ qh ∈ Vh or Wh , Z (12) ph dx = 0 Ω
with uh = curl ψh , where
Z
ωθ dx dy ∀ ω, θ ∈ Vh or Wh , ah (ω, θ) = Ω Z X bh (ω, ϕ) = − curl ω curl ϕ dx dy ∀ ω ∈ Vh or Wh , ϕ ∈ V0h or W0h , e∈T
sh (ψ, ϕ) =
h X Z
e∈Th
e
curl ψ curl ϕ dx dy
∀ ψ ∈ V0h or W0h , ϕ ∈ V0h or W0h .
e
5
In order to get the error expansion of the eigenvalue problem we analyze the original problem first. The original problem is: Find (ψ, ω) ∈ H01 (Ω) × H 1 (Ω) such that (
(13)
a(ω, θ) + b(θ, ψ) = 0 ∀ θ ∈ H 1 (Ω), ∀ ϕ ∈ H01 (Ω).
b(ω, ϕ) = −s(g, ϕ)
The finite element approximation for (13) is: Find (Rh ψ, Rh ω) ∈ V0h × Vh or (Rh ψ, Rh ω) ∈ W0h × Wh such that (14)
(
ah (Rh ω, θ) + bh (θ, Rh ψ) = 0
∀ θ ∈ Vh or Wh ,
bh (Rh ω, ϕ) = −sh (g, ϕ)
∀ ϕ ∈ V0h or W0h .
Lemma 3.1. Assume ψ ∈ H 5 (Ω), Th is a square mesh for Qrot and a uniform 1 . Then mesh for EQrot 1 (15)
kψI − Rh ψk1,h + kωI − Rh ωk0 6 Ch2 ,
(16)
kψ − Rh ψk0 + hkψ − Rh ψk1,h + kω − Rh ωk0 6 Ch2 .
P r o o f. Since ω = −∆ψ, we have kωI − Rh ωk20 = ah (ωI − Rh ω, ωI − Rh ω) = ah (ωI − ω, ωI − Rh ω) + ah (ω − Rh ω, ωI − Rh ω) X Z = ah (ωI − ω, ωI − Rh ω) − ∆ψ(ωI − Rh ω) dx dy e∈Th
e
− ah (Rh ω, ωI − Rh ω)
= ah (ωI − ω, ωI − Rh ω) +
X Z
e∈Th
+
X Z
e∈Th
n × curl ψ(ωI − Rh ω) ds
∂e
curl ψ curl(ωI − Rh ω) dx dy + bh (ωI − Rh ω, Rh ψ)
e
= ah (ωI − ω, ωI − Rh ω) +
X Z
e∈Th
− bh (ωI − Rh ω, ψ − Rh ψ) X Z = ah (ωI − ω, ωI − Rh ω) + e∈Th
n × curl ψ(ωI − Rh ω) ds
∂e
n × curl ψ(ωI − Rh ω) ds
∂e
− bh (ωI − Rh ω, ψ − ψI ) − bh (ωI − Rh ω, ψI − Rh ψ).
6
Let us define I := ah (ωI − ω, ωI − Rh ω) +
X Z
e∈Th
n × curl ψ(ωI − Rh ω) ds
∂e
− bh (ωI − Rh ω, ψ − ψI ),
II := − bh (ωI − Rh ω, ψI − Rh ψ). Then II = − bh (ωI − ω, ψI − Rh ψ) − bh (ω − Rh ω, ψI − Rh ψ) X Z = − bh (ωI − ω, ψI − Rh ψ) + curl ω curl(ψI − Rh ψ) dx dy e
e∈Th
+ bh (Rh ω, ψI − Rh ψ)
= − bh (ωI − ω, ψI − Rh ψ) −
X Z
−
X Z
e∈Th
∆ω(ψ − Rh ψ) dx dy − sh (g, ψI − Rh ψ)
e
= − bh (ωI − ω, ψI − Rh ψ) −
X Z
X Z
e∈Th
∆g(ψI − Rh ψ) dx dy − sh (g, ψI − Rh ψ)
e
= − bh (ωI − ω, ψI − Rh ψ) −
X Z
X Z
e∈Th
n × curl ω(ψI − Rh ψ) ds
∂e
e∈Th
+
n × curl ω(ψI − Rh ψ) ds
∂e
e∈Th
−
n × curl ω(ψI − Rh ψ) ds
∂e
e∈Th
n × curl g(ψI − Rh ψ) ds +
∂e
e∈Th
− sh (g, ψI − Rh ψ) = − bh (ωI − ω, ψI − Rh ψ) −
X Z
+
e∈Th
curl g curl(ψI − Rh ψ) ds
e
n × curl ω(ψI − Rh ψ) ds
∂e
e∈Th
X Z
X Z
n × curl g(ψI − Rh ψ) ds.
∂e
Therefore, (17) kωI − Rh ωk20 = ah (ωI − ω, ωI − Rh ω) +
X Z
e∈Th
n × curl ψ(ωI − Rh ω) ds
∂e
− bh (ωI − Rh ω, ψ − ψI ) − bh (ωI − ω, ψI − Rh ψ) X Z X Z − n × curl ω(ψI − Rh ψ) ds + n × curl g(ψI − Rh ψ) ds. e∈Th
∂e
e∈Th
∂e
7
Similarly, we have CkψI − Rh ψk21,h 6 bh (ψI − Rh ψ, ψI − Rh ψ) = bh (ψI − Rh ψ, ψI − ψ) + bh (ψI − Rh ψ, ψ − Rh ψ), where bh (ψI − Rh ψ, ψ − Rh ψ) = bh (ψI − Rh ψ, ψ) − bh (ψI − Rh ψ, Rh ψ) X Z = − curl(ψI − Rh ψ) curl ψ dx dy + ah (Rh ω, ψI − Rh ψ) e∈T
=
h X Z
e∈Th
e
n × curl ψ(ψI − Rh ψ) ds +
∂e
X Z
e∈Th
∆ψ(ψI − Rh ψ) dx dy e
+ ah (Rh ω, ψI − Rh ψ) X Z X Z = n × curl ψ(ψI − Rh ψ) ds − ω(ψI − Rh ψ) dx dy e∈Th
∂e
e∈Th
e
+ ah (Rh ω, ψI − Rh ψ) X Z = n × curl ψ(ψI − Rh ψ) ds − ah (ω − Rh ω, ψI − Rh ψ) e∈Th
=
∂e
X Z
e∈Th
n × curl ψ(ψI − Rh ψ) ds − ah (ω − ωI , ψI − Rh ψ)
∂e
− ah (ωI − Rh ω, ψI − Rh ψ). Then, (18) CkψI − Rh ψk21,h 6 bh (ψI − Rh ψ, ψI − ψ) X Z + n × curl ψ(ψI − Rh ψ) ds e∈Th
∂e
− ah (ω − ωI , ψI − Rh ψ) − ah (ωI − Rh ω, ψI − Rh ψ).
From (17), (18) and Lemmas 2.1–2.3 we can prove the assertion of Lemma 3.1. Here we assume that all eigenvalues have ascent and their geometric multiplicity is one. From Lemma 3.1 and the results of [6], [12], [20], and [24] we have the following theorem.
8
Theorem 3.1. Under the conditions of Lemma 3.1 let us assume that (λ, ψ, ω) ∈ R × H01 (Ω) × H 1 (Ω) is an eigenpair of (9) and (λh , ψh , ωh ) ∈ R × V0h × Vh or R × W0h × Wh is an eigenpair of (11). Then (19)
|λh − λ| 6 Ch2 ,
(20)
kω − ωh k0 6 Ch2 ,
(21)
kψ − ψh k0 + hkψ − ψh k1,h 6 Ch2 .
rot 4. Asymptotic eigenvalue error expansions by Qrot 1 and EQ1 rot Theorem 4.1. When we use the finite element spaces Qrot 1 and EQ1 and under the conditions of Theorem 3.1, we have
(22)
λh − λ = λh sh (ψ − ψI , ψ h ) + bh (ω h , ψ − ψI ) X Z − ωh n × curl ψ ds + ah (ω − ωI , ωh ) e∈Th
∂e
+ bh (ω − ωI , ψ h ) −
X Z
e∈Th
+λ
X Z
e∈Th
where ψh =
∂e
∂e
n × curl ωψ h ds
n × curl ψψ h ds,
ψh , sh (ψ, ψ h )
ωh =
ωh . sh (ψ, ψh )
P r o o f. It is obvious that sh (ψ, ψ h ) = 1. Then λh = λh sh (ψ, ψ h ) = λh sh (Rh ψ, ψ h ) + λh sh (ψ − Rh ψ, ψ h ) := I + Π, where (Rh ψ, Rh ω) is the solution of (14) with g = λψ. Then I = λh sh (Rh ψ, ψ h ) = −bh (ω h , Rh ψ) = ah (Rh ω, ωh ) = ah (ω h , Rh ω) = − bh (Rh ω, ψ h ) = λsh (ψ, ψ h ) = λ, 9
Π = λh sh (ψ − Rh ψ, ψ h ) = λh (ψ − ψI , ψ h ) + λh sh (ψI − Rh ψ, ψ h ) = λh (ψ − ψI , ψ h ) − bh (ω h , ψI − Rh ψ) = λh (ψ − ψI , ψ h ) − bh (ω h , ψI − ψ) − bh (ω h , ψ − Rh ψ) := Π1 + Π2 , where Π1 = λh sh (ψ − ψI , ψ h ) − bh (ωh , ψI − ψ), and since ω = −∆ψ, Π2 = − bh (ω h , ψ − Rh ψ) X Z = curl ω h curl ψ dx dy + bh (ω h , Rh ψ) e∈Th
= −
e
X Z
e∈Th
ωh n × curl ψ ds +
∂e
X Z
e∈Th
ωh curl(curl ψ) dx dy
e
− ah (Rh ω, ω h ) X Z X Z = − ωh n × curl ψ ds − ωh ∆ψ dx dy − ah (Rh ω, ωh ) e∈Th
= −
e∈Th
= −
= − = −
ωωh dx dy − ah (Rh ω, ωh )
e
ωh n × curl ψ ds + ah (ω − ωI , ω h ) + ah (ωI − Rh ω, ωh )
∂e
∂e
X Z
e∈Th
e∈Th
ωh n × curl ψ ds + ah (ω − Rh ω, ωh )
X Z
e∈Th
e
X Z
∂e
X Z
e∈Th
e∈Th
ωh n × curl ψ ds +
∂e
X Z
e∈Th
= −
∂e
X Z
ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − Rh ω, ψ h ) ωh n × curl ψ ds + ah (ω − ωI , ω h )
∂e
− bh (ωI − ω, ψ h ) − bh (ω − Rh ω, ψh ) X Z = − ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − ω, ψh ) e∈Th
+
e∈Th
10
∂e
X Z
e
curl ω curl ψ h dx dy + bh (Rh ω, ψh )
= −
X Z
e∈Th
−
X Z
e∈Th
+
−
∂e
∂e
X Z
e∈Th
∂e
X Z
e∈Th
+λ
∂e
X Z
X Z
e∈Th
−
∂e
X Z
e∈Th
n × curl ωψ h ds
ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − ω, ψh ) n × curl ωψ h ds −
∂e
X Z
e∈Th
e
∆ωψh dx dy − λsh (ψ, ψ h )
ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − ω, ψh ) n × curl ωψ h ds
e
e∈Th
= −
ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − ω, ψh )
curl(curl ω)ψ h dx dy + bh (Rh ω, ψ h )
X Z
e∈Th
= −
e
X Z
e∈Th
−
∂e
X Z
e∈Th
= −
∂e
curl(curl ψ)ψ h dx dy − λsh (ψ, ψ h )
ωh n × curl ψ ds + ah (ω − ωI , ω h ) − bh (ωI − ω, ψh ) n × curl ωψ h ds + λ
X Z
e∈Th
∂e
n × curl ψψ h ds.
Thus we get the theorem.
Theorem 4.2. If ψ ∈ H 5 (Ω) and ω ∈ H 5 (Ω) then for the finite element Qrot 1 , when Th is a square mesh, we have (23)
Z Z h2 4h2 ∆2 ψ∆ψ dx dy + ∆ψψxxyy dx dy 6 Ω 3 Ω Z 2λh2 ψxxyy ψ dx dy + O(h4 ), + 3 Ω
λh − λ = −
and for the finite element EQrot 1 , if Th is uniform, we have (24)
Z 2(h2 + k 2 ) ψxxyy ∆ψ dx dy λh − λ = − 3 Ω Z λ(h2 + k 2 ) + ψxxyy ψ dx dy + O(h4 ). 3 Ω 11
P r o o f. It is obvious that ([15]) kψ h − ψh k0 6 ch2 ,
kωh − ωh k0 6 ch2 .
Then Lemma 2.1–2.3, Theorem 3.1 and Theorem 4.1 yield the assertion of this theorem.
5. Extrapolation and an a posteriori error estimate for eigenvalues In order to use the extrapolation method, we assume that Th/2 has been obtained from Th by dividing each element into four congruent rectangles by connecting the midpoints of its edges. Let (λh/2 , ψh/2 , ωh/2 ) be the eigensolution approximation on the mesh Th/2 . Denote by (25)
λextra = h
4λh/2 − λh 3
the extrapolation of λ. Then by Theorem 4.2 we get the following error estimate for ˜h and an a posteriori error estimate for the eigenvalue. the extrapolation λ Theorem 5.1. Under the conditions of Theorem 4.2 we have (26)
λ − λextra = O(h4 ) h
and thus, (27)
λh/2 − λ =
λh − λh/2 + O(h4 ) 3
provides an a posteriori error estimate 31 (λh − λh/2 ) for λh/2 − λ.
12
6. Numerical results First, we introduce some notation errh = λh − λ, 1 λextra = (4λh/2 − λh ), h 3 extra errextra = λ − λ, h h log |errh |/|errh/2 | Rh = , log(2) Rhextra
=
log |errextra |/|errextra h h/2 | log(2)
.
We compute the first eigenvalue and take λ = 52.3446911 (accurate enough). M ×N λh λextra h
4×4
8×8
16 × 16
32 × 32
64 × 64
52.15082488284 52.31809045313 52.34015032048 52.34368098538 52.34444610834 52.37384564323 52.34750360960 52.34485787368 52.34470114933
−
errh −0.19386621716 −0.02660064687 −0.00454077952 −0.00101011462 −0.00024499166 errextra h
−
0.02915454323
0.00281250960
0.00016677368
0.00001004933
Rh
−
2.86552818903
2.55044943728
2.16842097918
2.04371449127
extra Rh
−
−
3.37379079440
4.07589449578
4.0527200264
Table 1. Computation of the first eigenvalue by Qrot 1 . M ×N λh λextra h errh
4×4
8×8
16 × 16
32 × 32
47.18962964955835 50.70035642020135 51.90518931765972 52.23288933622509 −
51.87059867708234 52.30680028347918 52.34212267574687
−5.15506145044165 −1.64433467979865 −0.43950178234028 −0.11180176377491
errextra h
−
Rh
−
1.64848565724684
1.90356304681561
1.97492606784368
extra Rh
−
−
3.64524820130709
3.88289279718122
−0.47409242291766 −0.03789081652081 −0.00256842425313
Table 2. Computation of the first eigenvalue by
EQrot 1 .
From Tabs. 1 and 2 we can find that with the extrapolation the approximation accuracy can be improved from O(h2 ) to O(h4 ), which validates the corresponding theoretical result in Theorem 5.1 computationally. The extrapolation of the eigenvalue gives a more efficient approximation.
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7. Concluding remarks rot The nonconforming finite elements, Qrot 1 and EQ1 , can give optimal error estimates under some conditions for the stream function-vorticity-pressure method of the Stokes eigenvalue problems. The eigenvalue extrapolation can improve the accuracy of the eigenvalue approximations and give an a posteriori error estimate. We can also apply Qrot and EQrot elements to other problems and also use the 1 1 extrapolation method to improve the accuracy order. We also need to notice that the extrapolation method may give “good” results even though the true solution does not satisfy regularity assumptions guaranteeing superconvergence theoretically.
Acknowledgement. The authors would like to express their grateful thanks to their supervisor Prof. Qun Lin for his supervision. References [1] I. Babuška, J. F. Osborn: Estimate for the errors in eigenvalue and eigenvector approximation by Galerkin methods with particular attention to the case of multiple eigenvalue. SIAM J. Numer. Anal. 24 (1987), 1249–1276. [2] I. Babuška, J. F. Osborn: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52 (1989), 275–297. [3] M. Bercovier, O. Pironneau: Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979), 211–224. [4] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol. 15. Springer, New York, 1991. [5] W. Chen, Q. Lin: Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method. Appl. Math. 51 (2006), 73–88. [6] W. Chen, Q. Lin: Asymptotic expansion and extrapolation for the eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme. Adv. Comput. Math. 27 (2007), 95–106. [7] Z. Chen: Finite Element Methods and Their Applications. Springer, Berlin, 2005. [8] P. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. [9] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Berlin, 1986. [10] R. Glowinski, O. Pironneau: On a mixed finite element approximation of the Stokes problem. I. Convergence of the approximate solution. Numer. Math. 33 (1979), 397–424. [11] H. Han: Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984), 223–233. [12] S. Jia, H. Xie, X. Yin, S. Gao: Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods. Numer. Methods Partial Differ. Equations 24 (2008), 435–448. [13] M. Křížek: Conforming finite element approximation of the Stokes problem. Banach Cent. Publ. 24 (1990), 389–396.
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[14] Q. Lin, H. Huang, Z. Li: New expansion of numerical eigenvalue for −∆u = λ̺u by nonconforming elements. Math. Comput. 77 (2008), 2061–2084. [15] Q. Lin, J. Lin: Finite Element Methods: Accuracy and Improvement. China Sci. Press, Beijing, 2006. [16] Q. Lin, T. Lü: Asymptotic expansions for finite element eigenvalues and finite element solution. Bonn. Math. Schrift 158 (1984), 1–10. [17] Q. Lin, N. Yan: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Press, Hebei, 1996. (In Chinese.) [18] Q. Lin, S. Zhang, N. Yan: Extrapolation and defect correction for diffusion equations with boundary integral conditions. Acta Math. Sci. 17 (1997), 405–412. [19] Q. Lin, Q. Zhu: Preprocessing and Postprocessing for the Finite Element Method. Shanghai Sci. Tech. Publishers, Shanghai, 1994. (In Chinese.) [20] B. Mercier, J. Osborn, J. Rappaz, P.-A. Raviat: Eigenvalue approximation by mixed and hybrid method. Math. Comput. 36 (1981), 427–453. [21] R. Rannacher, S. Turek: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111. [22] V. Shaidurov: Multigrid Methods for Finite Elements. Kluwer Academic Publishers, Dordrecht, 1995. [23] J. Wang, X. Ye: Superconvergence of finite element approximations for the Stokes problem by projection methods. SIAM J. Numer. Anal. 39 (2001), 1001–1013. [24] Y. Yang: An Analysis of the Finite Element Method for Eigenvalue Problems. Guizhou People Public Press, Guizhou, 2004. (In Chinese.) [25] X. Ye: Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Partial Differ. Equations 18 (2002), 143–154. [26] A. Zhou, J. Li: The full approximation accuracy for the stream function-vorticitypressure method. Numer. Math. 68 (1994), 427–435. Authors’ addresses: S. Jia, School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, P. R. China, e-mail: shjia lsec.cc.ac.cn; H. Xie, X. Yin, LSEC, ICMSEC, Academy of Mathematics and Systems Science, CAS, Beijing 100190, P. R. China, e-mail: hhxie lsec.cc.ac.cn, yinxb lsec.cc.ac.cn; S. Gao, College of Mathematics and Computer, Hebei University, Baoding 071002, P. R. China, e-mail: gaoshq amss.ac.cn.
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