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CHAI Zhao YANG ZhiHua ZHANG RuiYan [email protected] [email protected] [email protected] SCIENCE CHINA Mathematics

. ARTICLES . June 2013 Vol. 56 No. 6: 1313–1330 doi: 10.1007/s11425-013-4582-4

Upper spectral bounds and a posteriori error analysis of several mixed ﬁnite element approximations for the Stokes eigenvalue problem

YANG YiDu1,∗ & JIANG Wei2

1School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001,China; 2School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005,China Email: [email protected], [email protected]

Received November 21, 2011; accepted January 9, 2013; published online January 30, 2013

Abstract This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. (2) As for the P1-P1, Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact a posteriori error indicators are presented. (3) The reliable and efficient a posteriori error estimator proposed by Verfürth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis.

Keywords the Stokes eigenvalue, conforming mixed ﬁnite elements, upper spectral bounds, a posteriori error estimates

MSC(2010) 65N25, 65N30

Citation: Yang Y D, Jiang W. Upper spectral bounds and a posteriori error analysis of several mixed ﬁnite element approximations for the Stokes eigenvalue problem. Sci China Math, 2013, 56: 1313–1330, doi: 10.1007/s11425-013-4582-4

1 Introduction

The finite element eigenvalues approximate the exact ones from above or from below is always an active topic in academic community. It is well known that the minimum-maximum principle insures that the conforming finite element eigenvalues approximate the exact eigenvalues from above (see [34]). Unfor- tunately, the classic result is not valid for the nonconforming finite elements. It has been proved that the numerical eigenvalues by many nonconforming element/nonconforming mixed elements approximate exact eigenvalues from below (see [1, 14, 22–24, 27, 40, 42, 43], etc.). Then how about the conforming mixed finite elements? It can be observed from some numerical examples that the numerical eigenvalues obtained by some conforming mixed finite elements have the properties of approximations from above, but there also have converse results. For example, as for the Stokes eigenvalue problem, the numerical results in [44] indicate that the mixed Bernadi-Raugel element eigenvalues approximate from below, while the mixed Q2-P1 element eigenvalues approximate from above. Moreover, the numerical results in [21] demonstrate that, for the Laplace operator eigenvalue problem, the approximate eigenvalues obtained by low order mixed Raviart-Thomas element approximate exact eigenvalues from above. However, there is

∗Corresponding author

c Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com www.springerlink.com 1314 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 no theoretical proof for these phenomena. The first objective of this paper is to study the approximate properties of numerical eigenvalues obtained by some mixed finite elements for the Stokes eigenvalue problem. The a posteriori error estimates and adaptive computation in finite element methods were first proposed by Babuˇska and Rheinboldt in 1978 (see [3]), and have been developed into one of the main fields in finite element method. In the development of this field, a posteriori error estimate of finite element approximation for eigenvalue problem is also an important topic. For elliptic eigenvalue problems, there are many research results (see, for example, [5,10,11,13,17,18,28,30,39]). For the Stokes eigenvalue problem, Lovadina et al. [26] have presented a suitable error indicator based on residual type. The second objective of this paper is to study a posteriori error estimates of some mixed finite element approximations for the Stokes eigenvalue problem further. To fulfill these two goals, the theory of Babuˇska-Brezzi (see [7]) and the theory of spectral approximations (see [4, 8]) are adopted in this paper. Firstly, several mixed finite element identities for the Stokes eigenvalue problem are proved. Based on these identities, the following results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. (2) As for the P1-P1, Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact recovery type by post-processing interpolation a posteriori error indicators are presented, this is a new work, about which we have not seen the report until now. (3) The reliable and efficient a posteriori error estimator proposed by Verfürth [35] is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis. The rest of this paper is organized as follows. In the next section, some preliminaries needed in this paper are presented, and several mixed element identities are proved. In Sections 3 and 4, we prove the property of upper spectral bounds and present a posteriori error of mini-element and P1-P1 element, respectively. In Section 5, we analyze a posteriori error of Q1-P0 element. Finally, numerical experiments are presented to support the theoretical analysis.

2 The mixed ﬁnite element identities for the Stokes eigenvalue problem

Consider the Stokes eigenvalue problem

− Δu + ∇p = λu, in Ω, divu =0, in Ω, (2.1) u =0, on ∂Ω,

⊂ R2 where Ω is a polygonal domain, u =(u1,u2) is the velocity of the ﬂow, and p is the pressure. 1 × 1 2 { ∈ 2 } 2 × 2 Denote U = H0 (Ω) H0 (Ω),V= L0(Ω) = v L (Ω) : Ω v =0 and H = L (Ω) L (Ω). In this paper, the above symbol denotes the element in product space U or H.SetD ⊂ Ω. For any function p in Hm(D), denote

1 2 α 2 pm,D = |∂ p| ,α=(α1,α2), |α| = α1 + α2. |α|m D

For simplicity, write pm,Ω = pm. For the vector function u =(u1,u2), denote

1 1 2 2 2 2 2 2 um,D =(u1m,D + u2m,D) , |u|m,D =(|u1|m,D + |u2|m,D) .

Write um,Ω = um, |u|m,Ω = |u|m.Denote

2 2 a(u, v)= ui ·vi,b(v, q)=− divvq, (u,v)= uivi. i i Ω =1 Ω Ω =1 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1315

| |2 2 Obviously, a(u, u)= u 1, u 0 =(u, u). The mixed variational form of (2.1) is given by: ﬁnd eigenpair (λ, u, p) ∈ R × U × V , u0 =1,such that

a(u,v)+b(v, p)=λ(u,v), ∀ v ∈ U, (2.2) b(u, q)=0, ∀ q ∈ V. (2.3)

In order to solve (2.2)–(2.3), the mixed ﬁnite element space Uh ⊂ U and Vh ⊂ V are constructed. Then

the discrete problem of (2.2)–(2.3) reads: ﬁnd eigenpair (λh,uh,ph) ∈ R × Uh × Vh, uh0 = 1, such that

a(uh,v)+b(v, ph)=λh(uh,v), ∀ v ∈ Uh, (2.4)

b(uh,q)=0, ∀ q ∈ Vh. (2.5)

Consider the source problem associated with the Stokes eigenvalue problem (2.2)–(2.3) and its discrete mixed ﬁnite element form: ﬁnd (ψ,ϕ ) ∈ U × V , such that

a(ψ, v)+b(v,ϕ)=(f, v), ∀ v ∈ U, (2.6) b(ψ, q)=0, ∀ q ∈ V ; (2.7)

ﬁnd (ψh,ϕh) ∈ Uh × Vh, such that

a(ψh,v)+b(v, ϕh)=(f, v), ∀ v ∈ Uh, (2.8)

b(ψh,q)=0, ∀ q ∈ Vh, (2.9)

where f ∈ H. As Brezzi Theorem pointed out, (2.6)–(2.7) has a unique solution (ψ,ϕ ) ∈ U × V , and the following a prior error estimate is valid:

ψ1 + ϕ0 M1f0, (2.10)

where M1 is positive constant independent of f. Assume that the mixed ﬁnite element spaces Uh and Vh satisfy Babuˇska-Brezzi stability condition. Then (2.8)–(2.9) also has a unique solution (ψh,ϕh) ∈ Uh × Vh, and the following error estimate is valid (see [6, 7, 33]): ψ − ψh1 + ϕ − ϕh0 C inf ψ − v1 +infϕ − q0 . (2.11) v∈Uh q∈Vh

Throughout this paper, C denotes a generic positive constant independent of h,whichmaynotbethe same at each occurrence. 2 1+r 2 r We assume that the following regularity is valid: For any f ∈ L2(Ω) ,(ψ,ϕ ) ∈ H (Ω) × H (Ω) and there exists a positive constant M2 independent of f such that

ψ1+r + ϕr M2f0,

for some r ∈ (0, 1] depending on Ω. When Ω is a convex polygonal domain, we have r = 1 (see [16]). r When the approximate order of Uh and Vh is at least h under the meaning of norm ·1 and norm ·0, respectively, according to Aubin-Nitsche technique, we can prove (see [6,7,12,33]) r ψ − ψh0 Ch inf ψ − v1 +infϕ − q0 . (2.12) v∈Uh q∈Vh

Deﬁne linear bounded operators (see [4]):

T : H → U ⊂ H, S : H → V ⊂ L2(Ω), ∀ f ∈ H, 1316 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

a(T f, v)+b(v, Sf)=(f, v), ∀ v ∈ U, b(T f,q )=0, ∀ q ∈ V ; and

Th : H → Uh ⊂ H, Sh : H → Vh ⊂ L2(Ω), ∀f ∈ H,

a(Thf, v)+b(v, Shf)=(f, v), ∀ v ∈ Uh,

b(Thf,q )=0, ∀ q ∈ Vh.

Then (2.2)–(2.3) and (2.4)–(2.5) are equivalent to operator forms

λT u = u, p = S(λu) (2.13)

and

λhThuh = uh,ph = Sh(λhuh), (2.14)

respectively. One can prove that T and Th are all self-adjoint operators. In fact, for ∀ f, g ∈ H, taking v = Tg, q = Sg, we can infer that

a(T f,T g)+b(Tg, Sf)=(f,T g),b(T f,S g)=0; exchanging f and g,weget

a(Tg, Tf)+b(T f,S g)=(g,Tf),b(Tg, Sf)=0; thus we have

(f,T g)=a(T f,T g)+b(Tg, Sf)+b(T f,S g) = a(Tg, Tf)+b(T f,S g)+b(Tg, Sf)=(g,Tf)=(T f, g).

This implies that T : H → H is self-adjoint. In the similar way we can prove that Th : H → H is also self-adjoint. Taking f = λu in (2.6)–(2.7) and (2.8)–(2.9), then according to the deﬁnition of T, S, Th and Sh,itis easy to know that (T (λu),S(λu)) is the solution to (2.6)–(2.7), and (Th(λu),Sh(λu)) is the solution to (2.8)–(2.9). From (2.10) and the compact injection U→ H, we can conclude that T is completely continuous. Let (λ, u, p)and(λh,uh,ph) be the eigenpair of (2.2)–(2.3) and (2.4)–(2.5), respectively. Then taking v = u in (2.2), taking v = uh in (2.4), we have

a(u, u) a(uh,uh) λ = > 0,λh = > 0. (u, u) (uh,uh)

Therefore, the eigenvalues of (2.2)–(2.3) can be arranged by the increasing order, with each eigenvalue counted repeatedly according to its algebraic multiplicity:

0 <λ1 λ2 ··· λk ··· +∞,

and the corresponding eigenfunctions are

u1,p1,u2,p2, ..., uk,pk, ...,

where (ui,uj)=δij . The eigenvalues of (2.4)–(2.5) can be arranged as

··· ··· 0 <λ1,h λ2,h λk,h λNh,h, Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1317

and the corresponding eigenfunctions are

u1,h,p1,h,u2,h,p2,h, ..., uk,h,pk,h, ..., uNh,h,pNh,h,

where Nh =dimUh and (ui,h,uj,h)=δij .LetVλ be a function space spanned by all the eigenfunction u of (2.2)–(2.3) corresponding to the eigenvalue λ. From [4,29,38], we derive that the k-th eigenvalue λh of (2.4)–(2.5) converges to the k-th eigenvalue λ of (2.2)–(2.3), and the following lemma is valid.

Lemma 2.1. Assume that T − Th0 → 0(h → 0), (λh,uh,ph) is the k-th eigenpair of (2.4)–(2.5), and uh0 =1. Then there exists the k-th eigenpair (λ, u, p) of (2.2)–(2.3), u0 =1, such that λh → λ (h → 0),and | − | − − | λ λh + u uh 0 C (T Th) Vλ 0. (2.15)

Next, we make the more precise analysis for the error estimate. Lemma 2.2. Suppose that the conditions of Lemma 2.1 hold, then

2 λh − λ = λ ((T − Th)u, u)+R1, (2.16) 2 λh − λ = λh((T − Th)uh,uh)+R2, (2.17)

| | | | − | 2 where R1 , R2 C (T Th) Vλ 0. Proof. Since

−1 (Tuh − Thuh,u)=(Tuh,u) − (λh uh,u) −1 −1 −1 =(uh,Tu) − (λh uh,u)=(λ − λh )(uh,u),

we have

λλh λh − λ = ((T − Th)uh,u) (u, uh) λλh = (((T − Th)u, u)+((T − Th)(uh − u),u)) (u, uh) λλh 2 2 = − λ + λ (((T − Th)u, u)+((T − Th)(uh − u),u)) (u, uh) 2 ≡ λ ((T − Th)u, u)+R1.

Moreover, using the facts that T and Th are symmetric and (2.15), we deduce that

λλh 2 λλh |R1| = − λ ((T − Th)u, u)+ ((T − Th)(uh − u),u) (u, uh) (u, uh) − | 2 | − − | C (T Th) Vλ 0 + C (uh u, (T Th)u) − | 2 − − − | 2 C (T Th) Vλ 0 + C uh u 0 (T Th)u 0 C (T Th) Vλ 0. Therefore, we get (2.16) immediately. Using the same method, we may prove that (2.17) is valid. Consider the Stokes equation: ﬁnd (ψ,ϕ ) ∈ U × V , such that

a(ψ, v)+b(v, ϕ)=(λhuh,v), ∀ v ∈ U, (2.18) b(ψ, q)=0, ∀ q ∈ V. (2.19)

Evidently the generalized solution to this equation is (T (λhuh),S(λhuh)), the mixed finite element solution is (Th(λhuh),Sh(λhuh)), and (Th(λhuh),Sh(λhuh)) = (uh,ph). By the following Theorem 2.3, a posteriori error estimates of mixed finite element eigenpair (λh,uh,ph) are reduced to that of mixed finite element solution (Th(λhuh),Sh(λhuh)) corresponding to the source problem (2.18)–(2.19). 1318 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

Theorem 2.3. Assume that T − Th0 → 0(h → 0), (λh,uh,ph) is the k-th eigenpair of (2.4)–(2.5), and uh0 =1. Then there exists the k-th eigenpair (λ, u, p) of (2.2)–(2.3), u0 =1, such that

2 λh − λ = λh(a((T − Th)uh, (T − Th)uh)+2b((T − Th)uh,Suh − v)) + R1, ∀ v ∈ Vh, (2.20)

p − ph0,D = Sh(λhuh) − S(λhuh)0,D + R2, (2.21)

u − uh1,D = Th(λhuh) − T (λhuh)1,D + R3, (2.22)

| | − | 2 | | | | − | where R1 C (T Th) Vλ 0, R2 , R3 C (T Th) Vλ 0.

Proof. By the deﬁnition of T and Th, (2.7) and (2.9), we obtain

b(Tg, Sg − Shg)=0,

b((T − Th)g,v)=0, ∀ v ∈ Vh.

By the deﬁnition of T , S, Th, Sh, (2.6) and (2.8), we obtain

a(Tg − Thg, Thg)=−b(Thg,Sg − Shg)=b((T − Th)g,Sg).

Thus, for ∀g ∈ H, from the deﬁnition of T , S,weget

((T − Th)g,g)=(g,(T − Th)g)=a(Tg, (T − Th)g)+b((T − Th)g,Sg)

= a(Tg − Thg,(T − Th)g)+a((T − Th)g,Thg)+b((T − Th)g,Sg)

= a(Tg − Thg,(T − Th)g)+2b((T − Th)g,Sg − v), ∀ v ∈ Vh. (2.23)

Let (λ, u, p) satisfy Lemma 2.1. Next, we will prove this eigenpair satisﬁes (2.20)–(2.22). After taking g = λhuh in (2.23), and then substituting (2.23) into (2.17), we derive (2.20). From (2.10) and (2.15), we deduce

− − − | S(λu λhuh) 0 C λu λhuh 0 C (T Th) Vλ 0, (2.24) − − − | T (λu λhuh) 1 C λu λhuh 0 C (T Th) Vλ 0. (2.25)

Using triangle inequality and (2.24), we derive

|p − ph0,D −Sh(λhuh) − S(λhuh)0,D|

= |S(λu) − Sh(λhuh)0,D −Sh(λhuh) − S(λhuh)0,D| − − | S(λu λhuh) 0,D C (T Th) Vλ 0.

Writing R2 = p−ph0,D −Sh(λhuh)−S(λhuh)0,D, we get (2.21). Using triangle inequality and (2.25), we infer that

|u − uh1,D −Th(λhuh) − T (λhuh)1,D|

= |T (λu) − Th(λhuh)1,D −Th(λhuh) − T (λhuh)1,D| − − | T (λu λhuh) 1,D C (T Th) Vλ 0.

Writing R3 = u − uh1,D −Th(λhuh) − T (λhuh)1,D, we obtain (2.22). The proof is complete. Modifying the proof of Theorem 2.3 slightly, we get the following theorem: Theorem 2.4. Suppose that the conditions of Theorem 2.3 hold, then

2 λh − λ = λ (a((T − Th)u, (T − Th)u)+2b((T − Th)u, Su − v)) + R1, ∀ v ∈ Vh, (2.26)

p − ph0,D = Sh(λu) − S(λu)0,D + R2, (2.27)

u − uh1,D = Th(λu) − T (λu)1,D + R3, (2.28)

| | − | 2 | | | | − | where R1 C (T Th) Vλ 0, R2 , R3 C (T Th) Vλ 0. Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1319

Proof. Let (λ, u, p) satisfy Lemma 2.1. Next, we prove this eigenpair satisﬁes (2.26)–(2.28). After taking g = λu in (2.23), and then substituting it into (2.16), we get (2.26). From (2.10), we deduce

Th(λu − λhuh)1 + Sh(λu − λhuh)0 Cλu − λhuh0. (2.29)

Substituting (2.15) into (2.29), we obtain

− − − | Th(λu λhuh) 1 + Sh(λu λhuh) 0 C (T Th) Vλ 0. (2.30)

Using triangle inequality and (2.30), we have

|p − ph0,D −Sh(λu) − S(λu)0,D|

= |S(λu) − Sh(λhuh)0,D −Sh(λu) − S(λu)0,D| − − | Sh(λu λhuh) 0,D C (T Th) Vλ 0.

Writing R2 = p − ph0,D −Sh(λu) − S(λu)0,D, we get (2.27). By the triangle inequality and (2.30), we infer that

|u − uh1,D −Th(λu) − T (λu)1,D|

= |T (λu) − Th(λhuh)1,D −Th(λu) − T (λu)1,D| − − | Th(λu λhuh) 1,D C (T Th) Vλ 0.

Writing R3 = u − uh1,D −Th(λu) − T (λu)1,D, we get (2.28). The proof is complete.

Note that, (2.22) and (2.28) are still valid when we change ·1,D in Theorems 2.3 and 2.4 into |·|1,D. Deﬁnition 2.5. Deﬁne Rayleigh quotient a(u∗,u∗)+2b(u∗,p∗) λr = . (2.31) (u∗,u∗)

The following theorem is a generalization of Lemma 9.1 in [4]. Theorem 2.6. Let (λ, u, p) be an eigenpair of eigenvalue problem (2.2)–(2.3). Then for ∀ (u∗,p∗) ∈ U × V,u∗ =(0 , 0), its Rayleigh quotient satisﬁes

a(u∗ − u, u∗ − u)+2b(u∗ − u, p∗ − p) (u∗ − u, u∗ − u) λr − λ = − λ . (2.32) (u∗,u∗) (u∗,u∗)

Proof. From (2.2)–(2.3), we deduce

a(u∗ − u, u∗ − u)+2b(u∗ − u, p∗ − p) − λ(u∗ − u, u∗ − u) = a(u∗,u∗) − 2a(u∗,u)+a(u, u)+2b(u∗,p∗) − 2b(u∗,p) − 2b(u, p∗) +2b(u, p) − λ(u∗,u∗)+2λ(u∗,u) − λ(u, u) = a(u∗,u∗)+2b(u∗,p∗) − λ(u∗,u∗) − 2(a(u∗,u)+b(u∗,p) − λ(u∗,u)) − 2b(u, p∗)+2b(u, p)+(a(u, u) − λ(u, u)) = a(u∗,u∗)+2b(u∗,p∗) − λ(u∗,u∗) − 0 − 0+0+0,

dividing both sides by (u∗,u∗), we obtain (2.32) immediately.

3 The mini-element approximation

The mini-element was proposed by Arnold, Brezzi and Fortin in 1984 (see [2]). Let πh be a regular triangulation of Ω (see [9]),

h { ∈ ¯ | ∈ ∈ } h h ∩ 1 S = v C(Ω) : v κ P1,κ πh ,S0 = S H0 (Ω). 1320 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

For ∀ κ ∈ πh, N1,N2 and N3 denote barycentric coordinates. Set Bh = {v : v|κ ∈ span{N1N2N3},κ ∈ πh}, deﬁne h ⊕ 2 h ∩ 2 Uh =(S0 Bh) ,Vh = S L0(Ω). From [2], we know that the mini-element satisﬁes Babuˇska-Brezzi stability condition. When Ω is a convex 1 1 polygonal domain, from (2.11) and (2.12), in addition observing that Tu = λ u, Su = λ p, we obtain the following error estimate:

2 Tu − Thu0 + hTu − Thu1 + hSu − Shu0 Ch (u2 + p1). (3.1)

From (3.1) and (2.10) we obtain Th − T 0 → 0, and by Lemma 2.1 and Theorem 2.4 we derive 2 |λ − λh| + u − uh0 + hu − uh1 + hp − ph0 Ch (u2 + p1). (3.2)

According to approximation theory, we know that Tu − Thu1 Ch is optimal, generally, it cannot be improved further (see [25]). This section will discuss the properties of approximate eigenvalue from above and the a posteriori error estimates of mini-element approximation for the Stokes eigenvalue problem.

Theorem 3.1. Suppose that λh is the k-th mini-element eigenvalue of (2.4)–(2.5), λ is the k-th eigenvalue of (2.2)–(2.3) and the eigenfunctions corresponding to λ are in (H2(Ω))2 × Hs(Ω) (1

~~λh λ. (3.3) Proof. We use Theorem 2.4 to complete the proof. From Lemma 2.2 in [33], there exists p ∈ Vh such that s s Su − p 0 Ch Sus Ch ps. (3.4) Taking v = p in (2.26), we have~~

2 2 λh − λ = λ a((T − Th)u, (T − Th)u)+2λ b((T − Th)u, Su − p )+R1. When h is sufficiently small, from (3.1), (3.2) and (3.4), we find that in three terms on the right-hand side in the above identity, the first term is the infinitesimal quantity of h2, the absolute value of the second term satisfies

Therefore the ﬁrst term is the dominant term and so the sign of λh − λ is positive. Thus, (3.3) holds. Consider the Stokes equation (2.6)–(2.7) and its mini-element approximation (2.8)–(2.9). Evidently, 0 h the solution to (2.6)–(2.7) is (T f,S f), and the solution to (2.8)–(2.9) is (Thf,S hf). Let Il : C → S be a piecewise linear interpolation operator. Write

eh = T f − Thf, eh,l = T f − IlThf, εh = Sf − Shf. (3.5) Verfürth [35] has discussed the a posteriori error estimates of mini-element for the Stokes equation. For ∀ κ ∈ πh, define residual indicator 2 1/2 2 1 ∂(IlThf) 2 ηR,κ = |κ|P0f −∇Shf ,κ + |E| + ∇ · (IlThf) ,κ , (3.6) 0 2 ∂n 0 E⊂∂κ∩Ω J 0,E ∂(IlThf) ∂(IlThf) where |κ| and |E| are the area of κ and the length of E, respectively, [ ∂n ]J is the jump of ∂n through the edge E,andP0 is L2-projection onto the space of all functions which are piecewise polynomials of degree = 0. Verfürth proved the following conclusion: Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1321

Lemma 3.2. There exists two positive constants C0,C1 only depending on Ω and the minimum interior angle of πh, such that 1/2 | | 2 | | − 2 eh,l 1 + εh 0 C0 [ηR,κ + κ f P0f 0,κ] , (3.7) κ∈πh 1/2 ηR,κ C1{|eh,l|1,κ + εh0,κ + |κ| f − P0f0,κ}, ∀ κ ∈ πh. (3.8)

~~Denote eeh = u − uh, eeh,l = u − Iluh and eεh = p − ph. Now we prove the following theorem.~~

Theorem 3.3. Assume that (λh,uh,ph) is the k-th mini-element eigenpair of (2.4)–(2.5),anduh0 =1. Then there exists the k-th eigenpair (λ, u, p) of (2.2)–(2.3) with u0 =1, such that 1/2 | | 2 | | − 2 − eeh,l 1 + eεh 0 C0 [ηR,κ + κ λhuh P0(λhuh) 0,κ] + C λu λhuh 0, (3.9) κ∈πh 1 2 √ 2 1/2 ηR,κ C1 3{|eeh,l|1,D + eεh0,D + |κ| λhuh − P0(λhuh)0,D}

~~κ∈πh,κ⊂D~~

~~+ Cλu − λhuh0, (3.10)~~

~~where C0,C1 are two positive constants only dependent on Ω and the minimum interior angle of πh,and f = λhuh in ηR,κ.~~

Proof. Consider (2.6)–(2.7) and (2.8)–(2.9), with f = λhuh. Then the solution to (2.6)–(2.7) is (T (λhuh),S(λhuh)), and the solution to (2.8)–(2.9) is (Th(λhuh),Sh(λhuh)) = (uh,ph). By a simple calculation, we have

~~||eeh,l|1,D + eεh0,D − (|T (λhuh) − Iluh|1,D + S(λhuh) − ph0,D)|~~

~~= ||T (λu) − Iluh|1,D + S(λu) − ph0,D~~

~~− (|T (λhuh) − Iluh|1,D + S(λhuh) − ph0,D)|~~

~~|T (λu) − T (λhuh)|1,D + S(λu) − S(λhuh)0,D~~

~~Cλu − λhuh0. Therefore,~~

~~|eeh,l|1,D + eεh0,D = |T (λhuh) − Iluh|1,D + S(λhuh) − ph0,D + ϑ(λu − λhuh0). (3.11)~~

~~By Lemma 3.2 (take f = λhuh), we infer that~~

~~|T (λhuh) − Iluh|1 + S(λhuh) − ph0 1/2 2 | | − 2 C0 [ηR,κ + κ λhuh P0(λhuh) 0,κ] . (3.12) κ∈πh 1 2 √ 2 ηR,κ C1 3{|T (λhuh) − Iluh|1,D~~

κ∈πh,κ⊂D 1/2 + S(λhuh) − ph0,D + |κ| λhuh − P0(λhuh)0,D}. (3.13) Combining (3.11) (take D = Ω) with (3.12), we obtain (3.9). Combining (3.11) with (3.13), we get (3.10). 2 Note that, in all terms on the right-hand side of (3.9), κ∈πh ηR,κ is the dominant term, the other terms are higher order infinitesimals. In all terms on the right-hand side of (3.10), |eeh,l|1,D + eεh0,D 1 2 2 is the dominant term, the other terms are higher order infinitesimals. Hence, ( κ∈πh ηR,κ) is a reliable and efficient a posteriori error indicator of Iluh and ph. Remark 3.4. In this section, we used a posteriori error estimator proposed by Verfürth, the error estimators are only based on the linear part of the error (see [35, p. 312]), that is different from the one in [26]. 1322 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

~~4TheP1-P1 element approximation~~

Consider the Stokes eigenvalue problem (2.2)–(2.3) and the P1-P1 element approximation (2.4)–(2.5). We assume that Ω is a convex polygonal domain. Let π2h be a regular triangulation of Ω, πh be the result of reﬁning π2h by connecting the middle points on the edges. The P1-P1 element space is deﬁned by

~~h ∩ 1 2 2h ∩ 2 Uh =[S H0 (Ω)] ,Vh = S L0(Ω),~~

~~2h h where S and S are continuous piecewise linear polynomial spaces deﬁned on the π2h and πh, respectively.~~

Verf¨urth [36] has proved that P1-P1 element satisﬁes Babuˇska-Brezzi stability condition (see Proposi- 1 1 tion 3.3 in [36]). Therefore from (2.11) and (2.12), in addition observing that Tu = λ u and Su = λ p,we obtain the following error estimate:

~~2 Tu − Thu0 + hTu − Thu1 + hSu − Shu0 Ch (u2 + p1). (4.1)~~

~~From (4.1) and (2.10) we obtain Th − T 0 → 0, and by Lemma 2.1 and Theorem 2.4 we derive~~

~~2 |λ − λh| + u − uh0 + hu − uh1 + hp − ph0 Ch (u2 + p1). (4.2)~~

According to approximation theory, we know that Tu − Thu1 Ch is optimal, generally, it cannot be improved further (see [25]). Lin and Lin [20] has discussed the property of the global superconvergence. Deﬁne the piecewise 2 2 2 quadratic node interpolation operator: I2hv =(Π2hv1, Π2hv2), 2 → 2 ∀ ∈ Π2h : C(τ) P2(τ), Π2hv(zi)=v(zi), τ π2h,i=1, 2,...,6, where P2(τ) is quadratic polynomial space on τ, interpolation nodes zi are three vertexes and the midpoints of three edges on τ. Lin and Lin [20] proved that let π2h be a uniform isosceles right triangle mesh of Ω, then

~~3 3 2 − − 2 2 I2hThu Tu 1 + Shu Su 0 Ch ( Tu 3 + Su 2) Ch ( u 3 + p 2). (4.3)~~

Based on the above-mentioned work, we discuss the properties of approximate eigenvalue from above and the recovery type by post-processing interpolation a posteriori error estimates for P1-P1 element approximation of the Stokes eigenvalue problem.

~~Theorem 4.1. Suppose that λh is the k-th P1-P1 element eigenvalue of (2.4)–(2.5), λ is the k-th eigenvalue of (2.2)–(2.3) and the eigenfunctions corresponding to λ are in (H2(Ω))2 ×Hs(Ω) (1~~

~~λh λ. (4.4)~~

~~Proof. We use Theorem 2.4 to complete the proof. From Lemma 2.2 in [33], there exists p ∈ Vh such that~~

~~s Su − p 0 Ch ps. (4.5)~~

~~Taking v = p in (2.26), we have~~

~~2 2 λh − λ = λ a((T − Th)u, (T − Th)u)+2λ b((T − Th)u, Su − p )+R1.~~

When h is small enough, from (4.1), (4.2) and (4.5), we find that in three terms of the right-hand side in the above identity, the first term is infinitesimal quantity of h2, the absolute value of the second term satisfies

Therefore the ﬁrst term is the dominant term and so the sign of λh − λ is positive. Thus, (4.4) holds. Theorem 4.2. Suppose that λ is the k-th eigenvalue of (2.2)–(2.3), and the eigenfunctions associated 3 2 2 with λ belong to (H (Ω)) × H (Ω), and suppose that πh is a uniform isosceles right triangle mesh of Ω, (λh,uh,ph) is the k-th P1-P1 element eigenpair of (2.4)–(2.5),anduh0 =1. Then there exists an eigenfunction (u, p) associated with λ,withu0 =1, such that

~~3 2 − − 2 I2huh u 1 + ph p 0 Ch ( u 3 + p 2), (4.6) − − 2 u uh 1 = uh I2huh 1 + R1, (4.7) − | − 2 |2 λh λ = uh I2huh 1 + R2, (4.8)~~

3 5 where |R1| Ch2 , |R2| Ch2 . Proof. Consider (2.6)–(2.7) and (2.8)–(2.9), with f = λu. The solution to (2.6)–(2.7) is (T (λu),S(λu)) = (u, p), the solution to (2.8)–(2.9) is (Th(λu),Sh(λu)). From [31], we know that 2 ∀ ∈ I2hv 1 C v 1, v Uh. By the above inequality, (2.29) and (4.2), we deduce 2 − 2 − I2huh I2hTh(λu) 1 + ph Sh(λu) 0 2 − 2 − = I2hTh(λhuh) I2hTh(λu) 1 + Sh(λhuh) Sh(λu) 0

~~CTh(λhuh − λu)1 + Sh(λhuh − λu)0 2 Cλhuh − λu0 Ch (u2 + p1). (4.9)~~

~~From (4.3), we deduce~~

3 2 − − 2 I2hTh(λu) T (λu) 1 + Sh(λu) S(λu) 0 Ch ( u 3 + p 2). (4.10) Combining (4.9) with (4.10), we obtain (4.6). From (4.6), we see that (4.7) holds. A simple calculation shows − − 2 2 − − 2 2 − − Th(λhuh) T (λhuh)=uh I2huh + I2huh T (λhuh)=uh I2huh + I2huh u + u T (λhuh). Combining the above identity with (4.6) and

~~2 u − T (λhuh)1 = T (λu − λhuh)1 Ch , we obtain~~

~~3 | − | | − 2 | 2 Th(λhuh) T (λhuh) 1 = uh I2huh 1 + ϑ(h ). (4.11) From (2.11), (4.2) and (4.5) with s = 2, we deduce~~

Combining the above two formulas with (2.20) (take v = p), we obtain (4.8) immediately. − 2 | − 2 |2 Theorem 4.2 shows that uh I2huh 1 and uh I2huh 1 are the asymptotically exact a posteriori error indicators of uh and λh, respectively. Denote Rayleigh quotient

~~2 2 2 r a(I2huh,I2huh)+2b(I2huh,ph) λh = 2 2 . (I2huh,I2huh) 1324 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6~~

~~Theorem 4.3. Under the assumptions of Theorem 4.2, the following conclusion is valid:~~

~~r 3 |λh − λ| Ch , (4.12)~~

r thus λh − λh is an asymptotically exact a posteriori error indicator of λh. r r ∗ 2 ∗ Proof. We use (2.32) to complete the proof. In this case, λ = λh, u = I2huh, p = ph.After estimating each term on the right-hand of (2.32) by using (4.6), we get (4.12). From (4.12), we have

~~r r r 3 λh − λ = λh − λh + λh − λ = λh − λh + ϑ(h ),~~

r i.e., λh − λh is an asymptotically exact a posteriori error indicator of λh. Remark 4.4. Using the methods in the proofs of Theorems 4.1–4.3, we can prove that the conclusions of Theorems 4.1–4.3 hold for the Q1-Q1 element.

~~5TheQ1-P0 element approximation~~

Consider the Stokes eigenvalue problem (2.2)–(2.3) and the Q1-P0 element approximation (2.4)–(2.5). Here we assume that Ω is a convex polygon with boundaries parallel to the axes, let π2h = {τ} be a rectangular mesh of Ω,

~~τ =[xτ − hτ ,xτ + hτ ] × [yτ − kτ ,yτ + kτ ],~~

~~where (xτ ,yτ ) represents the center of τ. The subdivision πh is obtained from π2h by dividing each element τ of π2h into four small congruent rectangles κ1,κ2,κ3 and κ4 (see Figure 1).~~

~~The Q1-P0 element space is deﬁned by~~

2 2 Uh = {v ∈ C(Ω)¯ : v|κ ∈ (Q1(κ)) , ∀ κ ∈ πh,v|∂Ω =0}, 3 ∈ 2 | τ τ τ ∈ Vh = q L0(Ω) : q τ = ξi ϕi , ξ1 =0,τ π2h , i=1 τ∈π2h { 1 i i ∈ } τ where Q1(κ)= v : v = i,j=0 αij x y , (x, y) κ , ϕi is base function consisting of piecewise constant on τ,and

~~ϕτ (x, y)=1, ∀ (x, y) ∈ τ, 1 τ 1, if (x, y) ∈ κ1 ∪ κ2, ϕ2 (x, y)= −1, if (x, y) ∈ κ3 ∪ κ4, τ 1, if (x, y) ∈ κ1 ∪ κ4, ϕ3 (x, y)= −1, if (x, y) ∈ κ2 ∪ κ3.~~

Girault and Raviart [12] have proved that the Q1-P0 element satisﬁes Babuˇska-Brezzi stability condi- 1 1 tion. Therefore from (2.11) and (2.12), in addition noting that Tu = λ u, Su = λ p, we get the following error estimate:

~~2 Tu − Thu0 + hTu − Thu1 + hSu − Shu0 Ch (u2 + p1). (5.1)~~

~~From (5.1) and (2.10) we obtain Th − T 0 → 0, and by Lemma 2.1 and Theorem 2.4 we derive~~

~~2 |λ − λh| + u − uh0 + hu − uh1 + hp − ph0 Ch (u2 + p1). (5.2) Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1325~~

~~κ1 κ2~~

~~κ3 κ4~~

~~Figure 1 The element τ~~

Pan [31] has discussed the property of the global superconvergence. Let π2h = {τ} be a uniform rect- 2 2 2 angular mesh. Let I2hv =(Π2hv1, Π2hv2) denote the piecewise biquadratic node interpolation polynomial of v on π2h, and let the interpolation nodes be four vertexes, the midpoints of four edges and the center 1 on τ. The deﬁnition of J2h is as follows: 1 ∈ J2hp Q1(τ), 1 1 1 (J p − p)=0 or (J p)(xκ ,yκ )= p, i =1, 2, 3, 4, 2h 2h i i | | κi κi κi 4 where, τ = i=1 κi,(xκi ,yκi ) denotes the center of κi. Pan [31] has proved that 2 − 1 − 2 2 I2hThu Tu 1 + J2hShu Su 0 Ch ( Tu 3 + Su 2) Ch ( u 3 + p 2). (5.3) Based on the above-mentioned work, we discuss the recovery type by post-processing interpolation a posteriori error estimates of Q1-P0 element approximation for the Stokes eigenvalue problem. Theorem 5.1. Suppose that λ is the k-th eigenvalue of (2.2)–(2.3), and the eigenfunctions associated 3 2 2 with λ belong to (H (Ω)) × H (Ω), and suppose that πh is a uniform rectangular mesh of Ω, (λh,uh,ph) is the k-th Q1-P0 element eigenpair of (2.4)–(2.5),anduh0 =1. Then there exists an eigenfunction (u, p) associated with λ,withu0 =1, such that 2 − 1 − 2 I2huh u 1 + J2hph p 0 Ch ( u 3 + p 2), (5.4) − − 2 u uh 1 = uh I2huh 1 + R1, (5.5) − − 1 p ph 0 = ph J2hph 0 + R2, (5.6) − | − 2 |2 − 2 − 1 λh λ = uh I2huh 1 +2b(uh I2huh,ph J2hph)+R3, (5.7) 2 3 where |R1|, |R2| Ch , |R3| Ch . Proof. Consider (2.6)–(2.7) and (2.8)–(2.9), with f = λu. The solution to (2.6)–(2.7) is (T (λu),S(λu)) = (u, p), the solution to (2.8)–(2.9) is (Th(λu),Sh(λu)). From [31], we know that 2 ∀ ∈ 1 ∀ ∈ I2hv 1 C v 1, v Uh, J2hq 0 C q 0, q Vh. By the above two inequalities, (2.29) and (5.2), we deduce 2 − 2 − − 2 I2huh I2hTh(λu) 1 C Th(λhuh) Th(λu) 1 C λhuh λu 0 Ch ( u 2 + p 1), (5.8) 1 − 1 − − 2 J2hph J2hSh(λu) 0 C Sh(λhuh) Sh(λu) 0 C λhuh λu 0 Ch ( u 2 + p 1). (5.9) From (5.3), we get 2 − 1 − 2 I2hTh(λu) T (λu) 1 + J2hSh(λu) S(λu) 0 Ch ( u 3 + p 2). (5.10) Combining (5.8) with (5.9) and (5.10), we obtain (5.4), from which we conclude that (5.5) and (5.6) are valid. Using the method to deduce (4.11) we can get | − | | − 2 | 2 Th(λhuh) T (λhuh) 1 = uh I2huh 1 + ϑ(h ), − | − 1 | 2 S(λhuh) ph 0 = ph J2hph 0 + ϑ(h ). − | 2 Combining the above two identities, (T Th) Vλ 0 Ch and (2.20), we obtain (5.7). 1326 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

~~Theorem 5.1 shows that~~

− 2 | − 1 | | − 2 |2 − 2 − 1 uh I2huh 1, ph J2hph 0 and uh I2huh 1 + b(uh I2huh,ph J2hph) are the asymptotically exact a posteriori error indicators for uh, ph and λh, respectively. Denote Rayleigh quotient

~~2 2 2 1 r a(I2huh,I2huh)+2b(I2huh,J2hph) λh = 2 2 . (I2huh,I2huh)~~

~~Theorem 5.2. Under the assumptions of Theorem 5.1, the following conclusion is valid:~~

~~r 4 |λh − λ| Ch , (5.11)~~

~~r thus λh − λh is an asymptotically exact a posteriori error indicator of λh.~~

r r ∗ 2 ∗ 1 Proof. We use (2.32) to complete the proof. In this case, λ = λh, u = I2huh and p = J2hph.After we estimate each term on the right-hand of (2.32) by using (5.4), we get (5.11). From (5.11), we conclude r that λh − λh is an asymptotically exact a posteriori error indicator of λh. Remark 5.3. From 1989 to 1991, Lin and Yang pointed out and proved that the higher order interpolation functions of lower order finite element solutions, obtained by using nodes of lower order element as interpolation nodes, can have global gradient superconvergence (see [19] and the references therein). The technique was called finite element interpolation postprocessing. In Sections 4 and 5 of this paper, we extended the technique to the mixed finite element method for the Stokes eigenvalue problem. In recent years, many postprocessing techniques have also been researched, for example, the semi-discrete defect-correction mixed finite element method (see [32]), the gradient recovery schemes of higher order for the linear interpolation (see [15]), and the shifted inverse iteration (see [41]). It is a feasible and meaningful work to apply these to the Stokes eigenvalue problem .

~~6 Numerical experiments~~

Consider the Stokes eigenvalue problem (2.1), where Ω ⊂ R2 is a unit square domain. The smallest eigenvalue satisfies λ1 ≈ 52.3446911 for this problem (see [37,44]). Example 6.1. We adopt a uniform isosceles right triangle mesh of Ω. In Figure 2, we present the initial triangulation of the domain Ω, the mesh πh is obtained by using a uniform refinement (each triangle is divided into four small congruent triangles). We make use of Matlab 7.1 to compute the first five eigenvalues by using the mini-element on πh. Numerical results are presented in Table 1. From Table 1, we see that the numerical eigenvalues obtained by the mini-element approximate the exact eigenvalues from above, which is in accordance with Theorem 3.1. Denote λ1,h − λ1 ratio(λ1,h)=lg lg 2. (6.1) λ h − λ 1, 2 1

~~Computing the convergent order of λ1,h by using formula (6.1), we obtain~~

√ √ √ ratio(λ 2 )=2.0171, ratio(λ 2 )=2.0092, ratio(λ 2 )=2.0047, 1, 20 1, 40 1, 80 which coincide with (3.2) basically. Example 6.2. For the initial mesh in Figure 2, we adopt a uniform refinement (each triangle is divided into four small congruent triangles) to obtain the meshes π2h and πh. We compute the first five eigenvalues by using the P1-P1 element with Matlab 7.1. Numerical results are presented in Table 2. Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6 1327

~~@@ @@ @@ @@ @ @ @ @ @@ @ @@ @ @ @ @ @ @@@ @ @@ @ @ @ @ @@ @ @ @ @ @ @ @@ - 1~~

~~Figure 2 The initial triangulation of the domain Ω~~

From Table 2, we see that the numerical eigenvalues obtained by the P1-P1 element approximate the exact eigenvalues from above, which is in accordance with Theorem 4.1. Using the formula (6.1) to compute the convergent order of λ1,h,weget

~~√ √ √ ratio(λ 2 )=2.00109, ratio(λ 2 )=2.00181, ratio(λ 2 )=2.00114, 1, 20 1, 40 1, 80 which coincides with (4.2) basically.~~

~~Example 6.3. Let πh, π2h be the uniform square meshes, we compute the ﬁrst ﬁve eigenvalues by using the Q1-P0 element with Matlab 7.1, numerical results are presented in Tables 3–5.~~

From Table 3, we see that the numerical eigenvalues obtained by the Q1-P0 element approximate the exact eigenvalues from above, but theoretical proof has not been seen for this phenomenon. Using the formula (6.1) to compute the convergent order of λ1,h,andget

~~√ √ √ ratio(λ 2 )=2.00054, ratio(λ 2 )=2.00029, ratio(λ 2 )=2.00009, 1, 20 1, 40 1, 80 which coincide with (5.2) basically.~~

~~Let (λ1,h,uh,ph) be the smallest Q1-P0 element eigenpair. Write~~

~~| − 2 |2 − 2 − 1 η1(λ1,h)= uh I2huh 1 +2b(uh I2huh,ph J2hph).~~

The eigenvalue error λ1,h − λ1 and the a posteriori error indicator η1(λ1,h) are presented in Table 4. We see that η1(λ1,h) is the asymptotically exact a posteriori error indicator of Q1-P0 element eigenvalue λ1,h. Let (λ1,h,uh,ph) be the smallest Q1-P0 element eigenpair,

~~2 2 2 1 r a(I2huh,I2huh)+2b(I2huh,J2hph) λ1,h = 2 2 . (I2huh,I2huh)~~

~~Table 1 Mini-element eigenvalues approximate the exact eigenvalues from above~~

hλ1,h λ2,h λ3,h λ4,h λ5,h trend √ 2 53.14417 93.90394 94.94800 133.3861 159.8579 √20 2 52.54220 92.56453 92.82091 129.4928 155.5440 √40 2 52.39375 92.23375 92.29755 128.5284 154.4784 √80 2 160 52.35692 92.15164 92.16758 128.2890 154.2135

~~Table 2 P1-P1 element eigenvalues approximate the exact eigenvalues from above~~

hλ1,h λ2,h λ3,h λ4,h λ5,h trend √ 2 53.22782 94.17990 95.22234 133.8788 160.6570 √20 2 52.56531 92.63753 92.89480 129.6316 155.7486 √40 2 52.39978 92.25254 92.31635 128.5649 154.5303 √80 2 160 52.35845 92.15641 92.17234 128.2984 154.2266 1328 Yang Y D et al. Sci China Math June 2013 Vol. 56 No. 6

~~Table 3 Q1-P0 element eigenvalues approximate the exact eigenvalues from above~~

hλ1,h λ2,h λ3,h λ4,h λ5,h trend √ 2 52.64899 93.44996 93.44996 130.0009 158.5885 √20 2 52.42074 92.45367 92.45367 128.6566 155.2275 √40 2 52.36370 92.20658 92.20658 128.3213 154.4001 √80 2 160 52.34944 92.14493 92.14493 128.2375 154.1941

~~Table 4 η1(λ1,h) is the asymptotically exact error indicator of Q1-P0 element eigenvalue λ1,h~~

hλ λ − λ η λ η1(λ1,h) 1,h 1,h 1 1( 1,h) λ −λ trend √ 1,h 1 2 52.64899 0.304303 0.308614 1.01416 √20 2 52.42074 0.076047 0.076285 1.00312 √40 2 52.36370 0.019008 0.019022 1.00071 √80 2 160 52.34944 0.004752 0.004752 1.00016 1

~~Table 5 η2(λ1,h) is the asymptotically exact error indicator of Q1-P0 element eigenvalue λ1,h~~

hλr λ − λ η λ η2(λ1,h) 1,h 1,h 1 2( 1,h ) λ −λ trend √ 1,h 1 2 52.3493622 0.3043032 0.2996320 0.9846496 √20 2 52.3450156 0.0760474 0.0757223 0.9957321 √40 2 52.3447127 0.0190080 0.0189864 0.9988624 √80 2 160 52.3446925 0.0047517 0.0047503 0.9996964 1

− r − Write η2(λ1,h)=λ1,h λ1,h. The eigenvalue error λ1,h λ1 and the a posteriori error indicator η2(λ1,h) are presented in Table 5. We see that η2(λ1,h) is the asymptotically exact a posteriori error indicator of Q1-P0 element eigenvalue λ1,h.Denote r − r λ1,h λ1 ratio(λ1,h)=lg r lg 2. (6.2) λ h − λ1 1, 2

~~r We compute the convergent order of λ1,h by using formula the (6.2), and get~~

~~r √ r √ r √ ratio(λ 2 )=3.8472, ratio(λ 2 )=3.9079, ratio(λ 2 )=3.9060, 1, 20 1, 40 1, 80 which is basically in accordance with (5.11).~~

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 10761003) and Science and Technology Foundation of Guizhou Province of China (Grant No. [2011] 2111). The authors cordially thank the referees and Professor Hai Bi for their valuable comments which led to the improvement of this paper.

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~~Mathematics~~

~~CONTENTS Vol. 56 No. 6 June 2013~~

Progress of Projects Supported by NSFC An approach of constructing mixed-level orthogonal arrays of strength ³ 3 ...... 1109 JIANG Ling & YIN JianXing The bounds of restricted isometry constants for low rank matrices recovery ...... 1117 WANG HuiMin & LI Song

~~Articles~~

Positive divisors in symplectic geometry ...... 1129 HU JianXun & RUAN YongBin 2 ...... Reverse Bonnesen style inequalities in a surface   of constant curvature 1145 XIA YunWei, XU WenXue, ZHOU JiaZu & ZHU BaoCheng An efficient algorithm for factoring polynomials over algebraic extension field ...... 1155 SUN Yao & WANG DingKang Further results on existence-uniqueness for stochastic functional differential equations ...... 1169 XU DaoYi, WANG XiaoHu & YANG ZhiGuo Asymptotically or super linear cooperative elliptic systems in the whole space ...... 1181 CHEN GuanWei & MA ShiWang Ergodicity of stochastic Boussinesq equations driven by Lévy processes ...... 1195 ZHENG Yan & HUANG JianHua Samuel multiplicity and the structure of essentially semi-regular operators: A note on a paper of Fang ...... 1213 ZENG QingPing, ZHONG HuaiJie & WU ZhenYing Empirical likelihood inference for estimating equation with missing data ...... 1233 WANG XiuLi, CHEN Fang & LIN Lu Empirical likelihood inference for semi-parametric estimating equations ...... 1247 WANG ShanShan, CUI HengJian & LI RunZe Statistical inference for right-censored data with nonignorable missing censoring indicators ...... 1263 SUN ZhiHua, XIE TianFa & LIANG Hua Proof of a conjecture on a discretized elliptic equation with cubic nonlinearity ...... 1279 ZHANG XuPing, YU Bo & ZHANG JinTao Moving finite element methods for time fractional partial differential equations ...... 1287 JIANG YingJun & MA JingTang rot EQ 1 nonconforming finite element approximation to Signorini problem ...... 1301 SHI DongYang & XU Chao Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem ...... 1313 YANG YiDu & JIANG Wei

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