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Mathematical Problems in Engineering
Volume 2011, Article ID 873152, 13 pages
http://dx.doi.org/10.1155/2011/873152
Research Article

Conforming Finite Element Approximations for a Fourth-Order Steklov Eigenvalue Problem

School of Mathematics and Computer Science, Guizhou Normal University, GuiYang 550001, China

Received 30 May 2011; Revised 5 August 2011; Accepted 13 September 2011

Academic Editor: Bin Liu

Copyright © 2011 Hai Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper characterizes the spectrum of a fourth-order Steklov eigenvalue problem by using the spectral theory of completely continuous operator. The conforming finite element approximation for this problem is analyzed, and the error estimate is given. Finally, the bounds for Steklov eigenvalues on the square domain are provided by Bogner-Fox-Schmit element and Morley element.

1. Introduction

Steklov eigenvalue problems, in which the eigenvalue parameter appears in the boundary condition, have several deep applications both in maths and physics. For instance, they are found in the study of surface waves (see [1]), the analysis of stability of mechanical oscillators immersed in a viscous fluid (see [2]), and the study of the vibration modes of a structure in contact with an incompressible fluid (see, e.g., [3]), and the first eigenvalue also plays a crucial role in the positivity preserving property for the biharmonic operator Δ2 under the boundary conditions 𝑢=Δ𝑢𝜆𝑢𝜈=0 on 𝜕Ω (see [4]) and so forth.

Thus, numerical methods for approximate Steklov eigenvalues become a concerned problem by mathematics and engineering community. Many scholars have investigated the finite element methods for second-order Steklov eigenvalue problem and achieved many results; for example, see [512] and so on.

However, for fourth-order Steklov eigenvalue problems the existing references are mostly qualitative analysis: [13] studied the bound for the first eigenvalue on the square and proved that the first eigenvalue is simple and its eigenfunction does not change sign, [14] discussed the smallest nonzero Steklov eigenvalue by the method of a posteriori-a priori inequalities, [15, 16] studied the spectrum of a fourth-order Steklov eigenvalue problem on a bounded domain in 𝑛 and gave the explicit form of the spectrum in the case where the domain is a ball, and [17] provided bounds for the first non-zero Steklov eigenvalues when Ω is isometric to an n-dimensional Euclidean ball. More recently, [18] proved the existence of an optimal convex shape among domains of given measure, and [19] by a new method established Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues. But as for the finite element methods for fourth-order Steklov eigenvalue problem, to the best of our knowledge, there are no reports.

This paper discusses conforming finite element approximations for a fourth-order Steklov eigenvalue problem, and the main work is as follows.

(1) We define the operator 𝑇𝐻2(Ω)𝐻10(Ω)𝐻2(Ω)𝐻10(Ω) and prove that 𝑇 is completely continuous thus characterize the spectrum of a fourth-order Steklov eigenvalue problem by the spectrum of 𝑇. Note that [16] analyzed the spectrum of this problem by introducing an orthogonal decomposition,𝐻2(Ω)𝐻10𝐻(Ω)=𝑊20(Ω), and conducted the research on the space 𝑊. Compared with the argument used in [16], our approach is more direct and lays the foundation for further discussion of the finite element approximation.

(2) We study for the first time conforming finite element approximations for a fourth-order Steklov eigenvalue problem by using the spectral approximation theory (e.g., see [20, 21]) and prove a priori error estimates of finite element eigenvalues and eigenfunctions. Hence, in principle, we can compute approximate eigenvalues and eigenfunctions of a fourth-order Steklov eigenvalue problem on any bounded domain by finite element methods. As an example, we compute the approximate eigenvalues by the conforming Bogner-Fox-Schmit element, and numerical results indicate that the numerical eigenvalues of Bogner-Fox-Schmit element approximate the exact eigenvalues from above. We also compute the nonconforming Morley element eigenvalues, and the numerical results show that the Morley element eigenvalues approximate the exact eigenvalues from below. Thus, we provide bounds for the exact Steklov eigenvalues on a square, which are more precise than those given in [16].

The rest of this paper is organized as follows. In the next section, the spectrum of the fourth-order Steklov eigenvalue problem is characterized by using the spectral theory of completely continuous operator. In Section 3, the error estimate of conforming finite element approximation for the fourth-order Steklov eigenvalue problem is proved. Numerical experiments of the Bogner-Fox-Schmit element and Morley element are presented in Section 4 to give bounds for Steklov eigenvalues on the square.

Let W𝑠,𝑞(Ω) denote the usual Sobolev space with real-order 𝑠 with norm 𝑠,𝑞. For simplicity, we write 𝐻𝑠(Ω) for 𝑊𝑠,2(Ω) with norm 𝑠 and 𝐻𝑠(𝜕Ω) for 𝑊𝑠,2(𝜕Ω) with norm 𝑠,𝜕Ω. 𝐻0(Ω)=𝐿2(Ω),𝐻0(𝜕Ω)=𝐿2(𝜕Ω). Throughout this paper, 𝐶 denotes a generic positive constant independent of , which may not be the same at each occurrence.

2. The Spectrum of the Fourth-Order Steklov Eigenvalue Problem

We consider the fourth-order Steklov eigenvalue problemΔ2𝑢=0,inΩ,(2.1)𝑢=0,on𝜕Ω,(2.2)Δ𝑢=𝜆𝑢𝜈,on𝜕Ω,(2.3) where Ω𝑛(𝑛2) is a bounded domain and 𝜕Ω is smooth, or Ω2 is a convex domain, 𝑢𝜈 denotes the outer normal derivative of 𝑢 on 𝜕Ω.

Denote 𝑉=𝐻2(Ω)𝐻10(Ω).

The weak form of (2.1)–(2.3) is given by the following. Find 𝜆, 0𝑢𝑉, such that𝑎(𝑢,𝑣)=𝜆𝑏(𝑢,𝑣),𝑣𝑉,(2.4) where𝑎(𝑢,𝑣)=ΩΔ𝑢Δ𝑣𝑑𝑥,𝑏(𝑢,𝑣)=𝜕Ω𝑢𝜈𝑣𝜈𝑑𝑠.(2.5)

Lemma 2.1. Assume that Ω is a Lipschitz bounded domain which satisfies the uniform outer ball condition. Then the space 𝑉 becomes a Hilbert space when endowed with the scalar product𝑎(𝑢,𝑣)=ΩΔ𝑢Δ𝑣𝑑𝑥,𝑢,𝑣𝑉, and 𝑢𝑎=𝑎(𝑢,𝑢) is equivalent to the norm 2 induced by 𝐻2(Ω).

Proof. See Lemma  1 and its proof in [15].

It is obvious that the condition of Lemma 2.1 holds for Ω𝑛 with 𝜕Ω2 or a convex domain Ω2.

The source problem associated with (2.4) is as follows. Find 𝑢𝑉, such that 𝑎(𝑢,𝑣)=𝑏(𝑓,𝑣),𝑣𝑉.(2.6)

It follows from Lemma 2.1 that 𝑎(,) is a symmetric, continuous, and 𝑉-elliptic bilinear form on 𝑉. By Schwarz inequality and trace theorem we have ||𝑏||=||||(𝑓,𝑣)𝜕Ω𝑓𝜈𝑣𝜈||||𝑓𝑑𝑠𝜈0,𝜕Ω𝑣𝜈0,𝜕Ω𝑓𝜈0,𝜕Ω𝑣2,𝑣𝑉.(2.7) Hence, from Lax-Milgram Theorem we know that (2.6) has one and only one solution.

Therefore, according to the source problem (2.6) we define the operator 𝑇𝑉𝑉 by 𝑎(𝑇𝑓,𝑣)=𝑏(𝑓,𝑣),𝑣𝑉.(2.8) From [20], we know that (2.4) has the equivalent operator form: 1𝑇𝑢=𝜆𝑢.(2.9)

Lemma 2.2. The operator 𝑇𝑉𝑉 is self-adjoint and completely continuous.

Proof. By the definition of 𝑇, for any 𝑢,𝑣𝑉, there holds 𝑎(𝑇𝑢,𝑣)=𝑏(𝑢,𝑣)=𝑏(𝑣,𝑢)=𝑎(𝑇𝑣,𝑢)=𝑎(𝑢,𝑇𝑣),(2.10) that is, 𝑇 is self-adjoint with respect to 𝑎(,).
Next we will prove that 𝑇 is completely continuous. By Schwarz inequality and trace theorem, for any 𝑓𝑉 we have 𝑓𝑎(𝑇𝑓,𝑇𝑓)=𝑏(𝑓,𝑇𝑓)𝜈0,𝜕Ω(𝑇𝑓)𝜈0,𝜕Ω𝑓𝐶𝜈0,𝜕Ω𝑇𝑓2;(2.11) it follows from the fact that 𝑎(,) is 𝑉-elliptic that 𝑇𝑓2𝑓𝐶𝜈0,𝜕Ω.(2.12) Let {𝑓𝑚} be a bounded sequence in 2; then by trace theorem we have {(𝑓𝑚)𝜈} is a bounded sequence in 1/2,𝜕Ω. And, by compact embedding 𝐻1/2(𝜕Ω)𝐿2(𝜕Ω), we know that there is a subsequence {(𝑓𝑚𝑙)𝜈} that is a Cauchy sequence in 0,𝜕Ω. From (2.12) we conclude that {𝑇𝑓𝑚𝑙} is a Cauchy sequence in 2, which implies that 𝑇 is completely continuous.

Let (𝜆,𝑢) be an eigenpair of (2.4); then 𝑢𝑎0. Since 𝑎(𝑢,𝑢)=𝜆𝑏(𝑢,𝑢), we see that 𝑢𝜈0,𝜕Ω0,𝜆0. And, by trace theorem we have 𝑢𝜈0,𝜕Ω𝐶𝑢2𝐶𝑢𝑎;(2.13) thus, 𝜆=𝑎(𝑢,𝑢)=𝑏(𝑢,𝑢)𝑢2𝑎𝑢𝜈20,𝜕Ω1𝐶2>0.(2.14)

Therefore, from the spectral theory of completely continuous operator we know that all eigenvalues of 𝑇 are real and have finite algebraic multiplicity. We arrange the eigenvalues of 𝑇 by increasing order: 0<𝜆1𝜆2𝜆3+.(2.15)

The eigenfunctions corresponding to two arbitrary different eigenvalues of 𝑇 must be orthogonal. And there must exist a standard orthogonal basis with respect to 𝑎 in eigenspace corresponding to the same eigenvalue. Hence, we can construct a complete orthonormal system of 𝑉 by using the eigenfunctions of 𝑇 corresponding to {𝜆𝑗}:𝑢1,𝑢2,𝑢3,.(2.16)

Remark 2.3. Reference [16] first discussed the property of the spectrum and obtained the above results by compact embedding 𝐻1/2(𝜕Ω)𝐿2(𝜕Ω). But [16] conducted the study in such a space 𝑊: 𝐻2(Ω)𝐻10𝐻(Ω)=𝑊20(Ω), while we define the operator 𝑇 on the space 𝑉=𝐻2(Ω)𝐻10(Ω) directly. Our method is convenient for constructing conforming finite element space 𝑉𝑉 and analyzing the finite element error estimates.

3. Finite Element Method and Its Error Estimates

Let 𝑉𝑉 be a conforming finite element space; for example, 𝑉 is the finite element space associated with one of the Argyris element, Bell element, and Bogner-Fox-Schmit element (see [22]).

The conforming finite element approximation of (2.4) is given by the following. Find 𝜆, 0𝑢𝑉, such that𝑎𝑢,𝑣=𝜆𝑏𝑢,𝑣,𝑣𝑉.(3.1) The source problem associated with (3.1) is as follows. Find 𝑢𝑉, such that𝑎𝑢,𝑣=𝑏(𝑓,𝑣),𝑣𝑉.(3.2) Likewise, from Lax-Milgram theorem we know that (3.2) has a unique solution.

Thus, we can define the operator 𝑇𝑉𝑉 by 𝑎𝑇𝑓,𝑣=𝑏(𝑓,𝑣),𝑣𝑉.(3.3) From [20], we know that (3.1) has the equivalent operator form:𝑇𝑢=1𝜆𝑢.(3.4) Let 𝑃𝑉𝑉 be Ritz projection operator; then𝑎𝑢𝑃𝑢,𝑣=0,𝑣𝑉.(3.5) Combining (2.8) with (3.3), we deduce that, for any 𝑢𝑉,𝑣𝑉, there holds𝑎𝑃𝑇𝑢𝑇𝑃𝑢,𝑣=𝑎𝑇𝑢𝑇𝑢,𝑣+𝑎𝑇𝑢𝑇𝑢,𝑣=0,(3.6) hence,𝑇𝑢𝑃𝑇𝑢𝑎=0,𝑢𝑉;(3.7) thus we get 𝑇=𝑃𝑇. It is clear that 𝑇|𝑉𝑉𝑉(3.8) is a self-adjoint finite rank operator with respect to the inner product 𝑎(,), and the eigenvalues of (3.1) can be arranged as0<𝜆1,𝜆2,𝜆3,𝜆𝑁,𝑁=dim𝑉.(3.9)

As for the regularity of source problem (2.6) it has been reported in the literatures; for example, see [4]. Here, we prove the following regular estimates which will be used in the sequel.

Lemma 3.1. If Ω𝑛(𝑛2) with 𝜕Ω𝑟 (𝑟3), 𝑓𝜈𝐻𝑟(5/2)(𝜕Ω), and 𝑇𝑓 is the solution of (2.6), then 𝑇𝑓𝐻𝑟(Ω) and 𝑇𝑓𝑟𝐶𝑝𝑓𝜈𝑟(5/2),𝜕Ω.(3.10)

Proof. Let 𝑢=𝑇𝑓, Δ𝑢=𝑣; then the boundary value problem (2.6) is transformed to Δ𝑣=0,inΩ,(3.11)𝑣=𝑓𝜈,on𝜕Ω,(3.12)Δ𝑢=𝑣,inΩ,(3.13)𝑢=0,on𝜕Ω.(3.14) Note that (3.11)-(3.12) and (3.13)-(3.14) are two second-order problems. From [23, 24], we know that, when 𝑓𝜈𝐻𝑟(5/2)(𝜕Ω), there exists a weak solution 𝑣𝐻𝑟2(Ω) to (3.11)-(3.12) and 𝑣𝑟2𝐶𝑝𝑓𝜈𝑟(5/2),𝜕Ω.(3.15) From [25] we have that, when 𝜕Ω𝑟, there exists a weak solution 𝑢𝐻𝑟(Ω) to (3.13)-(3.14) and 𝑢𝑟𝐶𝑝𝑣𝑟2.(3.16)
By combining the above two inequalities we obtain 𝑇𝑓𝐻𝑟(Ω) and (3.10).

In this paper, 𝐶𝑝 denotes the prior constant dependent on the equation and Ω and independent of the right-hand side of the equation and . Clearly, constants 𝐶𝑝 that appeared in Lemma 3.1 are not the same.

Lemma 3.2. Let (𝜆,𝑢) and (𝜆,𝑢) be the 𝑘th eigenpair of (2.4) and (3.1), respectively. Then, 𝜆𝑢𝜆=𝑢2𝑎(𝑢)𝜈20,𝜕Ω𝜆(𝑢𝑢)𝜈20,𝜕Ω(𝑢)𝜈20,𝜕Ω.(3.17)

Proof . See [20, 26].

Denote 𝜂=sup𝑓𝑉,𝑓𝑎=1inf𝑣𝑉𝑇𝑓𝑣𝑎.(3.18)

Theorem 3.3. Suppose that 𝜂0(0); then there holds 𝑇𝑇𝑎0(0).(3.19)

Proof. By the definition of operator norm we have 𝑇𝑇𝑎=sup𝑓𝑉,𝑓𝑎=1(𝑇𝑇)𝑓𝑎=sup𝑓𝑉,𝑓𝑎=1𝑇𝑓𝑃𝑇𝑓𝑎=sup𝑓𝑉,𝑓𝑎=1inf𝑣𝑉𝑇𝑓𝑣𝑎=𝜂0(0).(3.20)

Remark 3.4. It is satisfied naturally in conforming finite elements that 𝜂0(0); which is not a restriction. Since 𝑇 is a finite rank operator, it follows from the operator theory that the limit 𝑇 of 𝑇 must be completely continuous. Thus, we have provided another proof that 𝑇𝑉𝑉 is completely continuous.
Let 𝑀(𝜆) denote the eigenfunctions space of (2.4) corresponding to the eigenvalue 𝜆.

Theorem 3.5. Suppose that Ω𝑛(𝑛2) is a bounded domain with 𝜕Ω𝑟(𝑟3). Then, 𝑀(𝜆)𝐻𝑟(Ω).

Proof. Let 𝑢𝑀(𝜆); then (𝜆,𝑢)×𝑉 satisfying (2.4). In (2.4) let 𝜆𝑢=𝑓; then 𝑓𝜈=(𝜆𝑢)𝜈𝐻1/2(𝜕Ω). Therefore, from Lemma 3.1 we know that 𝑢𝐻3(Ω). And 𝑢𝐻3(Ω) leads to 𝑓𝜈=(𝜆𝑢)𝜈𝐻1+1/2(𝜕Ω); again from Lemma 3.1 it follows that 𝑢𝐻4(Ω). By using Lemma 3.1 repeatedly we deduce 𝑢𝐻𝑟(Ω). Thus, 𝑀(𝜆)𝐻𝑟(Ω).

Theorem 3.6. Suppose that 𝜕Ω𝑟(𝑟3), and 𝑉𝑉 is a piecewise 𝑚-degree finite element space. Let (𝜆,𝑢) be the 𝑘th conforming finite element eigenpair of (3.1) and 𝜆 the 𝑘th eigenvalue of (2.4). Then, there exists 𝑢𝑀(𝜆) such that 𝑢𝑢𝑎𝐶𝑡||𝜆,(3.21)||𝜆𝐶2𝑡,(3.22) where 𝑡=min{𝑟,𝑚+1}2.

Proof. By the interpolation error estimate of 𝑚-degree finite element and Lemma 3.1, we have 𝜂=sup𝑓𝑉,𝑓𝑎=1inf𝑣𝑉𝑇𝑓𝑣𝑎sup𝑓𝑉,𝑓𝑎=1𝐶𝑇𝑓3sup𝑓𝑉,𝑓𝑎=1𝑓𝐶𝜈1/2,𝜕Ωsup𝑓𝑉,𝑓𝑎=1𝐶𝑓𝑎𝐶0(0).(3.23) Then, from Theorem 3.3 we know that 𝑇𝑇𝑎0(0). Thus, according to Theorem  7.4 in [20] we have 𝑢𝑢𝑎𝐶𝑇𝑇𝑀(𝜆)𝑎.(3.24) From Theorem 3.5 we have 𝑀(𝜆)𝐻𝑟(Ω). Therefore, for any 𝑢𝑀(𝜆),𝑢𝑎=1, we deduce that (𝑇𝑇)𝑢𝑎=𝑇𝑢𝑃𝑇𝑢𝑎=1𝜆𝑢𝑃𝑢𝑎𝐶𝑡𝑢𝑡+2,𝑇𝑇|𝑀(𝜆)𝑎=sup𝑢𝑀(𝜆),𝑢𝑎=1𝑇𝑇𝑢𝑎;(3.25) combining the above two relations with (3.24), we get the desired result (3.21).
By Lemma 3.2, we get ||𝜆||𝑢𝜆𝐶𝑢2𝑎𝑢𝜈20,𝜕Ω,(3.26) which together with (3.21) yields (3.22).

Corollary 3.7. Suppose that 𝜕Ω6. Let (𝜆,𝑢) be the 𝑘th eigenpair of the Argyris element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that 𝑢𝑢𝑎𝐶4,||𝜆||𝜆𝐶8.(3.27)

Proof. Since the Argyris element contains the complete polynomials of degree 5, that is, 𝑚=5. From the assumption 𝑟=6, we have 𝑡=4. Then, by Theorem 3.6 we get the desired results.

Corollary 3.8. Suppose that 𝜕Ω5. Let (𝜆,𝑢) be the 𝑘th eigenpair of the Bell element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that 𝑢𝑢𝑎𝐶3,||𝜆||𝜆𝐶6.(3.28)

Proof. Since the Bell element contains the complete polynomials of degree 4, that is, 𝑚=4. From the assumption 𝑟=5, we have 𝑡=3. Then, by Theorem 3.6 we get the desired results.

Corollary 3.9. Suppose that 𝜕Ω4. Let (𝜆,𝑢) be the 𝑘th eigenpair of Bogner-Fox-Schmit element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that 𝑢𝑢𝑎𝐶2,||𝜆||𝜆𝐶4.(3.29)

Proof. Applying Theorem 3.6 with 𝑚=3 and noting that 𝑟=4 from the assumption and 𝑡=2, we complete the proof immediately.

Remark 3.10. Next we will discuss, when Ω2 is convex, the regularities of the boundary value problem (2.6) and the eigenvalue problem (2.4) and the error estimates of finite element approximations.
To complete the discussion we need the following regular estimate. Suppose that 𝑢𝑊3,𝑞(Ω)𝐻10(Ω); then 𝑢3,𝑞𝐶𝑝Δ𝑢1,(3.30) where 𝑞<2/(3𝜋/𝜔) while 𝑞 can be arbitrarily close to 2/(3𝜋/𝜔), and 𝜔 is the largest inner angle of Ω.
Reference [27] gave this estimate (see (1.2.9) in [27]) and used it as a fundamental result. Although we have not seen the proof of it, we believe that this estimate is correct.
Suppose that Ω2 is convex, (3.30) holds, and 𝑓𝜈𝐻1/2(𝜕Ω). Let 𝑢=𝑇𝑓, Δ𝑢=𝑣; then the boundary value problem (2.6) is transformed to (3.11)-(3.12) and (3.13)-(3.14). From [23, 24], we know that there exists a weak solution 𝑣𝐻1(Ω) to (3.11)-(3.12) and 𝑣1𝐶𝑝𝑓𝜈1/2,𝜕Ω.(3.31) Reference [28] proved that there exists a weak solution 𝑢𝑊3,𝑞(Ω) to (3.13)-(3.14); and from (3.30) we get𝑢3,𝑞𝐶𝑝𝑣1.(3.32) By combining the above two inequalities we obtain 𝑇𝑓𝑊3,𝑞(Ω) and the following regular estimate: 𝑇𝑓3,𝑞𝐶𝑝𝑓𝜈1/2,𝜕Ω.(3.33)
From the fact that the weak solution, to the boundary value problem, 𝑇𝑓𝑊3,𝑞(Ω), it is easy to know that 𝑀(𝜆)𝑊3,𝑞(Ω), namely, 𝑀(𝜆)𝐻𝑠+2(Ω), where 𝑠<(𝜋/𝜔)1 while 𝑠 can be arbitrarily close to 𝜋/𝜔1.
When Ω is a rectangle, it can be deduced that 𝑀(𝜆)𝐻𝑠+2(Ω), where 𝑠<1 while 𝑠 can be arbitrarily close to 1.
Similar to Theorem 3.6 and Corollaries 3.73.9, by using (3.33) we can prove the following error estimates. Let (𝜆,𝑢) be the 𝑘th eigenpair of the Argyris element, Bell element, or Bogner-Schmit element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that 𝑢𝑢𝑎𝐶𝑠,||𝜆||𝜆𝐶2𝑠.(3.34)

Remark 3.11. In this section we give a priori estimates of finite element approximations (see Theorem 3.6, Corollaries 3.73.9, and (3.34)). These estimates indicate that when the mesh size is small enough we can obtain sufficiently precise approximations of fourth-order Steklov eigenvalues and eigenfunctions (biharmonic function).

4. Numerical Examples

Consider the eigenvalue problem (2.1)–(2.3), where Ω=(0,𝜋/2)×(0,𝜋/2).

We illustrate the Bogner-Fox-Schmit element by Figure 1.

873152.fig.001
Figure 1

The degrees of freedom (interpolation conditions) of Bogner-Fox-Schmit element are function values and gradients (𝜕/𝜕𝑥1,𝜕/𝜕𝑥2) and the second derivatives 𝜕2/𝜕𝑥1𝜕𝑥2 at the four vertices of a rectangle. We adopt a uniform square partition 𝜋 with mesh diameter for Ω, and the Bogner-Fox-Schmit finite element space defined on 𝜋 is 𝑉=𝑣𝐶1(Ω)𝑣|𝜅𝑄3(𝜅),𝜅𝜋,𝑣,𝜕𝑣𝜕𝑥1,𝜕𝑣𝜕𝑥2,𝜕2𝑣𝜕𝑥1𝜕𝑥2,arecontinuousatelementvertices,and𝑣vanishesonboundarynodes(4.1) where 𝑄3(𝜅) is the bicubic polynomial space on an element 𝜅.

It is well known that the Bogner-Fox-Schmit element is a conforming plate element. We compute the first four eigenvalues of (2.1)–(2.3) by the Bogner-Fox-Schmit element by using MATLAB and list the numerical results in Table 1.

tab1
Table 1: Numerical eigenvalues on the square domain Ω=(0,𝜋/2)×(0,𝜋/2) using the Bogner-Fox-Schmit element.

It can be seen from Table 1 that the eigenvalues of Bogner-Fox-Schmit element decrease with the decrease of . This is not an accident. In fact, for conforming finite element approximations for many eigenvalue problems, the minimum-maximum principle is valid; therefore it insures that numerical eigenvalues approximate exact eigenvalues from above (see [20, 26]). Our numerical results coincide with this principle.

Is it possible to compute the lower bound of the eigenvalues of (2.1)–(2.3)? Reference [29] proved theoretically that the nonconforming Morley element can produce the lower bound for the eigenvalues of plate vibration problem, and [30] provided numerical example. These works inspire us to compute approximate eigenvalues of the fourth-order Steklov problem (2.1)–(2.3) by using the Morley element. We illustrate the Morley element by Figure 2.

873152.fig.002
Figure 2

The degrees of freedom (interpolation conditions) of Morley element are function values at the three vertices and outer normal derivatives at the three midpoints of the three edges. We adopt a uniform isosceles right triangulation 𝜋 along three directions for Ω (each triangle is divided into four congruent triangles), and the Morley finite element space defined on 𝜋 is𝑉=𝑣𝐿2(Ω)𝑣|𝜅𝑃2(𝜅),𝜅𝜋,𝑣,𝑣𝜈arecontinuousatelementverticesandmidpointsofthreeedges,resp.,and𝑣vanishesonboundarynodes},(4.2) where 𝑃2(𝜅) is the quadratic polynomial space on an element 𝜅.

The Morley element is a nonconforming plate element. We compute the first four eigenvalues of (2.1)–(2.3) by the Morley element by using MATLAB, and list the numerical results in Table 2.

tab2
Table 2: Numerical eigenvalues on the square domain Ω=(0,𝜋/2)×(0,𝜋/2) using Morley element.

From Table 2 it can be seen that the eigenvalues of Morley element increase with the decrease in . We have the reason to conjecture that the eigenvalues by Morley element approximate the exact ones from below.

From Tables 1 and 2 we can provide bounds for the exact eigenvalues:𝜆1(2.2126268,2.2126974),𝜆2𝜆(4.4156245,4.4161723),3(4.4157676,4.4161723),𝜆4(6.0803290,6.0817236).(4.3)

Remark 4.1. Reference [16] gave bounds for the smallest eigenvalue of (2.1)–(2.3) on Ω=(0,𝜋/2)×(0,𝜋/2): 𝜆1(2.2118,2.2133). By comparison, the bounds we give here are more precise; furthermore, we give the upper and lower bounds for the first four eigenvalues, and it is also confirmed by the numerical experiments that the smallest eigenvalue 𝜆1 is simple [13].

Acknowledgments

The authors cordially thank Professor Bin Liu and the referees for their valuable comments and suggestions that led to the great improvement of this paper. This work supported by the Science and Technology Fund of Guizhou Province of China (LKS [2010] no.01) and the National Science Foundation of China (Grant no.10761003).

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