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

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=Δ𝑢−𝜆𝑢_𝜈=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 [5–12] 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 ℝ^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 𝑇∶𝐻²(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)∩𝐻¹⁰(Ω)=𝑊⨁𝐻²⁰(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 W^𝑠,𝑞(Ω) denote the usual Sobolev space with real-order 𝑠 with norm ‖⋅‖_𝑠,𝑞. For simplicity, we write 𝐻^𝑠(Ω) for 𝑊^𝑠,2(Ω) with norm ‖⋅‖_𝑠 and 𝐻^𝑠(𝜕Ω) for 𝑊^𝑠,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 ℎ, 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 problemΔΔ𝑢𝑢²==𝑢=𝜆0,𝑢0_𝜈,oni𝜕nΩΩ,,((22..12)),on𝜕Ω,(2.3) where Ω⊂ℝ^𝑛(𝑛≥2) is a bounded domain and 𝜕Ω is smooth, or Ω⊂ℝ² is a convex domain, 𝑢_𝜈 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∈ℝ, 0≠𝑢∈𝑉, such that𝑎(𝑢,𝑣)=𝜆𝑏(𝑢,𝑣),∀𝑣∈𝑉,(2.4) where𝑎(𝑢,𝑣)=_ΩΔ𝑢Δ𝑣𝑑𝑥,𝑏(𝑢,𝑣)=_𝜕Ω𝑢_𝜈𝑣_𝜈𝑑𝑠.(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,𝑣∈𝑉, and ‖𝑢‖_𝑎=√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 Ω⊂ℝ^𝑛 with 𝜕Ω∈ℂ² or a convex domain Ω⊂ℝ².

The source problem associated with (2.4) is as follows. Find 𝑢∈𝑉, such that 𝑎(𝑢,𝑣)=𝑏(𝑓,𝑣),∀𝑣∈𝑉.(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)||=||||_𝜕Ω𝑓_𝜈𝑣_𝜈𝑑𝑠||||≤‖‖𝑓_𝜈‖‖_0,𝜕Ω‖‖𝑣_𝜈‖‖_0,𝜕Ω≤‖‖𝑓_𝜈‖‖_0,𝜕Ω‖𝑣‖₂,∀𝑣∈𝑉.(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 𝑇∶𝑉→𝑉 by 𝑎(𝑇𝑓,𝑣)=𝑏(𝑓,𝑣),∀𝑣∈𝑉.(2.8) From [20], we know that (2.4) has the equivalent operator form: 𝑇𝑢=1𝜆𝑢.(2.9)

Lemma 2.2. The operator 𝑇∶𝑉→𝑉 is self-adjoint and completely continuous.

Proof. By the definition of 𝑇, for any 𝑢,𝑣∈𝑉, 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 𝑓∈𝑉 we have 𝑎(𝑇𝑓,𝑇𝑓)=𝑏(𝑓,𝑇𝑓)≤‖‖𝑓_𝜈‖‖_0,𝜕Ω‖‖(𝑇𝑓)_𝜈‖‖_0,𝜕Ω≤𝐶‖‖𝑓_𝜈‖‖_0,𝜕Ω‖𝑇𝑓‖₂;(2.11) it follows from the fact that 𝑎(⋅,⋅) is 𝑉-elliptic that ‖𝑇𝑓‖₂≤𝐶‖‖𝑓_𝜈‖‖_0,𝜕Ω.(2.12) Let {𝑓_𝑚} be a bounded sequence in ‖⋅‖₂; then by trace theorem we have {(𝑓_𝑚)_𝜈} is a bounded sequence in ‖⋅‖_1/2,𝜕Ω. And, by compact embedding 𝐻^1/2(𝜕Ω)⊂𝐿₂(𝜕Ω), we know that there is a subsequence {(𝑓_𝑚𝑙)_𝜈} that is a Cauchy sequence in ‖⋅‖_0,𝜕Ω. From (2.12) we conclude that {𝑇𝑓_𝑚𝑙} is a Cauchy sequence in ‖⋅‖₂, which implies that 𝑇 is completely continuous.

Let (𝜆,𝑢) be an eigenpair of (2.4); then ‖𝑢‖_𝑎≠0. Since 𝑎(𝑢,𝑢)=𝜆𝑏(𝑢,𝑢), we see that ‖𝑢_𝜈‖_0,𝜕Ω≠0,𝜆≠0. And, by trace theorem we have ‖‖𝑢_𝜈‖‖_0,𝜕Ω≤𝐶‖𝑢‖₂≤𝐶‖𝑢‖_𝑎;(2.13) thus, 𝜆=𝑎(𝑢,𝑢)𝑏(𝑢,𝑢)=‖‖𝑢‖_𝜈𝑢‖‖‖^{2𝑎20,𝜕Ω}≥1𝐶²>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<𝜆₁≤𝜆₂≤𝜆₃≤⋯↗+∞.(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 {𝜆_𝑗}: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 𝐻^1/2(𝜕Ω)⊂𝐿₂(𝜕Ω). But [16] conducted the study in such a space 𝑊: 𝐻²(Ω)∩𝐻¹⁰(Ω)=𝑊⨁𝐻²⁰(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 𝑉_ℎ⊂𝑉 and analyzing the finite element error estimates.

3. Finite Element Method and Its Error Estimates

Let 𝑉_ℎ⊂𝑉 be a conforming finite element space; for example, 𝑉_ℎ 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_ℎ∈ℝ, 0≠𝑢_ℎ∈𝑉_ℎ, such that𝑎𝑢_ℎ,𝑣=𝜆_ℎ𝑏𝑢_ℎ,𝑣,∀𝑣∈𝑉_ℎ.(3.1) The source problem associated with (3.1) is as follows. Find 𝑢_ℎ∈𝑉_ℎ, such that𝑎𝑢_ℎ,𝑣=𝑏(𝑓,𝑣),∀𝑣∈𝑉_ℎ.(3.2) Likewise, from Lax-Milgram theorem we know that (3.2) has a unique solution.

Thus, we can define the operator 𝑇_ℎ∶𝑉→𝑉_ℎ by 𝑎𝑇_ℎ𝑓,𝑣=𝑏(𝑓,𝑣),∀𝑣∈𝑉_ℎ.(3.3) From [20], we know that (3.1) has the equivalent operator form:𝑇_ℎ𝑢_ℎ=1𝜆_ℎ𝑢_ℎ.(3.4) Let 𝑃_ℎ∶𝑉→𝑉_ℎ be Ritz projection operator; then𝑎𝑢−𝑃_ℎ𝑢,𝑣=0,∀𝑣∈𝑉_ℎ.(3.5) Combining (2.8) with (3.3), we deduce that, for any 𝑢∈𝑉,𝑣∈𝑉_ℎ, there holds𝑎𝑃_ℎ𝑇𝑢−𝑇_ℎ𝑢,𝑣=𝑎𝑃_ℎ𝑇𝑢−𝑇𝑢,𝑣+𝑎𝑇𝑢−𝑇_ℎ𝑢,𝑣=0,(3.6) hence,‖‖𝑇_ℎ𝑢−𝑃_ℎ𝑇𝑢‖‖_𝑎=0,∀𝑢∈𝑉;(3.7) thus we get 𝑇_ℎ=𝑃_ℎ𝑇. It is clear that 𝑇_ℎ|_𝑉ℎ∶𝑉_ℎ→𝑉_ℎ(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<𝜆_1,ℎ≤𝜆_2,ℎ≤𝜆_3,ℎ≤⋯𝜆_𝑁ℎ,ℎ𝑁_ℎ=dim𝑉_ℎ.(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 Ω⊂ℝ^𝑛(𝑛≥2) with 𝜕Ω∈ℂ^𝑟 (𝑟≥3), 𝑓_𝜈∈𝐻^𝑟−(5/2)(𝜕Ω), and 𝑇𝑓 is the solution of (2.6), then 𝑇𝑓∈𝐻^𝑟(Ω) and ‖𝑇𝑓‖_𝑟≤𝐶_𝑝‖‖𝑓_𝜈‖‖_{𝑟−(5/2),𝜕Ω}.(3.10)

Proof. Let 𝑢=𝑇𝑓, Δ𝑢=𝑣; then the boundary value problem (2.6) is transformed to 𝑣Δ=𝑣=𝑓_𝜈0,inΩ,(3.11)Δ𝑢𝑢==0,𝑣,,ooninn𝜕𝜕ΩΩΩ,.,(((333...111234))) Note that (3.11)-(3.12) and (3.13)-(3.14) are two second-order problems. From [23, 24], we know that, when 𝑓_𝜈∈𝐻^𝑟−(5/2)(𝜕Ω), there exists a weak solution 𝑣∈𝐻^𝑟−2(Ω) to (3.11)-(3.12) and ‖𝑣‖_𝑟−2≤𝐶_𝑝‖‖𝑓_𝜈‖‖_{𝑟−(5/2),𝜕Ω}.(3.15) From [25] we have that, when 𝜕Ω∈ℂ^𝑟, there exists a weak solution 𝑢∈𝐻^𝑟(Ω) to (3.13)-(3.14) and ‖𝑢‖_𝑟≤𝐶_𝑝‖𝑣‖_𝑟−2.(3.16)
By combining the above two inequalities we obtain 𝑇𝑓∈𝐻^𝑟(Ω) and (3.10).

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

Lemma 3.2. Let (𝜆,𝑢) and (𝜆_ℎ,𝑢_ℎ) be the 𝑘th eigenpair of (2.4) and (3.1), respectively. Then, 𝜆_ℎ−𝜆=‖‖𝑢^ℎ−𝑢‖‖^2𝑎‖‖(𝑢_ℎ)_𝜈‖‖^20,𝜕Ω−𝜆‖‖(𝑢^ℎ−𝑢)_𝜈‖‖^20,𝜕Ω‖‖(𝑢_ℎ)_𝜈‖‖^20,𝜕Ω.(3.17)

Proof . See [20, 26].

Denote 𝜂_ℎ=sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}inf_{𝑣∈𝑉ℎ}‖𝑇𝑓−𝑣‖_𝑎.(3.18)

Theorem 3.3. Suppose that 𝜂_ℎ→0(ℎ→0); then there holds ‖‖𝑇−𝑇_ℎ‖‖_𝑎⟶0(ℎ⟶0).(3.19)

Proof. By the definition of operator norm we have ‖‖𝑇−𝑇_ℎ‖‖_𝑎=sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}‖‖(𝑇−𝑇_ℎ)𝑓‖‖_𝑎=sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}‖‖𝑇𝑓−𝑃_ℎ𝑇𝑓‖‖_𝑎=sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}inf_{𝑣∈𝑉ℎ}‖𝑇𝑓−𝑣‖_𝑎=𝜂_ℎ⟶0(ℎ⟶0).(3.20)

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

Theorem 3.5. Suppose that Ω⊂ℝ^𝑛(𝑛≥2) is a bounded domain with 𝜕Ω∈ℂ^𝑟(𝑟≥3). Then, 𝑀(𝜆)⊂𝐻^𝑟(Ω).

Proof. Let 𝑢∈𝑀(𝜆); then (𝜆,𝑢)∈ℝ×𝑉 satisfying (2.4). In (2.4) let 𝜆𝑢=𝑓; then 𝑓_𝜈=(𝜆𝑢)_𝜈∈𝐻^1/2(𝜕Ω). Therefore, from Lemma 3.1 we know that 𝑢∈𝐻³(Ω). And 𝑢∈𝐻³(Ω) leads to 𝑓_𝜈=(𝜆𝑢)_𝜈∈𝐻^1+1/2(𝜕Ω); again from Lemma 3.1 it follows that 𝑢∈𝐻⁴(Ω). By using Lemma 3.1 repeatedly we deduce 𝑢∈𝐻^𝑟(Ω). Thus, 𝑀(𝜆)⊂𝐻^𝑟(Ω).

Theorem 3.6. Suppose that 𝜕Ω∈ℂ^𝑟(𝑟≥3), and 𝑉_ℎ⊂𝑉 is a piecewise 𝑚-degree finite element space. Let (𝜆_ℎ,𝑢_ℎ) be the 𝑘th conforming finite element eigenpair of (3.1) and 𝜆 the 𝑘th eigenvalue of (2.4). Then, there exists 𝑢∈𝑀(𝜆) such that ‖‖||𝜆𝑢−𝑢_ℎ‖‖_𝑎≤𝐶ℎ^𝑡,(3.21)_ℎ−𝜆||≤𝐶ℎ^2𝑡,(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_ℎ=sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}inf_{𝑣∈𝑉ℎ}‖𝑇𝑓−𝑣‖_𝑎≤sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}𝐶ℎ‖𝑇𝑓‖₃≤sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}𝐶ℎ‖‖𝑓_𝜈‖‖_1/2,𝜕Ω≤sup_{𝑓∈𝑉,‖𝑓‖𝑎=1}𝐶ℎ‖𝑓‖_𝑎≤𝐶ℎ⟶0(ℎ⟶0).(3.23) Then, from Theorem 3.3 we know that ‖𝑇−𝑇_ℎ‖_𝑎→0(ℎ→0). Thus, according to Theorem  7.4 in [20] we have ‖‖𝑢−𝑢_ℎ‖‖_𝑎≤𝐶‖‖𝑇−𝑇_ℎ∣_𝑀(𝜆)‖‖_𝑎.(3.24) From Theorem 3.5 we have 𝑀(𝜆)⊂𝐻^𝑟(Ω). Therefore, for any 𝑢∈𝑀(𝜆),‖𝑢‖_𝑎=1, we deduce that ‖‖(𝑇−𝑇_ℎ)𝑢‖‖_𝑎=‖‖𝑇𝑢−𝑃_ℎ𝑇𝑢‖‖_𝑎=1𝜆‖‖𝑢−𝑃^ℎ‖‖𝑇−𝑇_ℎ𝑢‖‖_𝑎≤𝐶ℎ^𝑡‖𝑢‖_𝑡+2,|_𝑀(𝜆)‖‖_𝑎=sup_{𝑢∈𝑀(𝜆),‖𝑢‖𝑎=1}‖‖𝑇−𝑇_ℎ𝑢‖‖_𝑎;(3.25) combining the above two relations with (3.24), we get the desired result (3.21).
By Lemma 3.2, we get ||𝜆_ℎ−𝜆||≤𝐶‖‖𝑢^ℎ−𝑢‖‖^2𝑎‖‖𝑢_ℎ_𝜈‖‖^20,𝜕Ω,(3.26) which together with (3.21) yields (3.22).

Corollary 3.7. Suppose that 𝜕Ω∈ℂ⁶. Let (𝜆_ℎ,𝑢_ℎ) be the 𝑘th eigenpair of the Argyris element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that ‖‖𝑢||−𝜆_ℎ𝑢_ℎ‖‖_𝑎≤𝐶ℎ⁴,−𝜆||≤𝐶ℎ⁸.(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 𝜕Ω∈ℂ⁵. Let (𝜆_ℎ,𝑢_ℎ) be the 𝑘th eigenpair of the Bell element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that ‖‖𝑢||−𝜆_ℎ𝑢_ℎ‖‖_𝑎≤𝐶ℎ³,−𝜆||≤𝐶ℎ⁶.(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 𝜕Ω∈ℂ⁴. Let (𝜆_ℎ,𝑢_ℎ) be the 𝑘th eigenpair of Bogner-Fox-Schmit element. Then, there exists the 𝑘th eigenpair (𝜆,𝑢) of (2.4) such that ‖‖𝑢||−𝜆_ℎ𝑢_ℎ‖‖_𝑎≤𝐶ℎ²,−𝜆||≤𝐶ℎ⁴.(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 Ω⊂ℝ² 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 𝑢∈𝑊^3,𝑞(Ω)∩𝐻¹⁰(Ω); then ‖𝑢‖_3,𝑞≤𝐶_𝑝‖Δ𝑢‖₁,(3.30) where 𝑞<2/(3−𝜋/𝜔) while 𝑞 can be arbitrarily close to 2/(3−𝜋/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 Ω⊂ℝ² is convex, (3.30) holds, and 𝑓_𝜈∈𝐻^1/2(𝜕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 𝑣∈𝐻¹(Ω) to (3.11)-(3.12) and ‖𝑣‖₁≤𝐶_𝑝‖‖𝑓_𝜈‖‖_1/2,𝜕Ω.(3.31) Reference [28] proved that there exists a weak solution 𝑢∈𝑊^3,𝑞(Ω) to (3.13)-(3.14); and from (3.30) we get‖𝑢‖_3,𝑞≤𝐶_𝑝‖𝑣‖₁.(3.32) By combining the above two inequalities we obtain 𝑇𝑓∈𝑊^3,𝑞(Ω) and the following regular estimate: ‖𝑇𝑓‖_3,𝑞≤𝐶_𝑝‖‖𝑓_𝜈‖‖_1/2,𝜕Ω.(3.33)
From the fact that the weak solution, to the boundary value problem, 𝑇𝑓∈𝑊^3,𝑞(Ω), it is easy to know that 𝑀(𝜆)⊂𝑊^3,𝑞(Ω), namely, 𝑀(𝜆)⊂𝐻^𝑠+2(Ω), where 𝑠<(𝜋/𝜔)−1 while 𝑠 can be arbitrarily close to 𝜋/𝜔−1.
When Ω is a rectangle, it can be deduced that 𝑀(𝜆)⊂𝐻^𝑠+2(Ω), where 𝑠<1 while 𝑠 can be arbitrarily close to 1.
Similar to Theorem 3.6 and Corollaries 3.7–3.9, by using (3.33) we can prove the following error estimates. Let (𝜆_ℎ,𝑢_ℎ) 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 ‖‖||𝑢𝜆_ℎ−𝑢_ℎ‖‖_𝑎≤𝐶ℎ^𝑠,−𝜆||≤𝐶ℎ^2𝑠.(3.34)

Remark 3.11. In this section we give a priori estimates of finite element approximations (see Theorem 3.6, Corollaries 3.7–3.9, and (3.34)). These estimates indicate that when the mesh size ℎ 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.

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_ℎ with mesh diameter ℎ for Ω, and the Bogner-Fox-Schmit finite element space defined on 𝜋_ℎ is 𝑉_ℎ=𝑣∈𝐶¹(Ω)∶𝑣|_𝜅∈𝑄₃(𝜅),∀𝜅∈𝜋_ℎ,𝑣,𝜕𝑣𝜕𝑥₁,𝜕𝑣𝜕𝑥₂,𝜕²𝑣𝜕𝑥₁𝜕𝑥₂arecontinuousatelementvertices,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.

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 ℎ. 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.

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_ℎ along three directions for Ω (each triangle is divided into four congruent triangles), and the Morley finite element space defined on 𝜋_ℎ is𝑉_ℎ=𝑣∈𝐿₂(Ω)∶𝑣|_𝜅∈𝑃₂(𝜅),∀𝜅∈𝜋_ℎ,𝑣,𝑣_𝜈aorfetchorneetineudoguess,atreeslpe.m,eanntdv𝑣ervtiacneisshaensdomnidbpoouinntdsarynodes},(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.

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 ℎ. 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:𝜆𝜆₁∈(2.2126268,2.2126974),𝜆₂∈(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).