`Mathematical Problems in EngineeringVolume 2011, Article ID 873152, 13 pageshttp://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

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 under the boundary conditions on (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 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 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 and prove that is completely continuous thus characterize the spectrum of a fourth-order Steklov eigenvalue problem by the spectrum of . Note that [16] analyzed the spectrum of this problem by introducing an orthogonal decomposition,, and conducted the research on the space . 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 denote the usual Sobolev space with real-order with norm . For simplicity, we write for with norm and for with norm . . Throughout this paper, 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 problem where is a bounded domain and is smooth, or is a convex domain, denotes the outer normal derivative of on .

Denote .

The weak form of (2.1)–(2.3) is given by the following. Find , , such that where

Lemma 2.1. Assume that is a Lipschitz bounded domain which satisfies the uniform outer ball condition. Then the space becomes a Hilbert space when endowed with the scalar product and is equivalent to the norm induced by .

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

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

The source problem associated with (2.4) is as follows. Find , such that

It follows from Lemma 2.1 that is a symmetric, continuous, and -elliptic bilinear form on . By Schwarz inequality and trace theorem we have 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 by From [20], we know that (2.4) has the equivalent operator form:

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

Proof. By the definition of , for any , there holds that is, is self-adjoint with respect to .
Next we will prove that is completely continuous. By Schwarz inequality and trace theorem, for any we have it follows from the fact that is -elliptic that Let be a bounded sequence in ; then by trace theorem we have is a bounded sequence in . And, by compact embedding , we know that there is a subsequence that is a Cauchy sequence in . From (2.12) we conclude that is a Cauchy sequence in , which implies that is completely continuous.

Let be an eigenpair of (2.4); then . Since , we see that . And, by trace theorem we have thus,

Therefore, from the spectral theory of completely continuous operator we know that all eigenvalues of are real and have finite algebraic multiplicity. We arrange the eigenvalues of by increasing order:

The eigenfunctions corresponding to two arbitrary different eigenvalues of must be orthogonal. And there must exist a standard orthogonal basis with respect to in eigenspace corresponding to the same eigenvalue. Hence, we can construct a complete orthonormal system of by using the eigenfunctions of corresponding to :

Remark 2.3. Reference [16] first discussed the property of the spectrum and obtained the above results by compact embedding . But [16] conducted the study in such a space : , while we define the operator on the space directly. Our method is convenient for constructing conforming finite element space and analyzing the finite element error estimates.

3. Finite Element Method and Its Error Estimates

Let be a conforming finite element space; for example, 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 , , such that The source problem associated with (3.1) is as follows. Find , such that Likewise, from Lax-Milgram theorem we know that (3.2) has a unique solution.

Thus, we can define the operator by From [20], we know that (3.1) has the equivalent operator form: Let be Ritz projection operator; then Combining (2.8) with (3.3), we deduce that, for any , there holds hence, thus we get . It is clear that is a self-adjoint finite rank operator with respect to the inner product , and the eigenvalues of (3.1) can be arranged as

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 with (), , and is the solution of (2.6), then and

Proof. Let , ; then the boundary value problem (2.6) is transformed to Note that (3.11)-(3.12) and (3.13)-(3.14) are two second-order problems. From [23, 24], we know that, when , there exists a weak solution to (3.11)-(3.12) and From [25] we have that, when , there exists a weak solution to (3.13)-(3.14) and
By combining the above two inequalities we obtain and (3.10).

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

Lemma 3.2. Let and be the th eigenpair of (2.4) and (3.1), respectively. Then,

Proof . See [20, 26].

Denote

Theorem 3.3. Suppose that ; then there holds

Proof. By the definition of operator norm we have

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

Theorem 3.5. Suppose that is a bounded domain with . Then, .

Proof. Let ; then satisfying (2.4). In (2.4) let ; then . Therefore, from Lemma 3.1 we know that . And leads to ; again from Lemma 3.1 it follows that . By using Lemma 3.1 repeatedly we deduce . Thus, .

Theorem 3.6. Suppose that , and is a piecewise -degree finite element space. Let be the th conforming finite element eigenpair of (3.1) and the th eigenvalue of (2.4). Then, there exists such that where .

Proof. By the interpolation error estimate of -degree finite element and Lemma 3.1, we have Then, from Theorem 3.3 we know that . Thus, according to Theorem  7.4 in [20] we have From Theorem 3.5 we have . Therefore, for any , we deduce that combining the above two relations with (3.24), we get the desired result (3.21).
By Lemma 3.2, we get which together with (3.21) yields (3.22).

Corollary 3.7. Suppose that . Let be the th eigenpair of the Argyris element. Then, there exists the th eigenpair of (2.4) such that

Proof. Since the Argyris element contains the complete polynomials of degree , that is, . From the assumption , we have . Then, by Theorem 3.6 we get the desired results.

Corollary 3.8. Suppose that . Let be the th eigenpair of the Bell element. Then, there exists the th eigenpair of (2.4) such that

Proof. Since the Bell element contains the complete polynomials of degree , that is, . From the assumption , we have . Then, by Theorem 3.6 we get the desired results.

Corollary 3.9. Suppose that . Let be the th eigenpair of Bogner-Fox-Schmit element. Then, there exists the th eigenpair of (2.4) such that

Proof. Applying Theorem 3.6 with and noting that from the assumption and , we complete the proof immediately.

Remark 3.10. Next we will discuss, when 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 ; then where while can be arbitrarily close to , and is the largest inner angle of .
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 is convex, (3.30) holds, and . Let , ; 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 to (3.11)-(3.12) and Reference [28] proved that there exists a weak solution to (3.13)-(3.14); and from (3.30) we get By combining the above two inequalities we obtain and the following regular estimate:
From the fact that the weak solution, to the boundary value problem, , it is easy to know that , namely, , where while can be arbitrarily close to .
When is a rectangle, it can be deduced that , where while 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 be the th eigenpair of the Argyris element, Bell element, or Bogner-Schmit element. Then, there exists the th eigenpair of (2.4) such that

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 .

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 and the second derivatives at the four vertices of a rectangle. We adopt a uniform square partition with mesh diameter for , and the Bogner-Fox-Schmit finite element space defined on is where is the bicubic polynomial space on an element .

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 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.

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 along three directions for (each triangle is divided into four congruent triangles), and the Morley finite element space defined on is where is the quadratic polynomial space on an element .

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 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:

Remark 4.1. Reference [16] gave bounds for the smallest eigenvalue of (2.1)–(2.3) on : . 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 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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