Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 650530, 9 pages

http://dx.doi.org/10.1155/2013/650530

Research Article

Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem

Xuqing Zhang, Yidu Yang, and Hai Bi

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

Received 18 January 2013; Accepted 3 September 2013

Academic Editor: Daoyi Dong

Copyright © 2013 Xuqing Zhang 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 discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001), a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.

1. Introduction

Since Steklov eigenvalue problems have important physical background and many applications, for instance, they appear in the analysis of stability of mechanical oscillators immersed in a viscous fluid (see [1] and the references therein), in the analysis of the antiplane shearing on a system of collinear faults under slip-dependent friction law (see [2]), in the study of surface waves (see [3]), in the study of the vibration modes of a structure in contact with an incompressible fluid (see [4]) and vibration of a pendulum (see [5]), and in the eigen-oscillations of mechanical systems with boundary conditions containing frequency [6], and thus, numerical methods for solving Steklov eigenvalue problems have received increasing attention in recent years. Bramble and Osborn [7] studied the Galerkin approximation of a Steklov eigenvalue problem of nonself-adjoint second order elliptic operators in smooth domain, Andreev and Todorov [8] discussed the isoparametric finite element method for the approximation of the Steklov eigenvalue problem of second-order self-adjoint elliptic differential operators, Armentano and Padra [9] introduced and analyzed the conforming linear finite element approximation of the Steklov eigenvalue problem in a bounded polygonal domain, Alonso and Russo [10], Yang et al. [11], and Li et al. [12] studied nonconforming finite elements approximation of the Steklov eigenvalue problem, Li and Yang [13] and Bi and Yang [14] discussed a two-grid method of the conforming and non-conforming finite element method, respectively, Li et al. [15] studied the extrapolation and superconvergence of the Steklov eigenvalue problem, Tang et al. [16] studied the boundary element approximation, and Cao et al. [17] discussed multiscale asymptotic method for Steklov eigenvalue equations in composite medias. However, to the best of our knowledge, there have been no reports on spectral method for Steklov eigenvalue problems.

Spectral method is an important numerical method for solving differential equations developed after finite-difference and finite-element methods. Because spectral method has the characteristics of superior accuracy, it is widely used in the field of meteorology, physics, and mechanics (see [18, 19]). This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto (LGL for short) points for solving the Steklov eigenvalue problem. We use spectral approximation theory (see [20]) to give a priori error estimates. Based on the work of [21], we discuss a posterior error estimator of the residual type. Moreover, inspired by the work of [22] this paper combines the shifted-inverse iterative method and spectral method to propose an efficient computation scheme. Finally, by using program with Matlab, we implement the numerical experiments and get satisfactory results.

The rest of the paper is organized as follows. In Section 2, some preliminaries needed in this paper are presented. In Section 3, we discuss a priori and a posterior error estimates for spectral method. In Section 4, we establish an efficient scheme combining the shifted-inverse iteration and spectral method. In Section 5, the numerical experiments on the square domain are reported.

2. Preliminaries

We consider the following boundary value problem:

where is a rectangular domain and is the outward normal derivative on . We denote the Sobolev space with norm and seminorm by . Let and be the norms in the space and , respectively.

Throughout this paper, denotes a generic positive constant independent of the polynomial degrees , which may not be the same at each occurrence.

The variational problem associated with (1) is given by the following.

Find , such that

where

Obviously, the bilinear form is continuous and -elliptic, that is, there exists constants and independent of , , such that

From the Lax-Milgram theorem we know that there exists a unique solution to (2).

The following regularity results (see [4]) will be needed in the sequel: for , the solution of problem (1) , and

furthermore, if , then and

where , are constants independent of , and if is convex and if is concave (with being the largest inner angle of ).

Define

The spectral-Galerkin approximation to (2) is the following.

Find , such that

For and each integer , we define the interpolation operator

with the LGL interpolation nodes in . We quote from [18] the following interpolation estimates for spectral method at the tensor-product LGL points

Let and be the solution of (2) and (8), respectively. Then we derive from Céa Lemma that

Notice that (11) is also valid for special method with modal basis (see [18, 19]).

Using Aubin-Nitsche technique, we deduce the a priori error estimate from (5) and (10)-(11):

3. Spectral Method and Its Error Estimates for the Steklov Eigenvalue Problem

3.1. Spectral Method for the Steklov Eigenvalue Problem

We consider the following Steklov eigenvalue problem:

where is a rectangular domain.

The variational problem associated with (13) is given by the following.

Find , , such that

Let be an eigenpair of (14); then from (6) we know that .

The spectral-Galerkin approximation to (14) is as the follows.

Find , , such that

Let and be the solution operators defined by

Obviously, is completely continuous and is a finite ranked operator. Equations (14) and (15) have the following equivalent operator forms, respectively:

3.2. A Prior Error Estimates of Spectral Method for the Steklov Eigenvalue Problem

Let be an eigenpair of (14), let be the space spanned by all eigenfunctions corresponding to the eigenvalue , and let be the direct sum of the eigenspace corresponding to the eigenvalues of (15) that converge to .

Denote

where characterizes the degree of the space approximating , and characterize the degrees of approximating the generalized solution of (2).

Lemma 1. Let ; then

Proof. We deduce from the interpolation estimate (10) that

Using (10) and (6) we have

From (10) and regularity estimate (5) we get

The proof is completed.

Theorem 2. Let ; then

let and ; then there exists , such that

let and ; then there exists , such that

Proof. From [20, 23] we have that

let and ; then there exists , such that

let and ; then there exists , such that

Combining Lemma 1 we obtain the desired results.

3.3. A Posterior Error Estimates for the Steklov Eigenvalue Problem

Theorem 2.1 in [24] gave the following results.

Lemma 3. There exists a positive constant and a linear bounded operator , such that for all and an arbitrary edge of  , and there holds

Analysis of the error indicators requires polynomial inverse estimates in weighted Sobolev spaces. Define on the interval the weight function ; then there holds the following polynomial inverse estimates.

Lemma 4. Let , ; then there exists , , and such that for every univariate polynomial of degree , there holds

Proof. See Lemma 2.4 in [21].

Consider the domain and the weight function . From [21] we have the two-dimensional polynomial inverse estimates.

Lemma 5. Let , ; then there exists , , and such that for all polynomials there hold

Proof. See Lemma 2.5 in [21].

Lemma 6. Let , set , and let be defined as the same as , that is, . Then there exists , such that for every and every univariate polynomial of degree , there exists an extension , such that the following holds:

Proof. See Lemma 2.6 in [21].

Remark 7. Because there has an affine transformation from rectangular domain to the unit square, Lemma 5 is suited to rectangular domain, and Lemmas 4 and 6 are valid for all edges of rectangular.

Define the global error indicator , , which is expressed as the sum of two terms:

where the first term is the weighted internal residual given by

and the second term is the weighted edge residual given by

where the weight functions and are scaled transformations of the weight functions and , respectively.

By using the proof of [21], we can get the following results.

Theorem 8. Let . Then there exists independent of , such that

Proof. We denote where is an interpolation operator satisfying Lemma 3. We derive from -ellipticity of that

Since

thus

By using Cauchy-Schwartz inequality, the trace theorem, and Lemma 3, we get

Plugging (47) into (46) gives

Setting , in (32) and (35), respectively, we obtain and , and thus

The proof is completed.

Lemma 9. Let . Then there exists independent of , such that

Proof. Denote , ; then

Now, we consider . Using (35)-(36), we find for that

noticing that (35) is valid since the assumption implies , hence we take , in (35) and the above third inequality holds. By the definition , we obtain

To get an upper bound of the error indicator in the case of , using (35) we find for that

Therefore,

We obtain the desired result immediately.

Lemma 10. Let , . Then there exists independent of , such that

Proof. In order to obtain an upper bound for the error indicator , we will use weight function and a proper extension operator. Let be an edge of ; we construct a function and .

Let . We extend to by using Lemma 6 where the polynomial corresponds to , and from (37) we know that is zero on . We extend to such that on ; then

Now we estimate , , in turn. First,

In the case of we have

Combining (58) and (59), we get

By the definition of and setting in Lemma 9, we have that

Setting yields the assertion for in (61).

For the case , setting and using (35), we find

therefore,

From the above discussion we complete the proof.

Combining Lemmas 9 and 10, we obtain the following theorem.

Theorem 11. Let . Then there exists independent of , such that

Remark 12. The term is a term of higher order. In fact, for any , we have

Thus, taking and using the a priori estimates (23) and (25), we conclude that

4. Spectral Method Based on the Shifted-Inverse Iteration

High efficient finite element schemes have been researched further in recent years; for example, see [22, 25–31]. Based on these works, this paper combines the shifted-inverse iteration and spectral method to propose an efficient scheme.

Scheme 13 (spectral method based on the shifted-inverse iteration).

Step 1. Solve (13) in : Find , , such that

Step 2. Solve a linear system in . Find , such that

Step 3. Compute the Rayleigh quotient

Let be the th eigenpair of (67) then is regarded as the th approximate eigenpair of (13).

Theorem 14. Suppose that and is properly large. Let be the th approximate eigenpair obtained by Scheme 13; then there exists the th eigenpair of (13), such that

Proof. From the proof of Theorem 4.1 in [22] we can deduce that

which together with Lemma 1 yields (70).

Theorem 14 tells us, when , the degree of precision (DOP) of and obtained by Scheme 13 is and , respectively; that is to say, have the same DOP with .

5. Numerical Experiments

Consider the Steklov eigenvalue problem (13) where .

We use spectral method with tensor-product nodal basis at the LGL points (LGL spectral method) to solve (13). We construct two-dimensional basis functions by using the tensor product of one-dimensional basis functions. More precisely, let be the Lagrange basis polynomials associated with the LGL points; then , and let ; then (15) is equivalent to the following linear system:

where is called stiffness matrix, is called mass matrix, and

We program with Matlab to solve (72), and the numerical results are presented in Table 1.

Table 1: The eigenvalues of LGL spectral method.

DOF

3
16
0.24007936507936
1.49252393896074
2.08264705882353

4
25
0.24007908542791
1.49230719085030
2.08264705882353

5
36
0.24007908542792
1.49230316584831
2.08264705403320

6
49
0.24007908542722
1.49230313468790
2.08264705403320

7
64
0.24007908542723
1.49230313453462
2.08264705403320

8
81
0.24007908542723
1.49230313453360
2.08264705403320

9
100
0.24007908542722
1.49230313453358
2.08264705403318

10
121
0.24007908542721
1.49230313453357
2.08264705403318

11
144
0.24007908542722
1.49230313453359
2.08264705403321

12
169
0.24007908542721
1.49230313453357
2.08264705403318

13
196
0.24007908542723
1.49230313453360
2.08264705403323

14
225
0.24007908542722
1.49230313453357
2.08264705403318

15
256
0.24007908542724
1.49230313453359
2.08264705403320

16
289
0.24007908542723
1.49230313453359
2.08264705403319

17
324
0.24007908542728
1.49230313453361
2.08264705403321

18
361
0.24007908542725
1.49230313453359
2.08264705403319

19
400
0.24007908542721
1.49230313453356
2.08264705403317

20
441
0.24007908542718
1.49230313453349
2.08264705403311

21
484
0.24007908542705
1.49230313453341
2.08264705403305

22
529
0.24007908542721
1.49230313453356
2.08264705403317

23
576
0.24007908542727
1.49230313453364
2.08264705403326

24
625
0.24007908542700
1.49230313453339
2.08264705403301

25
676
0.24007908542723
1.49230313453356
2.08264705403317

27
784
0.24007908542747
1.49230313453375
2.08264705403338

29
900
0.24007908542735
1.49230313453373
2.08264705403335

30
961
0.24007908542716
1.49230313453351
2.08264705403311

31
1024
0.24007908542735
1.49230313453364
2.08264705403327

34
1225
0.24007908542735
1.49230313453361
2.08264705403323

In the tables in this section, DOF denotes the degree of freedom (number of nodes).

From Table 1 we can see that

We also use the bilinear finite element to solve (13), and the numerical results are listed in Table 2.

Table 2: The eigenvalues of the bilinear finite element.

DOF

1/8
81
0.24022930645534
1.49577435506528
2.08269976804992

1/16
289
0.24011662732054
1.49317640860702
2.08266035463158

1/32
1089
0.24008847006616
1.49252179566534
2.08265038681122

1/64
4225
0.24008143153493
1.49235782124229
2.08264788770449

1/128
16641
0.24007967195002
1.49231680754965
2.08264726248054

1/150
22801
0.24007951252139
1.49231309100913
2.08264720582238

1/200
40401
0.24007932566556
1.49230873510836
2.08264713941521

1/256
66049
0.24007923205310
1.49230655286905
2.08264710614529

1/300
90601
0.24007919219220
1.49230562369281
2.08264709197866

1/400
160801
0.24007914547718
1.49230453468660
2.08264707537565

1/512
263169
0.24007912210763
1.49230398913452
2.08264706706984

From Table 2 we see that

We use Scheme 13 to solve (13), and let be the th approximate eigenvalue , and the numerical results are presented in Table 3.

Table 3: The eigenvalues of shifted-inverse iterative method.

4
5
0.24007908542792
1.49230316584831
2.08264705403320

4
6
0.24007908542722
1.49230313468790
2.08264705403320

4
7
0.24007908542722
1.49230313453463
2.08264705403320

4
8
0.24007908542723
1.49230313453360
2.08264705403320

From Table 3 we find the eigenvalues obtained by Scheme 13 have the same accuracy with which is computed directly in by LGL spectral method.

Since we do not know the exact eigenvalues of the Steklov eigenvalue problem, many research studies reported the approximate eigenvalues; for example, see [15, 22]. This paper uses the LGL spectral method to solve (13), and in Table 1 we give

We think these approximate eigenvalues have high accuracy.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant no. 11201093).

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