Advances in Mathematical Physics
Volume 2016 (2016), Article ID 4691759, 13 pages
http://dx.doi.org/10.1155/2016/4691759

Research Article

A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem

Feiyan Li and Hai Bi

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China

Received 17 March 2016; Revised 9 May 2016; Accepted 10 May 2016

Academic Editor: Luigi C. Berselli

Copyright © 2016 Feiyan Li and Hai Bi. 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.

For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.

1. Introduction

Due to the wide applications in physical and mechanical field (see, e.g., [1–3]), there has been a lot of research on the numerical methods for Steklov eigenvalue problems; for instance, [4] studied the conforming linear finite element approximation, [5, 6] studied the nonconforming finite elements approximation, [7, 8] discussed a two-grid method of the conforming and nonconforming finite element method based on the inverse iteration, respectively, [9] studied multiscale asymptotic method, [10] studied multilevel method, [11] studied the spectral method, and [12] studied an adaptive algorithm based on the shifted inverse iteration.

In this paper we establish a type of multigrid discretizations based on the fixed-shift inverse iteration for the Steklov eigenvalue problem. The multilevel method in [10] made use of the inverse iteration and the extended finite element method. Compared with [10], our method has less computational complexity since we have no correction step in each iteration. On the other hand, compared with [12], we adopt the fixed-shift and thus avoid selecting appropriate shift to ensure the efficiency of shifted inverse iteration; meanwhile, we also do not face the difficulty of solving an almost singular algebraic system in the shifted inverse iteration.

We analyze elaborately the a priori and the a posteriori error estimates of the method proposed in this paper. Then, based on the a posteriori error estimates we design an adaptive algorithm of fixed-shift inverse iteration type. Moreover, we also compare the performance of three types of multigrid methods. Numerical results illustrate that our method is also an efficient method for solving the Steklov eigenvalue problem.

The rest of this paper is organized as follows. In the subsequent section, some preliminaries needed in this paper are presented. In Section 3, a scheme of the inverse iteration with fixed-shift based on multigrid discretizations is established, and the a priori error estimates are also given. The a posteriori error estimates of the inverse iteration with fixed-shift are analyzed in Section 4. Numerical experiments are presented in the final section.

In this paper, with or without subscript denotes a constant independent of mesh size and iterative times.

2. Preliminaries

Consider the Steklov eigenvalue problem

where is a polygonal domain with being the largest inner angle of and is the outward normal derivative.

We denote the real order Sobolev spaces with norm and by and , respectively; .

The variational form of (1) is given by the following: find and , , such that

where

As we know, is a symmetric, continuous, and -elliptic bilinear form on . Thus, we use and as the inner product and norm on , respectively.

Let be the dual space of with norm given by

where is the dual product on . When , .

Let be a family of regular triangulations of with the mesh diameter , and let be a space of piecewise polynomials defined on . For any , the following conclusion holds:

The conforming finite element approximation of (2) is the following: find and , , such that

Define the operators and satisfying

Define the Ritz projection by

From [13], we know that ; (2) and (6) have the equivalent operator forms and , respectively, where , , and .

Suppose that and are the th eigenvalue of (2) and (6), respectively, and the algebraic multiplicity of is equal to , . Let be the space spanned by all eigenfunctions corresponding to and let be the direct sum of eigenspaces corresponding to all eigenvalues of (6) that converge to . Let .

Denote

It is obvious that . It follows from Lemma in [14] that

By using the trace theorem we have

Moreover, if and we know that and, consequently,

For any , . Taking in (8) we deduce

and thus we get

The following lemmas are needed in our analysis.

Lemma 1. Let be an eigenpair of (2); then for any with , the Rayleigh quotient satisfies

Proof. See page 699 of [13].

Lemma 2. For any nonzero ,

Proof. See [15].

Lemma 3. Let and be the th eigenvalue of (2) and (6), respectively. Then for any eigenfunction corresponding to with , there exist and such that if ,

for any , there exists such that if ,

where constants and are positive and only depend on .

Proof. See page 699 of [13] and Lemma and () of [14].

If , , , and , then by Lemma 2 we have

From (2) we have ; then

Hence, from Lemma 1 we get

Denote

and then when and , (24) becomes

Since (6) implies , then combining (26) and (18) we deduce that

3. A Priori Error Estimates of the Inverse Iteration with Fixed-Shift

Let be a family of conforming finite element spaces that satisfy , (), and . Referring to [16], we establish the following scheme of the inverse iteration with fixed-shift based on multigrid discretizations.

Scheme 4 (the inverse iteration with fixed-shift based on multigrid discretizations). Given the iterative times and . Execute the following.

Step 1. Solve (2) on : find such that and

Step 2. Let , , .

Step 3. Solve a linear system on : find such that

set .

Step 4. Compute the Rayleigh quotient

Step 5. If , then , ; turn to Step ; else, , and return to Step .

Step 6. Solve a linear system on : find such that

set .

Step 7. Compute the Rayleigh quotient

Step 8. If , then output and stop; else, , and return to Step .

Let be the th eigenpair of (28); then derived from Scheme 4 is the th eigenpair approximation of (2).

In the following analysis, we also denote and .

Now, we will analyze the a priori error estimates of Scheme 4.

Denote .

Our analysis makes use of the following lemma (see Lemma in [16]) for the shifted inverse iteration method. Let and denote the th eigenpair of (2) and (6), respectively, and , , , and .

Lemma 5. Let be an approximation for , where is not an eigenvalue of , and with . Suppose that

; and , for , where is the separation constant of the eigenvalue ; satisfy

Then

Let be a positive constant satisfying the following inequalities:

Condition 6. There exists for such that

where is an approximate eigenvalue of , is an approximate eigenfunction obtained by Scheme 4, and is the separation constant of the eigenvalue .

Let the eigenvectors be an orthonormal basis of with respect to , and denote

From Lemma 3, we know that there exist eigenvectors making satisfy (18), (19), and (20). Let

and then and

To estimate the error, we split

Now, we will analyze the first term .

Theorem 7. Let be an approximate eigenpair obtained by Scheme 4 with . Assume that Lemma 3 and Condition 6 hold; then

where is independent of mesh parameters and .

Proof. We use Lemma 5 to complete the proof. First, we will verify that the conditions of Lemma 5 are satisfied.
From Lemma 3, we know that, for any given , there exists such that

where

Select and . Then, by (15) and (13) we have

noting that ; then using Lemma 2, (46), the Cauchy-Schwartz inequality, (26), Condition 6, and (36) we obtain

and then Condition in Lemma 5 holds.
By using the same arguments in [16], it is clear that the other two conditions in Lemma 5 are valid.
Hence, we see that the conditions of Lemma 5 hold.
Then, by the same proof method in [16], we derive that

Noting that the constants , , , , , and are independent of mesh parameters and and Condition 6 holds, then based on the above inequality we conclude that there exists a positive constant that is independent of mesh parameters and such that (44) holds. And we can have . The proof is completed.

Next, we will analyze the error .

Theorem 8. The error satisfies

Proof. The estimates (51) and (52) can be obtained by the proof arguments in [16].

Based on the above two theorems, we now analyze the a priori error estimates of Scheme 4.

Condition 9. For any given , there exist and such that and , respectively, .

In the practice Condition 9 is not a restrictive condition. For example, let be obtained from via regular refinement (producing 4 congruent elements) such that ; then, when and we have and (see [10]), where if is convex and if is concave.

Theorem 10. Let be an approximate eigenpair obtained by Scheme 4. Suppose that Condition 9 holds; then there exist and such that if it is valid that

Proof. We only prove the result (54) since (53) and (55) can be proved analogously by referring to [16].
The proof is completed by using induction, Theorems 7 and 8. Note that ; then there exists a proper small such that if , Lemma 3 and the following inequalities hold:

where .
When , it is easy to know that (53)–(55) are valid (see [12, 16]). Suppose that Theorem 10 holds for ; that is, there exists such that

Then we infer from (56) that the conditions of Theorem 7 hold.
From Theorems 7 and 8 we get

Therefore, for , from (59) we derive that

which together with (57) we get (54) immediately.

4. A Posteriori Error Estimates of the Inverse Iteration with Fixed-Shift

Based on the work of [4, 12, 17–19], in this section, we will discuss the a posteriori error estimates of Scheme 4 for the Steklov eigenvalue problem.

Consider the boundary value problem corresponding to (2): find such that

and its finite element approximation states: find such that

For any element with diameter , we denote by the set of edges, and

We decompose , where and refer to interior edges and edges on the boundary , respectively. For each , we choose an arbitrary unit normal vector and denote the two triangles sharing this edge by and , where   points outwards .

For we set

Let

For each we define the jump residual:

Now, the local error indicator is defined as

and then the global error estimator is given by

Substituting for , we can get the definitions of , , and similarly.

Now, we will estimate the error .

From [4, 12], we give the following two lemmas among which Lemma 11 provides the global upper bound of , while Lemma 12 provides the local lower bound of .

Lemma 11. The error satisfies

Lemma 12. The error satisfies the following:
(a) For , if , then

where denotes the union of and the triangles sharing an edge with .
(b) For , if , then

Next, we will analyze the error .

Theorem 13. Suppose that the conditions of Theorem 10 are satisfied; then

Proof. Note that ; then there exists a proper small such that if , the following inequality holds:

From (44), Theorem 10, and Condition 9, we have

which together with (73) yields (72) immediately.

We give the following lemma by referring to [12] (see Lemma in [12]).

Lemma 14. Suppose that the conditions of Theorem 10 are satisfied; then

In the following discussion, combining Lemmas 11, 12, and 14 and Theorem 13, we give the global upper bound and the local lower bound of the error.

Theorem 15. Suppose that the conditions of Theorem 10 are satisfied; then there exists such that

where .

Proof. Select which is given by (41); then from Lemma 11, Theorem 13, and (76) we get

The proof is completed.

It is obvious that is a higher order term. Hence, we obtain that is a global reliable error indicator of .

Theorem 16. Under the conditions of Theorem 10, there exists such that the following hold:
(a) For , if , then

where .
(b) For , if , then

where

Proof. We can prove the desired results by using the proof method of Theorem in [12].

According to Remark in [4] and Remark in [12] we know that the term is a higher order term. From Theorem 13, we know that and are also higher order terms. And it is obvious that is a higher order term. Therefore, from (79) and (80) we know that is an efficient local error indicator of and .

In the following theorem, we give the estimate for approximate eigenvalue.

Theorem 17. Suppose that the conditions of Theorem 10 are satisfied; then

Proof. From Theorem 10 and Lemma 1 it is easy to prove that

then combining with Theorem 16 and (77), we can get the desired result (82).

5. Numerical Experiments

In this section we first give an adaptive algorithm of the Rayleigh quotient iteration type and establish an adaptive algorithm of fixed-shift inverse iteration type for the Steklov eigenvalue problem.

The following Algorithm 1 of the Rayleigh quotient iteration type refers to Algorithm in [12] or Algorithm in [16].

Algorithm 1. Choose parameter .

Step 1. Pick any initial mesh .

Step 2. Solve (2) on for discrete solution .

Step 3. Let .

Step 4. Compute the local indicators .

Step 5. Construct by Marking Strategy E and .

Step 6. Refine to get a new mesh by Procedure REFINE.

Step 7. Find such that

denote and compute the Rayleigh quotient

Step 8. Let ,   and go to Step .
Marking Strategy E. Give parameter .

Step 1. Construct a minimal subset of by selecting some elements in such that

Step 2. Mark all the elements in .
and are defined by (67) and (68) with and replaced by and , respectively.
Note that when is too small, (84) is an almost singular linear equation. Although it has no difficulty in solving (84) numerically (see [12]), one would like to think of selecting a proper integer to establish the following adaptive algorithm.

Algorithm 2. Choose parameter .
Steps  1–7. Execute Steps  1–7 of Algorithm 1.
Step  8. If , go to Step ; else , go to Step .

Marking Strategy E in Algorithm 2 is the same as that in Algorithm 1.

Now, we will implement some numerical experiments to validate our theoretical analysis and show the efficiency of Algorithm 2 with . We use MATLAB 2012 together with the package of Chen [20] to solve Examples 1, 2, and 3, and we take .

For reading conveniently, we use the following notations in our tables:

  ⁡ : the th eigenvalue derived from the th iteration obtained by Algorithm ⁡ (). ⁡ : the error of obtained by Algorithm (). ⁡ : the degrees of freedom of the th iteration for (). ⁡ : the CPU time(s) from the program starting to calculate result of the th iteration appearing by using Algorithm ().

Example 1. We use Algorithms 1 and 2 to compute the approximations of the 1st and the 2nd eigenvalue of (1) with the triangle linear finite element on . The numerical results are listed in Table 1.
Since the exact eigenvalues are unknown, we use and obtained by the spectral element method  (see [21]) as the reference eigenvalues. We show the error curves and the a posteriori estimators obtained by two algorithms for and in Figure 1. It can be seen from Figure 1 that the error curves are approximately parallel to the line with slope , which indicates that Algorithm 2 achieves the optimal convergence rate of as well as Algorithm 1.
Observing the numerical results in Table 1, we can find that when the degrees of freedom are almost the same, the approximate eigenvalues obtained by Algorithm 2 are nearly as accurate as those obtained by Algorithm 1 and their CPU time are roughly the same.

Table 1: The 1st and the 2nd eigenvalues of Example 1 obtained by Algorithms 1 and 2 with .

1
5
5700
0.24008040
0.54
5700
0.24008040
0.57

1
10
27344
0.24007936
1.60
27343
0.24007936
1.66

1
15
117001
0.24007915
7.67
117001
0.24007915
7.82

1
19
408971
0.24007910
29.06
408971
0.24007910
29.25

1
20
509032
0.24007910
39.88
509032
0.24007910
40.07

1
21
764069
0.24007909
55.90
764069
0.24007909
55.96

2
5
4526
1.49245505
0.51
4529
1.49245448
0.54

2
12
37855
1.49231870
2.31
37941
1.49231871
2.33

2
18
223337
1.49230601
15.77
224277
1.49230600
15.89

2
24
1329617
1.49230362
109.68
1334272
1.49230362
109.90

2
25
1805637
1.49230347
151.36
1812726
1.49230347
151.02

2
26
2439573
1.49230337
211.29
2447212
1.49230337
210.12

(a)

(b)

Figure 1: The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 2nd (b) eigenvalues on the square domain.

Example 2. We use Algorithms 1 and 2 to compute the approximations of the 1st and the 3rd eigenvalue of (1) with the triangle linear finite element on . The numerical results are presented in Table 2.
In Figure 2 we depict the error curves and the a posteriori estimators obtained by two algorithms for and . Here we use and obtained by the spectral element method (see [21]) as the reference eigenvalues. It can be seen from Figure 2 that the error curves are approximately parallel to the line with slope , which indicates that Algorithm 2 achieves the optimal convergence rate of as well as Algorithm 1.
It also can be seen from Table 2 that when the degrees of freedom are the same, one can use Algorithms 1 and 2 to get the same accurate approximations with nearly the same CPU time.

Table 2: The 1st and the 3rd eigenvalues of Example 2 obtained by two algorithms with .

1
5
4369
0.18296573
0.46
4369
0.18296573
0.45

1
12
35119
0.18296446
2.00
35119
0.18296446
1.97

1
18
227719
0.18296426
14.90
227719
0.18296426
14.80

1
21
557370
0.18296425
39.77
557370
0.18296425
40.10

1
22
738108
0.18296425
55.04
738108
0.18296425
55.43

1
23
1105016
0.18296424
78.27
1105016
0.18296424
78.83

3
5
3690
1.68889444
0.49
3690
1.68889444
0.52

3
12
31089
1.68863207
1.79
31089
1.68863207
1.84

3
18
183089
1.68860633
11.84
183089
1.68860633
12.08

3
24
1098418
1.68860134
81.42
1098418
1.68860134
83.23

3
25
1485695
1.68860109
112.52
1485695
1.68860109
114.26

3
26
2002315
1.68860097
155.64
2002315
1.68860097
156.76

(a)

(b)

Figure 2: The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 3rd (b) eigenvalues on the -shaped domain.

Example 3. We use Algorithms 1 and 2 to compute the approximations of the 1st and the 5th eigenvalue of (1) with the triangle linear finite element on . The numerical results are presented in Table 3.
Since the exact eigenvalues are unknown, we compute the approximations of two exact eigenvalues of (1): and by the standard adaptive algorithm (see, e.g., [22]) with the degrees of freedom of more than 5000000. We show the curves of the error and the a posteriori estimators obtained by two algorithms for and in Figure 3. We can see from Figure 3 that the error curves are approximately parallel to the line with slope , which indicates that Algorithm 2 achieves the optimal convergence rate of as well as Algorithm 1.
From the numerical results in Table 3, we can conclude that Algorithm 2 is also an efficient approach like Algorithm 1 for solving the Steklov eigenvalue problem.

Table 3: The 1st and 5th eigenvalues of Example 3 obtained by two algorithms with .

1
5
9391
0.23957586
0.66
9391
0.23957586
0.75

1
10
42645
0.23957397
2.45
42645
0.23957397
2.56

1
15
189550
0.23957352
12.52
189550
0.23957352
12.72

1
19
634556
0.23957342
48.00
634556
0.23957342
47.09

1
20
860490
0.23957341
66.32
860490
0.23957341
65.38

1
21
1131274
0.23957340
91.04
1131274
0.23957340
90.05

5
5
9842
1.41254843
0.70
9842
1.41254843
0.74

5
12
77005
1.41241115
4.53
77005
1.41241115
4.68

5
18
453543
1.41238410
31.53
453543
1.41238410
31.74

5
21
1092642
1.41238245
82.23
1092642
1.41238245
82.74

5
22
1497488
1.41238165
112.92
1497488
1.41238165
113.82

5
23
1993327
1.41238104
155.36
1993327
1.41238104
156.14

(a)

(b)

Figure 3: The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 5th (b) eigenvalues on slit domain.

Example 4. We use the method in [10] (see Algorithms 4.1 and 7.2 there) to compute the numerical eigenvalues of (1) on ,  , and , respectively, and list the associated results in Tables 4–6 which are denoted by and .
From Tables 4–6 we can see that, with the same degrees of freedom , our method uses less CPU time to obtain the same accurate approximations, especially for multiple eigenvalue on , comparing with the one in [10].

Table 4: The results of Example 4 on .

1
1
1089
0.240088481
0.08
0.240088481
0.18

1
2
4225
0.240081438
0.17
0.240081438
0.22

1
3
16641
0.240079674
0.48
0.240079674
0.43

1
4
66049
0.240079233
1.92
0.240079233
1.45

1
5
263169
0.240079122
8.92
0.240079122
6.48

1
6
1050625
0.240079095
38.36
0.240079095
32.00

2
1
1089
1.492905398
0.11
1.492905378
0.19

2
2
4225
1.492454269
0.23
1.492454267
0.24

2
3
16641
1.492340958
0.65
1.492340958
0.46

2
4
66049
1.492312593
2.67
1.492312593
1.47

2
5
263169
1.492305499
12.19
1.492305499
6.57

2
6
1050625
1.492303726
56.05
1.492303726
32.00

Table 5: The results of Example 4 on .

1
1
833
0.182975157
0.04
0.182975157
0.19

1
2
3201
0.182966980
0.11
0.182966980
0.22

1
3
12545
0.182964924
0.31
0.182964924
0.36

1
4
49665
0.182964409
1.25
0.182964409
1.06

1
5
197633
0.182964280
5.88
0.182964280
4.60

1
6
788481
0.182964248
26.02
0.182964248
21.61

3
1
833
1.690165085
0.05
1.690165013
0.20

3
2
3201
1.688996545
0.12
1.688996536
0.22

3
3
12545
1.688700132
0.34
1.688700131
0.36

3
4
49665
1.688625481
1.27
1.688625481
1.04

3
5
197633
1.688606742
5.95
1.688606742
4.34

3
6
788481
1.688602046
26.11
1.688602046
20.20

Table 6: The results of Example 4 on ∖.

1
1
2145
0.239589697
0.12
0.239589697
0.22

1
2
8385
0.239577621
0.28
0.239577621
0.31

1
3
33153
0.239574482
0.96
0.239574482
0.77

1
4
131841
0.239573668
4.03
0.239573668
2.99

1
5
525825
0.239573457
17.43
0.239573457
13.80

1
6
2100225
0.239573403
78.31
0.239573403
71.35

5
1
2145
1.413086665
0.12
1.413086553
0.20

5
2
8385
1.412557485
0.28
1.412557472
0.29

5
3
33153
1.412424609
0.91
1.412424608
0.68

5
4
131841
1.412391332
3.92
1.412391332
2.77

5
5
525825
1.412383004
17.35
1.412383004
12.79

5
6
2100225
1.412380921
79.20
1.412380921
64.99

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11201093) and the Science and Technology Foundation of Guizhou Province of China (LKS[]06).

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