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

ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.


Introduction
Due to the wide applications in physical and mechanical field (see, e.g., [1][2][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.

Lemma 3.
Let  and  ℎ be the th eigenvalue of ( 2) and ( 6), respectively.Then for any eigenfunction  ℎ corresponding to  ℎ with ‖ ℎ ‖  = 1, there exist  ∈ () and for any  ∈ M(), there exists where constants  2 and  3 are positive and only depend on .

Advances in Mathematical Physics
Then Let  0 be a positive constant satisfying the following inequalities: where  0 is an approximate eigenvalue of   , is an approximate eigenfunction obtained by Scheme 4, and  is the separation constant of the eigenvalue   = 1/  .
Let the eigenvectors { ,ℎ  } +−1  be an orthonormal basis of  ℎ  (  ) with respect to (⋅, ⋅), and denote From Lemma 3, we know that there exist eigenvectors 18), (19), and (20).Let and then   ∈ (  ) and To estimate the error, we split where  0 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  ∈ M(  ), where Select  0 = 1/ 0 and  0 = Then, by ( 15) and ( 13) we have and then Condition (1) 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  1 ,  2 ,  3 ,  4 ,  5 , 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  0 that is independent of mesh parameters and  such that (44) holds.And we can have min{ 0 /4,  0  1 /4} > Proof.The estimates (51) and ( 52) can be obtained by the proof arguments in [16].
From Theorems 7 and 8 we get Therefore, for , from (59) we derive that which together with (57) we get (54) immediately.

A Posteriori Error Estimates of the Inverse Iteration with Fixed-Shift
Based on the work of [4,12,[17][18][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  ∈  1 (Ω) such that and its finite element approximation states: find  ℎ ∈  ℎ such that For any element  ∈  ℎ with diameter ℎ  , we denote by E  the set of edges, and We decompose E = E Ω ∪ E Γ , where E Ω and E Γ refer to interior edges and edges on the boundary Γ = Ω, respectively.For each ℓ ∈ E Ω , we choose an arbitrary unit normal vector  ℓ and denote the two triangles sharing this edge by  in and  out , where  ℓ points outwards  in .For V ℎ ∈  ℎ we set For each ℓ ∈ E we define the jump residual: Now, the local error indicator is defined as Proof.Note that   (  ) ≤   () → 0( → 0); then there exists a proper small  0 > 0 such that if  ≤  0 , 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 3.4 in [12]).

Lemma 14. Suppose that the conditions of
The proof is completed.
It is obvious that  1 is a higher order term.Hence, we obtain that  Ω ( Theorem 16.Under the conditions of Theorem 10, there exists   ∈ (  ) such that the following hold: (a) For  ∈  ℎ  , if  ∩ Γ = 0, then where where Proof.We can prove the desired results by using the proof method of Theorem 3.4 in [12].
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).

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.
Step 6. Refine  ℎ  to get a new mesh  ℎ +1 by Procedure REFINE.
Marking Strategy E. Give parameter 0 <  < 1.  Step 1. Construct a minimal subset πℎ  of  ℎ  by selecting some elements in  ℎ  such that Step  replaced by  ℎ  and  ℎ  , respectively.Note that when | 0 − | 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 0 ≥ 0 to establish the following adaptive algorithm.Algorithm 2. Choose parameter 0 <  < 1.
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  0 = 0. We use MATLAB 2012 together with the package of Chen [20] to solve Examples 1, 2, and 3, and we take  = 0.5.For reading conveniently, we use the following notations in our tables:  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 Ω = [0, 1] × [0, 1].The numerical results are listed in Table 1.
Since the exact eigenvalues are unknown, we use  1 ≈ 0.24007908542 and  2 ≈ 1.49230313453 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  1 and  2 in Figure 1.It can be seen from Figure 1 that the error curves are approximately parallel to the line with slope −1, which indicates that Algorithm 2 achieves the optimal convergence rate of O( −1  ) 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.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 Ω = ([0, 1] × [0, 1/2]) ∪ ([0, 1/2] × [1/2, 1]).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  1 and  3 .Here we use  1 ≈ 0.18296423687 and  3 ≈ 1.68860048358 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 −1, which indicates that Algorithm 2 achieves the optimal convergence rate of O( −1  ) 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.
Since the exact eigenvalues are unknown, we compute the approximations of two exact eigenvalues of (1):  1 ≈ 0.23957338768 and  5 ≈ 1.41238071918 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  1 and  5 in Figure 3.We can see from Figure 3 that the error curves are approximately parallel to the line with slope −1, which indicates that Algorithm 2 achieves the optimal convergence rate of O( −1  ) 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.
Example 4. We use the method in [10]   (3)  , .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  2 on [0, 1] × [0, 1], comparing with the one in [10].

Number of degrees of freedom Error The a posteriori estimator of Algorithm 1 Figure 1 :
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.

Figure 2 :
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.

Figure 3 :
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.

Table 2 :
The 1st and the 3rd eigenvalues of Example 2 obtained by two algorithms with  = √ 2/32.