Abstract
Combining the correction technique proposed by Lin and Xie and the shifted inverse iteration, a multilevel correction scheme for the Steklov eigenvalue problem is proposed in this paper. The theoretical analysis and numerical experiments indicate that the scheme proposed in this paper is efficient for both simple and multiple eigenvalues of the Steklov eigenvalue problem.
1. Introduction
Steklov eigenvalue problems have important applications in physics and engineering, for instance, in the study of surface waves (see [1]), in the analysis of stability of mechanical oscillators immersed in a viscous fluid (see [2]), and in the study of the vibration modes of a structure in contact with an incompressible fluid (see [3]). Thus, finite element methods for Steklov eigenvalue problems have attracted the attention of mathematics and physics community. Reference [4] first studied the convergence and error estimation of finite element approximations and [5–8] made in-depth research; after that, [9–15] discussed the highly efficient finite element methods.
Recently, an efficient multilevel method based on the correction step was proposed by Lin and Xie [16, 17] and further successfully applied to Helmholtz transmission eigenvalue problems [18], convection-diffusion eigenvalue problems [19], and the Steklov problem [15]. The multilevel correction method proposed by Lin and Xie can be regarded as a combination of two-grid method and the extended/generalized finite element method which was developed in 1990s (see [20, 21]).
The shifted inverse iteration method is a basic approach for solving matrix eigenvalue problems (see Algorithm 27.3 in [22]). Now, two-grid methods based on the shifted inverse iteration have been established (see [23, 24]) and applied to Steklov eigenvalue problems (see [11]).
In this paper, we combine the correction technique of Lin and Xie and the shifted inverse iteration to establish a new efficient multilevel scheme for the Steklov eigenvalue problem which is suitable not only for simple eigenvalues but also for multiple eigenvalues. Our scheme can be described as follows: (1) solve the Steklov eigenvalue problem in the coarsest finite element space; (2) implement the shifted inverse iteration once in an augmented space by using the previously obtained eigenvalue as the shift and eigenfunction as the iteration initial value; (3) solve the Steklov eigenvalue problem again in a new space which is constructed by combining the coarsest finite element space with the obtained eigenfunction approximation in (2), and then return to (2) for next loop. Furthermore, we prove the scheme can reach the optimal order that the same as solving the corresponding boundary value problem. Our scheme is easy to realize under the package of iFEM [25] with Matlab, and the numerical results are satisfactory.
In this paper, (with or without subscripts) denotes a positive constant independent of mesh diameters and correction times.
2. Preliminaries
Let and denote the usual Sobolev spaces with real-order with norms and on and , respectively. Here .
Consider the following Steklov eigenvalue problem: where is a bounded polygonal domain with the maximum interior angle and is the outward normal derivative on .
The variational form of (1) is given as follows: find and such that and where It is easy to know that is a symmetric, continuous, and -elliptic bilinear form on satisfying So we can use and as the inner product and norm on , respectively. We can also use and as the inner product and norm on , respectively.
From [4, 26] we know that problem (2) has an eigenvalue sequence satisfying and the corresponding eigenfunctions where . The eigenvalues are repeated according to their multiplicity in the sequence .
Let be a regular triangulation of . Denote the diameter of an element by and the mesh diameter . Let the finite element space be a piecewise polynomial space on .
Then the finite element approximation of (2) is given as follows: find and such that and It is well known that (7) has eigenvalues (see, e.g., [4, 26]) and the associated eigenfunctions where , .
Consider the following boundary value problems (10) and (11) associated with (2) and (7), respectively.
Find such that and
Find such that and
The following regularities of the Steklov eigenvalue problem are valid.
Lemma 1. If , there exists a unique solution to (10) satisfying If , there exists a unique solution to (10) satisfying where is a positive constant.
Proof. See (4.10) in [27], Proposition 4.4 in [3], and Lemma 2.1 in [11].
By (13), we know that , , if is convex; otherwise and can be closed to arbitrarily.
Then, from (10) and (11), we can define two linear bounded operators and such that It is obvious that and are the solution of (10) and (11), respectively. We know that and are completely continuous operators (see, e.g., [11]). From [4, 26], we know that (2) and (7) have the following equivalent operator forms: where and . From (15), we can deduce that is bounded; that is, where is a positive constant independent of and .
Denote the eigenfunction space corresponding to by Suppose that the multiplicity of is ; that is, . We use to denote the eigenpair approximation for , . Let and let and .
Denote From Lemma 1 and the interpolation error estimate, we get
For two linear spaces and , we define Define the gaps between and in as and in as
Lemma 2. The following estimates are valid: for . Here and hereafter is a positive constant depending on but independent of .
Proof. See [4] or P. 699 in [26].
We need the following result (cf. [11, 23]) for the shifted inverse iteration method.
Lemma 3. Let be an approximation for , where is not an eigenvalue of , and with . Suppose that , for , and , satisfy Then where is a separation constant of and is a positive constant independent of .
Proof. From (3.2) in [23], (2.25) in [11], and (27), we have The proof is completed.
3. One Correction Step Based on the Shifted Inverse Iteration
We first generate a coarse mesh with the mesh size . The coarse linear finite element space is defined on . Then we define a sequence of triangulations of determined as follows. Set and let be obtained from via regular refinement. Then we construct the linear finite element spaces such that where .
Assume that we have the eigenpair approximations (), where the eigenvalues are the approximations of the eigenvalue of (2) and is an extended space (see Algorithm 4). Now we give the following one correction step to improve the accuracy of the current eigenpair approximations .
Algorithm 4 (one correction step).
Step 1. For , do the following.
Find such that and set .
Step 2. Define a new finite element space and solve the following Steklov eigenvalue problem.
Find such that and Output the eigenvalues and the corresponding orthonormal eigenfunctions with respect to .
We adopt the notations in [15–17] to simplify and summarize Algorithm 4 as
Theorem 5. Suppose that is properly small. Assume the eigenpairs in Algorithm 4 have the following error estimates: Then, after one correction step, the resulting eigenpair approximations have the following error estimates: where is a positive constant dependent on but independent of and , and when is a simple eigenvalue, and when is a multiple eigenvalue is a positive constant independent of and (see Lemma 6 at the back).
Proof. From (35), we know there exists a basis of such that By (15), we know that (32) is equivalent to the following: and ; that is, Select ; then and . Note that differs from by only a constant; then (32) is equivalent to Let be the eigenvalue of (7) with ; then thanks to Lemma 3 we have Then, using the triangle inequality and Lemma 2, we get Combining (46) and (47), we obtain Let the eigenvectors be an orthonormal basis of with respect to . Denote then By (25), there exists satisfying Let Noting that is linearly independent, it can be deduced from (45), (49), and (52) that is linearly independent.
Using (49), (51), and (52), we deduce that Thus, using the triangle inequality, we have which together with (48) yields Noting that and using Lemma 3.1 in [23], we have Since and , from (17), (41), and (42), we have From (56) and (57), we get Combining (55) and (58), reminding that , , we can obtain When is properly small, we have then we obtain the following error estimate: Now we come to estimate the error for the eigenpairs . Based on the error estimate theory of eigenvalue problem by finite element methods (see, e.g., [4, 26]), from Lemmas 2 and 6 at the back, (61), and the definition of the space , we have the following estimates: where From (62) and (63), we obtain (36) and (37).
From (27) and (36), we have Hence, estimate (38) holds.
Lemma 6. For the multiple eigenvalue , assume that and , where , , and is a positive constant. Let be defined as (52); then is a basis of . For any , there holds where is a constant independent of and .
Proof. From the proof of Theorem 5 we know that is linearly independent, so it can be taken as a basis of . For any , , we have Noticing that the norm of self-adjoint operator is equal to its spectral radius, then where .
For in (40)-(41), by calculation, we deduce where Since , there exists such that Noting that , then, for , we have From (45), we have Since , we get Note that then, from Lemma 3.1 in [23], (57), and (68)–(73), we get For , from (76), we have By , from (40), we get thus then, from (61) and (79), we have Thus there exists such that Then, from (67), we have thus, we getthat is, are uniformly bounded from above with respect to
4. Multilevel Correction Scheme for the Steklov Eigenvalue Problem
Based on Algorithm 4, we introduce the following multilevel correction scheme.
Algorithm 7 (multilevel correction scheme).
Step 1. Construct a series of finite element spaces such that (31) holds.
Step 2. Compute eigenpair approximations for the eigenpairs , by solving the following Steklov eigenvalue problem.
Find such that and Step 3. and , for
For , do Finally, we obtain eigenpair approximations .
Theorem 8. After implementing Algorithm 7, if is small enough, the resultant eigenpair approximations have the following error estimates:
Proof. We use induction to prove Theorem 8.
First, for , since , we can obtain (86)–(88) from (84) and Lemma 2.
Suppose that (86)–(88) hold for . Then from Theorem 5 we know that (86)–(88) are valid for .
Corollary 9. Under the conditions of Theorem 8, further assume that is a series of linear or bilinear finite element spaces, and , . Then (86)–(88) are valid and
Remark 10. When , we obtain error estimates (86) and (88) for simple eigenvalues. Comparing (86) and (88) with the estimates (4.6), (4.7) in [15], we know that our algorithm achieves the optimal convergence order and is as efficient as Algorithm 4.1 in [15].
Remark 11. With the iteration times increasing, the approximate eigenvalues are closer to the exact ones which lead the equation (32) more closer to singular, although it is not difficult to solve numerically (see [22]). However, we fix the shift in Algorithms 4 and 7 and get a new scheme. In Section 5, we will report a numerical experiment when the shift is fixed as .
Remark 12. Reference [15, 17] analyzed that the computational costs of their algorithms are almost the same as those of solving the corresponding boundary value problem. Comparing our Algorithm 7 with Algorithm 7.2 in [15], it is not difficult to see that the computational work of these two algorithms are almost the same.
5. Numerical Examples
In this section, we consider the Steklov eigenvalue problem (1) on the unit square and the L-shape domain , respectively. In our numerical experiments, we use algorithms in [15], our Algorithm 7, and Algorithm 7 with fixed shift to solve (1), and the resulting approximations are denoted by , , and , respectively. Our programs are performed under the package of Chen (cf. [25]).
Example 1. We solve (1) on the unit square by three algorithms with the linear conforming element. Since the exact eigenvalues are unknown, we use and obtained by the spectral element method (see [28]) as the reference eigenvalues. We depict the error curves of three algorithms in Figure 1.
(a)
(b)
Example 2. We solve (1) on the L-shape domain by three algorithms with the linear conforming element. Since the exact eigenvalues are unknown, we use and obtained by the spectral element method (see [28]) as the reference values. The error curves of three algorithms are depicted in Figure 2.
(a)
(b)
From Figures 1 and 2 we can see that our algorithm achieves the optimal convergence order and is as efficient as that in [15].
6. Concluding Remark
In this paper, we combined the correction technique proposed by Lin and Xie and the shifted inverse iteration to establish a new efficient multilevel correction scheme for the Steklov eigenvalue problem which is suitable not only for simple eigenvalues but also for multiple eigenvalues. The algorithm and analysis in this paper are suitable for the Steklov eigenvalue problem in three dimensional domains and also suitable for general second-order self-adjoint elliptic eigenvalue problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant no. 11201093). The authors cordially thank the editor and the reviewers for their valuable comments and suggestions and thank Jiayu Han for his help which lead to the improvement of the paper.