Research Article | Open Access
A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem
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.
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,  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,  studied multiscale asymptotic method,  studied multilevel method,  studied the spectral method, and  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  made use of the inverse iteration and the extended finite element method. Compared with , our method has less computational complexity since we have no correction step in each iteration. On the other hand, compared with , 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.
Consider the Steklov eigenvalue problemwhere 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 thatwhere
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
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 .
It is obvious that . It follows from Lemma in  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 .
Lemma 2. For any nonzero ,
Proof. See .
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 .
Hence, from Lemma 1 we get
and then when and , (24) becomes
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 , 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 thatset .
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 thatset .
Step 7. Compute the Rayleigh quotient
Step 8. If , then output and stop; else, , and return to Step .
In the following analysis, we also denote and .
Now, we will analyze the a priori error estimates of Scheme 4.
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 ; satisfyThen
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 .
To estimate the error, we split
Now, we will analyze the first term .
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 thatwhereSelect 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 obtainand then Condition in Lemma 5 holds.
By using the same arguments in , 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 , 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
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 ), where if is convex and if is concave.
Proof. We only prove the result (54) since (53) and (55) can be proved analogously by referring to .
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 getTherefore, 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
Consider the boundary value problem corresponding to (2): find such thatand 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 asand then the global error estimator is given bySubstituting for , we can get the definitions of , , and similarly.
Now, we will estimate the error .
Lemma 11. The error satisfies
Lemma 12. The error satisfies the following:
(a) For , if , thenwhere 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
Lemma 14. Suppose that the conditions of Theorem 10 are satisfied; then
Theorem 15. Suppose that the conditions of Theorem 10 are satisfied; then there exists such thatwhere .
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 , thenwhere .
(b) For , if , thenwhere
Proof. We can prove the desired results by using the proof method of Theorem in .
According to Remark in  and Remark in  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
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.
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 thatdenote 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