Abstract and Applied Analysis

Volume 2012 (2012), Article ID 190768, 25 pages

doi:10.1155/2012/190768

Research Article

Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field

Yidu Yang, Yu Zhang, and Hai Bi

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

Received 9 July 2012; Revised 4 September 2012; Accepted 12 September 2012

Academic Editor: Xinan Hao

Copyright © 2012 Yidu Yang 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 highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

1. Introduction

The finite element method for eigenvalue problem of electric field has become a hot topic in the field of mathematics and physics (see, e.g., [1–7]). This paper discusses high efficient mixed finite element calculation schemes for the eigenvalue problem of electric field.

Kikuchi [6] introduced the first type of mixed variational formulation for the eigenvalue problem of electric field. Based on this formulation, in [3] Buffa et al.analyzed the approximation of nodal finite element. Boffi et al. [1] discussed the second type of mixed variational formulation for the eigenvalue problem of electric field and analyzed approximations of edge element and nodal element. Yang et al. [7] studied a two-grid discretization scheme of finite element for the first type of mixed variational formulation.

Based on the work mentioned above, in this paper a new multi-grid discretization scheme and an adaptive algorithm are proposed for the first type of mixed variational formulation of eigenvalue problem and applied to the eigenvalue problem of electric field. The main features of the research in this paper are as follows.

(1)Our multi-grid discretization scheme and adaptive algorithm, which are the extension of conforming finite element multi-grid discretization scheme (see scheme  3 in [8] and scheme  1 in [9]), are a combination of the mixed finite element method and the Rayleigh quotient iteration method (see the algorithm  27.3 in [10]). With our algorithm one solves an eigenvalue problem on a coarse grid just at the first step and always solves a linear algebraic system on finer and finer grids at each following step. We derive the error estimates for the algorithm and prove that the constants appeared in the error estimates are independent of the iteration degrees. Thus we prove the convergence of iterations.(2)The eigenvalue problem of electric field is so complicated that it is very difficult to obtain local a posteriori error estimates for the eigenfunctions of mixed finite element. As yet, there is no achievement reported in this field. Our adaptive algorithm substitutes the weight method established by Costabel and Dauge (see [3, 11]) for local refinement, which uses as a posteriori error estimator of instead of estimating local a posteriori error for the eigenfunction. And the results are satisfying. (3)We analyze the mixed finite element error for the eigenvalue problem of electric field (see Theorem 2.2 and Theorem 4.2). We refer to [12] to propose a new proof method which differs from the usual one in [13].

The rest of this paper is organized as follows. Some preliminaries of finite element approximations for eigenvalue problems which are needed in this paper are provided in the next section. In Section 3, for the first type of mixed variational formulation of eigenvalue problem, the finite element multi-grid discretization scheme and the adaptive algorithm are established and the validity of these schemes is proved theoretically. In Section 4, the multi-grid discretization scheme is applied to the eigenvalue problem of electric field. Finally, numerical experiments are presented in Section 5.

2. Preliminaries

Let , , and be three real Hilbert spaces with inner products and norms , , , , , and , respectively. Suppose that (continuously imbedded), is a symmetric, continuous, and -elliptic bilinear form on , that is,

is a continuous bilinear form on , that is,

It is obvious that is an inner product on and is an equivalent norm to .

In scientific and engineering computations, many eigenvalue problems have the following first type of mixed variational formulation: find , , such that

In order to solve problem (2.3)-(2.4), one should construct finite element spaces and . Restricting (2.3)-(2.4) on , we get the conforming mixed finite element approximation as follows: find , such that

Consider the associated source and approximate source problems. Given , find satisfying

Given , find satisfying

Note that the source term is independent of the solution.

As for the mixed finite element method for boundary value problems, Brezzi and Fortinand so forth have established a systematic theory (see [14]). By Brezzi-Babuska Theorem, we have the following.

Lemma 2.1 (Brezzi-Babuska). Suppose that

(C1) (2.1)-(2.2) hold;(C2) inf-sup condition holds, namely, there exists a constant , such that

 then there exists a unique solution to the problem (2.7) and

 where just depends on , and . Moreover, suppose; (C3) discrete inf-sup condition holds, namely, there exists a constant independent of , such that

 then there exists a unique solution to the problem (2.8) and the following error estimate is valid:

 where just depends on and .

Suppose conditions (C1)–(C3) hold in Lemma 2.1. Then there exist unique solutions to the problem (2.7) and (2.8), respectively. Thus, we can define linear bounded operators as follows: ,

,

Obviously, (2.3)-(2.4) has the following equivalent operator form

and (2.5)-(2.6) has the following equivalent operator form

It is easy to verify that are self-adjoint operators in the sense of inner product (see [7]).

Assume (compactly embedded), then it’s easy to prove that is completely continuous, is completely continuous, and is a finite rank operator. Combining (2.3)-(2.4), (2.5)-(2.6), and the -ellipticity of , we deduce

Then from the spectral theory of self-adjoint and completely continuous operator, we know that the eigenvalues of (2.3)-(2.4) can be sorted as

and the corresponding eigenfunctions are

where .

The eigenvalues of (2.5)-(2.6) can be sorted as

and the corresponding eigenfunctions are

where , .

From (2.5), by taking , , we have

and from (2.6) we see that , then

thus is a completely normal eigenvector system on in the sense of the inner product .

Denote , . In this paper, and , and are all called eigenvalues.

Let be the th eigenvalue with algebraic multiplicity , . is the space spanned by all eigenfunctions corresponding to of . is the space spanned by all eigenfunctions corresponding to all eigenvalues of that converge to . Let , . We call as the th eigenvalue, too. Denote , , and . Define

The convergence and error estimates about the mixed element method of eigenvalue problem have been studied in [7, 12, 13, 15, 16]. From [12], we know that the following results are valid.

Theorem 2.2. Suppose , is symmetric, and the conditions of Lemma 2.1 hold; moreover, for any ,

Then .

Proof. From , we derive that is a completely continuous operator. It is obvious that is a finite rank operator. From (2.12), (2.26), and (2.27), we deduce

It shows that pointwisely converges to . From (2.10) and (2.12) we derive that both and are linear bounded. Hence, from Banach-Steinhaus Theorem, we know that there exists a positive constant independent of , such that

Thus, is a bounded set in with respect to the unit ball of . From , we know that is a relatively compact set in , which proves that is collectively compact. From (2.28), we know that , and pointwisely converge to . From [7], are self-adjoint operators in the sense of inner product . Then by Lemma 3.7 or Table  3.1 in [17], we get . The proof is completed.

Lemma 2.3. Suppose that the conditions of Theorem 2.2 hold. Let be the th eigenpair of (2.5)-(2.6) with , let be the th eigenvalue of (2.3)-(2.4). Then , and there exists an eigenfunction corresponding to , such that

let , then there exists such that

where depends on in general, and , , and are constants independent of .

Proof. By Theorem 2.2, we know . Thus from Theorem  2.2 in [7], we see that the desired results are valid. The proof is completed.

For , , define the Rayleigh quotient

Lemma 2.4. Suppose is an eigenpair of (2.3)-(2.4), then for any , , the Rayleigh quotient satisfies

Proof. The proof is completed by using the same argument as that of Lemma 9.1 (see [7, 18]).

Since (continuously imbedded), there exists a constant independent of such that

Taking in (2.35) and using (2.4) and (2.6), we deduce the following.

Lemma 2.5. Suppose is an approximation of   and , then

3. Mixed Finite Element Multigrid Discretization Scheme Based on the Rayleigh Quotient Iteration

In this section, we develop the work in [8], noticing that in [8], when and sholud be modified to their absolute values, respectively, and establish the following mixed finite element multi-grid discretization scheme based on the Rayleigh quotient iteration, and give a rigorous theoretical analysis. Suppose the partition satisfies the following conditions.

Condition 1. is a family of regular meshes (see [19]) with the mesh diameter and , is arbitrarily chosen, , and .

Let and be finite element spaces on .

Scheme 1. Multigrid Discretization.

Step 1. Solve the eigenvalue problem (2.3)-(2.4) on : find , such that

Step 2. , , .

Step 3. Solve an equation on : find such that

Set , .

Step 4. Compute the Rayleigh quotient

Step 5. If , then output , stop; else, , and return to Step 3.

Let be the th eigenpair, and we use as the th approximation eigenpair of (2.3)-(2.4).

Next, we will discuss the validity of Scheme 1.

Lemma 3.1. For any nonzero ,

Proof. See [8].

Denote .

Consider the eigenvalue problem (2.16) on .

Lemma 3.2. Suppose that and are the th eigenvalue of and , respectively, and is an approximate eigenpair, where is not an eigenvalue of , , , , , and satisfy

Then

where is the separation constant of the eigenvalue .

Proof. See [8]

Since the convergence rate of and approximating eigenfunctions is lower than that of and approximating eigenfunctions, respectively, the approximation order of is lower than that of . However, in general, the accuracy order of will not exceed that of ; therefore in the following Theorem 3.3 we assume that the accuracy order of is lower than that of .

Theorem 3.3. Suppose that , is small properly, and Condition 1 holds. Let be the approximate eigenpair obtained by Scheme 1, and let approximate , approximate , and the accuracy order of be lower than that of . Then there exists such that

where and are determined, respectively, by (3.10) and (3.14) in the following proof.

Proof. Let , and . Since , by calculation we deduce

From Lemma 2.1, there exists a positive constant depending only on , and such that

Then

By Lemma 3.1, we derive

Using the triangle inequality and (2.33), we deduce

According to the hypotheses of the theorem, we know that and are an infinitesimal of lower order comparing with . Hence, there exists a positive constant independent of such that for we have

Note that is small enough and ; from (3.13) and (3.12), we obtain

Noticing that , we have

which together with (3.14), noting that is an infinitesimal of higher order comparing with , yields

Since is the separation constant, is small enough, and , there holds

For in Step 3 of Scheme 1, from the definition of and (taking ), we have

Hence, when , Step 3 of Scheme 1 is equivalent to the following: find such that

And set , .

From (3.23), we obtain

Combining (3.24), (3.20) and (3.22), we get

By (3.26) and taking in (3.25), we obtain

Thus

From (3.28) we know that the first term on the left-hand side of (3.25) is equal to , thus

then, using discrete - condition we get

Thus Step 3 of Scheme 1 is equivalent to (3.28), (3.30), , and .

Noting that differs from by only a constant and selecting , we have

By (3.15), (3.17), (3.18), and (3.31), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.13) and (3.14) into (3.6), we obtain

Let the eigenfunctions be an orthonormal system of in the sense of , then

Let

noting , from (3.32) we deduce

By Lemma 2.3, there exist such that satisfy (2.32). Let

then . Using (2.32), we deduce

Combining (3.35) with the above inequality, we have

Substituting (3.12) into (3.38), we get (3.7).

From Step 3 of Scheme 1, we know that , thus

Select , , and . From Lemma 2.4, we get

Noting that , , and , we have

Note . From (3.41) we obtain (3.8).

For , Scheme 1 is actually the two-grid discretization scheme established in [7]. By Theorem 3.3 we get the following conclusion directly.

Theorem 3.4. Suppose . Let be small properly. Let be an approximate eigenpair obtained by Scheme 1 (). Then there exists such that

Proof. Consider Scheme 1. Here , . By Lemma 2.3, we know that approximates , and the accuracy order of is lower than . Hence, for , the conditions of Theorem 3.3 hold. Select in the proof of Theorem 3.3 such that satisfies Lemma 2.3. Then from (2.30), we obtain

substituting it into (3.7), we obtain (3.42). From (3.8), we deduce (3.43).

Theorem 3.4 is actually Theorem  3.3 in [7], but we analyze in detail the constants appeared in the error estimates.

From Theorem 3.4 and Theorem 3.3, we know that () and the convergence rate is high. Thus, we use as a posteriori error indicator of (details can be seen in Remark 4.5). Then we establish the following adaptive algorithm.

Scheme 2 (Adaptive Algorithm). Give an error tolerance and choose the parameter , , , and .

Step 1. Solve (2.3)-(2.4) on : find , such that

Step 2. , , .

Step 3. Solve an equation on : find such that

Let , .

Step 4. Compute the Rayleigh quotient

Step 5. If , then select , , and return to Step 3; else output , stop.

Remark 3.5. In Scheme 2, we use as a posteriori error indicator which is global. In order to cope with difficulties caused by local singularity of a complicated problem in calculation, so far, most algorithms designed a local a posteriori error indicator to establish adaptive algorithm with local mesh refinement (e.g., see [20–22]). However, because the eigenvalue problem of electric field is so complicated, that it is very difficult to obtain a local a posteriori error indicator of eigenfunction. Fortunately, the influence of local singularity can be avoided by using the weight method which is established by Costabel and Dauge to discrete the eigenvalue problem of electric field. And the performance of the weigh method is very good (see [3, 11]). Hence, without local mesh refinement, by using the weight method mentioned above our algorithm can also guarantee its high efficiency.

4. The Eigenvalue Problem of Electric Field

Consider the eigenvalue problem of electric field:

where is a polyhedron in , is the unit outward normal to .

Physically denotes the electric field, denotes the time frequency, and is the speed of the light velocity. Usually, let named eigenvalue.

Let

When is a convex polyhedron, we define the following function space:

Denote

From [23, 24], we know that ; is a coercive bilinear form on , and is a norm.

When is a nonconvex polyhedron, the problem is relatively complicated. Let denote a set of edges of reentrant dihedral angles on , and denote the distance to the set : . We introduce a weight function which is a nonnegative smooth function of . It can be represented by in reentrant edge and angular domain. We shall write . Define the weighted functional spaces:

Denote

Let be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition:

From the regularity estimate, we know . Let .

From [11, 25], we see that for all , the seminorm is a norm on , and is dense in .

In the following discussion, we will use for both nonconvex and convex domains. We select for non-convex domain and for convex domain.

By introducing the Lagrange multiplier , [6] changed (4.1) into the mixed variational formulation: find such that

Let be a regular simplex partition (tetrahedral partition) of with the mesh diameter . Define the finite element space .

Restricting (4.8) on the above-mentioned finite space, we obtain the discrete mixed variational form: find such that

Set

Then (4.8) and (4.9) can be written as (2.3)-(2.4) and (2.5)-(2.6), respectively (it is needed to add    for the vector function, e.g., should be written in the forms of ).

We apply Schemes 1 and 2 to the eigenvalue problem of electric field (4.8). Adding the symbol    for the vector function, we get a multi-grid discretization scheme and adaptive algorithm for mixed finite element of the eigenvalue problem of electric field which are still called Schemes 1 and 2.

It is easy to know that and are continuous bilinear forms on and , respectively. . (It is true obviously when is convex; when is non-convex, see [25].)

Consider the source problem associated with (4.8): find such that

For (4.11) and its conforming finite element approximation, condition (C1) of Lemma 2.1 holds obviously; [3] has proved that condition (C2) holds; assume that the discrete inf-sup condition (C3) holds, then conditions of Lemma 2.1 hold. Thus we can define operators , and . (4.8) and (4.9) can be written as (2.15) and (2.16)-(2.17), respectively.

The following Lemma 4.1 is cited from [3, 25].

Lemma 4.1. Equation (4.1) is equivalent to (4.8), and the solutions of (4.8) and , .

Note that Lemma 4.1 shows that for the eigenvalue problem of electric field, the second term on the right-hand side in (3.8) is equal to 0.

Theorem 4.2. Suppose that the discrete inf-sup condition (C3) holds. Then . Let be the th eigenpair of (4.9) with , and let be the th eigenvalue of (4.8). Then , and there exists an eigenfunction corresponding to with , such that

let , then there exists such that

where , , and are constants independent of .

Proof. From the preceding discussion and hypotheses of the theorem, we know that , is symmetric, and the conditions of Lemma 2.1 hold. Besides, since is a regular partition, when is a convex polyhedron, (); when is a non-convex polyhedron, is dense in . Since is dense in , thus, no matter is convex or non-convex, is dense in . For any given , we have . Thus for any , according to the density, we know that there exists such that

Selecting being small properly, when , we have

where is an interpolation operator. Thus

Namely, . Hence (2.26) is true. Analogously, using the density of in , we deduce that , namely, (2.27), is true. Hence, from Theorem 2.2 and Lemma 2.3, we know that , , (4.13), and (4.14) hold. From (2.38), (4.13), and Lemma 4.1, we get (4.12). The proof is completed.

Denote

From Lemma 2.1, noting , we deduce

Theorem 4.3. Assume that the discrete inf-sup condition (C3) holds, is small properly, Condition 1 holds and . Let be an approximate eigenpair obtained by Scheme 1, then there exists such that

Proof. We use induction to complete the proof. Note that the conditions of Theorem 2.2 hold. For , Scheme 1 is actually two-grid discretization scheme. Substituting (4.19) into (4.12) and (4.13), we derive

Combining (3.7) with and the above two inequalities, we know that there exists such that

Since , substituting (4.23) into (3.43), we deduce

The above two inequalities show that Theorem 4.2 is true for .

Suppose that the theorem is true for , then by Theorem 3.3, we get

That is, (4.20) is valid.

From (4.20) and (3.8), we obtain (4.21). The proof is completed.

Assume that is a regular simplex partition (tetrahedral partition) of with the mesh diameter . Let and be the - finite element spaces as follows:

Here we set . means that is equal to on the tetrahedron where reentrant edge and angular point are adjacent. Considering finite element approximation of (4.11), for the 3- -iso- Taylor-Hood finite element, Ciarlet and Girault [26] have discussed that the discrete inf-sup condition (C3) holds when is a convex domain; for the - element, Ciarlet and Hechme have proved that the discrete inf-sup condition (C3) holds when is a polyhedron (see Section  2.2 in [4] and pp. 509 in [3]).

From the above mentioned, we know that the - element approximation of (4.11) satisfies the conditions of Theorem 4.2.

Let be the smallest singular exponent in the Laplace problem with homogenous Neumman boundary condition, then . Denote .

Corollary 4.4. Assume that is small properly, , and . Let be an approximate eigenpair of the - element obtained by Scheme 1. Then when is a convex domain, there exists such that

When is a non-convex domain, there exists such that

where and are determined by (4.31) and (4.32) in the proof, respectively.

Proof. The hypotheses of Corollary 4.4 imply that the conditions of Theorem 4.3 hold. When is convex, for any we have (see in [1]). Thus there exists independent of such that

Substituting the above inequality into (4.20) and (4.21), we get (4.27) and (4.28), respectively.

When is a non-convex domain, for any , by in [3] we know that there exists independent of , such that

Substituting the above inequality into (4.20) and (4.21), we derive (4.29) and (4.30), respectively.

Remark 4.5. From Corollary 4.4, we see that the constants in the error estimates are not only independent of the mesh diameter but also independent of the iterative degrees. Thus, when , we have . Suppose that the precision order of (4.28) is optimal which cannot be improved any more, then

where for convex domain , while for non-convex domain but approximates arbitrarily. Therefore we have that . Then

thus we can use as a posteriori error indicator of , where .

5. Numerical Experiments

Consider the eigenvalue problem (4.1) of electric field, where is a square domain or is an L-shaped domain. For the square domain, the first five exact eigenvalues are , , and ; for the L-shaped domain, the first five eigenvalues are , , , and , and the first eigenfunction has a strong singularity (see, e.g., [3]).

We adopt a uniform isosceles right triangulation for (the edge in each element is along three fixed directions, see Figure 1) to produce the meshes with mesh diameter .

Figure 1

The definition of - mixed finite element spaces is given by

We make use of Matlab to compute the first five approximate eigenvalues by using Scheme 1 with - element. The numerical results are listed in Tables 1, 2, and 3.

Table 1: The results on the square by Scheme 1 (- element) for the eigenvalue problem of electric field ().

1

1.0000001280750
1.00000012807483
248.41
248.41

1

1.0000000080355
1.00000000803501
14.94
14.94

1

—
1.00000000050340
—
14.96

3

2.0000017958515
2.00000179585163
245.62
245.62

3

2.0000001126165
2.00000011261761
14.95
14.95

3

—
2.00000000704840
—
14.98

4

4.0000081820714
4.00000818207150
242.08
242.08

4

4.0000005140035
4.00000051400385
14.92
14.92

4

—
4.00000003219500
—
14.97

Table 2: The results on the L-shaped domain by Scheme 1 (- element) for the eigenvalue problem of electric field ().

1

2.619901684020
2.619902307450
0.16
0.16

1

2.545994662814
2.545994677427
0.69
0.69

1

—
2.468072524066
—
0.78

2

3.540738971244
3.540738974231
5.75
5.75

2

3.536561194304
3.536561194305
1.65
1.65

2

—
3.534975905948
—
1.67

3

9.869612668412
9.869612668412
244.51
244.51

3

9.869604920374
9.869604920377
14.92
14.92

3

—
9.869604433619
—
14.92

5

11.392491049450
11.392491049225
7.11
7.11

5

11.390607786437
11.390607786437
1.67
1.67

5

—
11.389899921315
—
1.68

Table 3: The results on the L-shaped domain by Scheme 1 (- element) for the eigenvalue problem of electric field ().

1

1.550099678021
1.550100277590
1.88
1.88

1

1.512784318422
1.512784324775
1.00
1.00

1

—
1.492972425344
—
1.14

2

3.534598663496
3.534598663588
14.65
14.65

2

3.534154160497
3.534154160495
3.62
3.62

2

—
3.534055109225
—
4.17

3

9.869612641692
9.869612641692
243.23
243.23

3

9.869604919289
9.869604919288
14.90
14.90

3

—
9.869604433575
—
14.92

5

11.389671749018
11.389671749041
28.83
28.83

5

11.389519405478
11.389519405477
3.81
3.81

5

—
11.389487122250
—
4.18

In Tables 1–3, denote the first five eigenvalues obtained by using the mixed element on directly; denote the first five eigenvalues obtained by Scheme 1:

From Tables 1–3, we can see that    and have the same accuracy. It fails to find by direct computation by using the mixed element on with in the case of the square domain and in the case of the L-shaped domain (here Matlab shows that Sparse lu with 4 outputs (UMFPACK) failed), but Scheme 1 still works. is an efficient and reliable error indicator of .

It can be seen from Tables 1–3 that, in the calculation of error indicators, should be selected as follows: in the case of the square domain, ; in the case of the L-shaped domain, is equal to , and , respectively when , and is equal to and respectively, when .

Remark 5.1. Taking Table 1, for example, we illustrate how to select next. In Table 1, , , , and . According to , by calculation, we get that , , and .

Acknowledgments

The authors cordially thank the editor and the referees for their valuable comments which lead to the great improvement of this paper. This work is supported by National Natural Science Foundation of China (Grant no. 11161012).

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