Abstract and Applied Analysis

Volume 2014 (2014), Article ID 189490, 15 pages

http://dx.doi.org/10.1155/2014/189490

Research Article

An Adaptive Nonconforming Finite Element Algorithm for Laplace Eigenvalue Problem

Yuanyuan Yu, Yidu Yang, and Jiayu Han

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

Received 11 January 2014; Revised 2 April 2014; Accepted 29 April 2014; Published 29 May 2014

Academic Editor: Agacik Zafer

Copyright © 2014 Yuanyuan Yu 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.

We establish Crouzeix-Raviart element adaptive algorithm based on Rayleigh quotient iteration and give its a priori/a posteriori error estimates. Our algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

1. Introduction

A posteriori error estimates and adaptive methods of finite element approximation for eigenvalue problems are topics attracting more attention from mathematical and physical fields; see, for example, [1–8]. Basically, there are the following three ways of combining adaptivity and eigenvalue problems in which the a posteriori error estimators are more or less the same but different in the problem solved in each iteration: (1) solving the original eigenvalue problem (see Algorithm 10). The convergence and optimality of this adaptive procedure were proved in [2]; (2) inverse iteration type (with or without correction). The convergence has been studied in [1, 6, 7]; (3) Shifted-inverse iteration type (see [8–11]).

The triangular Crouzeix-Raviart element (C-R element) was first introduced by Crouzeix and Raviart [12] in 1973 to solve the stationary Stokes equation. After that, many scholars developed and applied it to eigenvalue problems, for instance, [13–16] discussed a posteriori error estimates and the adaptive methods of the C-R element. C-R element has important properties; for example, Armentano and Durán [17] discovered and proved that the C-R element eigenvalues approximate the exact ones of the Laplace operator from below, which is a very important property in engineering and mechanics computing.

Based on the above work, this paper further discusses the third kind of adaptive methods of the C-R finite element method for eigenvalue problems and obtains the following new results:

(1)we establish a multiscale discretization scheme of the C-R element based on Rayleigh quotient iteration and prove its convergence and a priori error estimates;(2)we give residual type a posteriori error estimator for our adaptive algorithm, as well as its reliability and efficiency;(3)we establish an adaptive algorithm (Algorithm 11), which is performed under the package of Chen (see [18]), and satisfactory numerical results are obtained.

As for the fundamental theory of finite elements and spectral approximation, we refer to [19–22].

Throughout this paper, denotes a positive constant independent of mesh parameter, which may not be the same constant in different places. For simplicity, we use the notation to mean that , and to mean that and .

2. Preliminaries

Consider Laplace eigenvalue problem

where is a polygonal domain with the maximum interior angle .

We denote the real order Sobolev spaces with norm by , . Let and be the inner product and the norm in the space , respectively.

The weak form of (1) is as follows: find , , such that

where

As we know, is a symmetric, continuous, and -elliptic bilinear form on , and is a symmetric, continuous, and positive definite bilinear form on .

Define the operator , satisfying

Then, (2) has the equivalent operator form , where the operators and are self-adjoint and completely continuous.

Let be a regular triangulation of the domain denote the set of all element edges in ,  denote the set of interior edges, denote the set of all boundary edges, and denote the set of the midside nodes of the edges . For the set of midpoints of the edges , we use the notation ; for any element , we let be the union set of edges of , and let   be the diameter of . The domain consists of all elements sharing at least a side with . For any edge ,  is the length of , and and are unit outward normal vector and unit tangential vector, respectively.

Given a nonnegative integer , the space consists of polynomials of total degree at most defined over . The C-R element space is given by , is continuous at each , and for .

The C-R element approximation of (2) is given as follows: find , ,  such that

where

is a symmetric, continuous, and -elliptic bilinear form on . is well known as the norm of the space ; let which consists of ,

Define the operator , satisfying

Then, (5) has the equivalent operator form , where the operators and are self-adjoint and completely continuous.

Suppose that and are the th eigenvalue of (2) and (5), 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 (5) that converge to . Let , .

Define

Define , which is the consistent item of nonconforming finite element. Let be an eigenpair of (2), and then

We need the lemmas as follows (see [11, 23]).

Lemma 1 (see [11, Lemma  2.5]). Let be an eigenpair of (2) and then, for any with , the Rayleigh quotient satisfies

Lemma 2 (see [11, Lemma  2.4] and [23, Lemma  2.3]). Let and be the th eigenvalue of (5) and (2), respectively. Then,

For any eigenfunction corresponding to , satisfying , there exists such that

For any , there exists such that

3. A Priori Error Estimates for Multiscale Discretization Scheme

In this section, we will discuss a priori error estimates of the C-R finite element multiscale discretization scheme based on the shift-inverse power method. Let be a family of shape-regular meshes and let be the C-R finite element spaces defined on . Besides, let , .

The following condition results from [10, 24].

Condition 1. There exists a properly small positive number , , , such that , .

The following scheme is proposed by Yang and Bi (see [11]).

Scheme 3 (multiscale discretization scheme). Consider the following steps.

Step 1. Solve (5) on : find such that and

Step 2. Execute ,  , .

Step 3. Solve a linear system 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 of (16), and then derived from Scheme 3 is the th eigenpair approximation of (5).

In the sequel, we also denote , .

Lemma 4 (see [11, Lemma  3.1]). For any nonzero ,

Denote .

Our analysis is based on the following crucial property of the shifted-inverse iteration in finite element method (see Lemma  4.2 of [24]), which is a development of Theorem 3.2 in [11]. Let , .

Lemma 5 (see [24, Lemma 4.2]). Let and be the th eigenvalue of and , respectively, be an approximation for the eigenpair , where is not an eigenvalue of , and with . Suppose that

(C1);(C2) for , , where is the separation constant of the eigenvalue ;(C3) satisfy

Then,

Let us construct the interpolation postprocessing operator (see [25]): on the vertex of elements,

where is the number of elements containing the vertex and is the union of elements containing the vertex .

Lemma 6. Suppose that Condition 1 holds and is properly small. Let be obtained by Scheme 3 for , and then there exists such that

Proof. Based on the proof of Theorem  5.1 in [11] and Lemma 5, we deduce that

and thus (23) holds. Using Strang Lemma and Lemma 3.1 of [25], we deduce that

From the above formula and in [11], we get

and thus (24) holds.

Based on [10, 11, 24], we will prove the following Theorems 7 and 8 for Scheme 3.

Theorem 7. Let be an approximate eigenpair obtained by Scheme 3, and and approximate and , respectively, and , . Suppose that is properly small and Condition 1 holds. Then, there exists such that

where .

Proof. Let , . Since , by calculation, we get

From the definition of , it is easy to know that

From , we get

and thus

By Lemma  3.1 in [25], we get that

Thus,

Using the above formula and (31), we can deduce that

Using Lemma 4, we get

Using triangle inequality and (15), we have

From (12), for , , we have

Noticing that is small enough and Condition 1 holds, then by (38) and (39), we can obtain

Since is the separation constant, is small enough, and Condition 1 holds, we have

From the definition of , we can see that Step 3 in Scheme 3 is equivalent to

where ; that is,

Noticing that differs from by only a constant, then Step 3 is equivalent to

From the above formulae, (41), (42), and (45), we can see that the conditions in Lemma 5 hold; therefore, substituting (39) and (40) into (21), we derive

Let eigenfunctions be an orthonormal basis of in the sense of inner product and then

Let

and then it follows directly from (46) that

By Lemma 2, there exists so that satisfies (14).

Let

Then, , and

By calculation,

where .

From (12) and (13), we deduce that

Substituting (52) into (51), we have

Let

By the above two equalities, we obtain

From (49) and (53), we have

Therefore,

By Lemma 1, we have

Since , using Strang Lemma and Lemma  3.1 of [25], we deduce that

Substituting (59) and (61) into (60), (29) holds.

By (56) and triangle inequality, we have

where .

By (57) and (29), we know that ; thus, (28) holds.

By calculation,

By the above formulae and (12), we deduce that

which together with (49) leads to (30). This completes the proof.

Theorem 8. Let be the th approximate eigenpair of (1) obtained by Scheme 3, let be the th eigenvalue of (1), and let be properly small. Suppose that Condition 1 holds, then there exists such that

Proof. The proof of (65) is completed by using induction. When , by Lemma 6, we know that Theorem 8 holds.

Suppose that Theorem 8 holds for ; that is,

which together with the assumptions in Theorem 8, we know that Theorem 7 holds. For , by (29) and (59), we get (65). The proof is completed.

4. A Posteriori Error Estimates for Multiscale Discretization Scheme

Based on the work of [14, 26], in this section, we will discuss a posteriori error estimates of the C-R element approximation for Laplace eigenvalue problem.

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

and its C-R element approximation: find such that

Let ,  be two elements sharing one edge . For any piecewise continuous function , we denote by the jump of across .

Let be the solution of (68), be the jump of across along , and be the jump of across along ; let be element residual; that is,

For , define the residual on the element as

and thus, for , the residual sum on is given by

For , define the date oscillation by

where stands for a piecewise polynomial approximation of over .

For the boundary value problem (67), Carstensen and Hu [27] have proved the following a posteriori error estimates:

where constant is only dependent on minimum angle of , and if the right-hand side of (67) is a piecewise linear polynomial over , then

Selecting in (67) and (68), then the generalized solution and the nonconforming finite element solution are and , respectively, and the a posteriori error indicator of is , which is defined by (71).

Define the element residual and the jump residual and for as follows:

For , define the residual on the element as

For , define the residual sum on as

Theorem 9. Suppose that the conditions in Theorem 7 hold and is a finite element space consisting of piecewise linear polynomials, then there exists a positive constant which is independent of mesh parameter, such that

Proof. Let , and by calculation

By triangle inequality, we have

From triangle inequality, (69)-(70), and (75)-(76), we deduce that

It is obvious that , and, by the trace theorem (see e.g., [28]) and the inverse estimates, we get

Thus,

Combining (82), (85), and (30), we get

Hence, from Condition 1, we know that is a small quantity of higher order than . Using (81), we obtain that is also a small quantity of higher order than .

Therefore, by (28), (73), (81), and (86), we have

which is (78).

Similarly, by (28) and (74), we get

and thus (79) holds.

By (61) and (28), we get

and, by substituting the above relation into (60), we obtain

which together with (78) yields (80). This completes the proof.

5. Adaptive Finite Element Algorithm Based on Multiscale Discretizations

As we know, The following Algorithm 10 is fundamental and important; see [14, 16] for its detailed theoretical results.

Algorithm 10. Choose parameter .

Step 1. Pick any initial mesh with mesh size .

Step 2. Solve (5) on for discrete solution .

Step 3. .

Step 4. Compute the local indicators .

Step 5. Construct by Marking Strategy and parameter .

Step 6. Refine to get a new mesh .

Step 7. Solve (5) on for discrete solution .

Step 8. , and go to Step 4.

Marking Strategy . Give parameter .

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

Step 2. Mark all the elements .

and are defined as (76) and (77), respectively, with and replaced by and .

We have the following adaptive algorithm on the basis of Scheme 3.

Algorithm 11. Choose parameter .

Step 1. Pick any initial mesh with mesh size .

Step 2. Solve (5) on for discrete solution .

Step 3. , .

Step 4. Compute the local indicators .

Step 5. Construct by Marking Strategy and parameter .

Step 6. Refine to get a new mesh

Step 7. Find such that

Set and compute the Rayleigh quotient

Step 8. , and go to Step 4.

Marking Strategy in Algorithm 11 will be the same as that in Algorithm 10, except for replacing with .

Note that when is too small, (92) is an almost singular linear equation. Although it has no difficulty in solving (92) numerically (see Lecture 27.4 in [29]), one would like to think of selecting a proper integer . When , set in (92). So, we can establish the following algorithm (see e.g., Scheme 3.2 in [24]).

Algorithm 12. Choose parameter .

Step 1–Step 7. Execute Step  1–Step  7 of Algorithm 11.

Step 8. If , , , go to Step 4; else , go to Step 4.

Marking Strategy in Algorithm 12 will be the same as that in Algorithm 11.

6. Numerical Experiments

In this section, we will report two numerical examples for Algorithms 10 and 11 to illustrate the theoretical results in this paper. We use MATLAB 2012 to solve Examples 1 and 2. Our program is compiled under the package of Chen. We take in two Algorithms.

For reading convenience, we use the following notations in our tables.

 : The th iteration of Algorithm 10 :The th approximate eigenvalue derived from the th iteration obtained by Algorithm 10 :The degrees of freedom of the th iteration for computing  : The total CPU time(s) for computing  : the error of th approximate eigenvalue  : The th iteration of Algorithm 11 :The th approximate eigenvalue derived from the th iteration obtained by Algorithm 11 : The degrees of freedom of the th iteration for computing  : The total CPU time(s) for computing  : The error of th approximate eigenvalue .

Example 1. We use Algorithms 10 and 11 to compute the approximate eigenvalues of (1) on the -shaped domain (see Figure 1(a)).

The first and fifth eigenvalues of (1) are and on this domain, respectively. The associated numerical results are presented in Table 1 and Figures 1(a), 2, 3, and 4. Figure 1(a) gives the uniform initial mesh with . Figures 2 and 3 show the adaptive meshes of the first and fifth eigenvalues after the fifth iteration by two algorithms, respectively. It is indicated in Figure 4 that the error curves of the first and fifth approximate eigenvalues and the curves of the associated a posteriori error estimators obtained by Algorithms 10 and 11 are approximately parallel to the line with slope −1, respectively; this coincides with our theory in Section 4.

But from Table 1, using Algorithm 11, we will spend much less time in the case of the same number of degrees of freedom but get the same accuracy to Algorithm 10. In addition, Algorithm 10, due to not having enough memory, can not proceed, while Algorithm 11 can have one more iteration; thus, more accurate numerical results will be obtained.

Table 1: The 1st and 5th eigenvalues obtained by two algorithms on -shaped domain with .

1
5
6615
9.624308
0.43
5
6615
9.624308
0.26

1
10
28350
9.636503
1.83
10
28350
9.636503
0.99

1
15
122122
9.638997
9.27
15
122122
9.638997
5.17

1
19
387527
9.639505
35.6
19
387527
9.639505
19.3

1
20
520999
9.639556
52.0
20
520999
9.639556
26.6

1
21
—
—
—
21
692323
9.639608
38.2

5
5
8319
31.83478
0.62
5
8319
31.83478
0.35

5
10
38601
31.89705
3.07
10
38601
31.89705
1.75

5
15
171591
31.90928
17.1
15
171591
31.90928
9.79

5
18
414327
31.91127
47.9
18
414327
31.91127
26.7

5
19
550331
31.91156
68.6
19
550331
31.91156
37.0

5
20
—
—
—
20
743563
31.91187
53.0

(a)

(b)

Figure 1: The initial mesh .

(a)

(b)

Figure 2: The adaptive meshes of 1st eigenvalue after 5th iteration by Algorithm 10 (a) and Algorithm 11 (b).

(a)

(b)

Figure 3: The adaptive meshes of 5th eigenvalue after 5th iteration by Algorithm 10 (a) and Algorithm 11 (b).

(a)

(b)

Figure 4: The error curves of two algorithms on -shaped domain.

Example 2. We use Algorithms 10 and 11 to compute the approximate eigenvalues of (1) on with a slit (see Figure 1(b)).

The first and sixth eigenvalues of (1) are and on this domain, respectively. The associated numerical results are presented in Table 2 and Figures 5, 6, and 7. Figure 7 show that the error curves of the first and sixth approximate eigenvalues and the curves of the associated a posteriori error estimators obtained by Algorithms 10 and 11 are approximately parallel to the line with slope −1, respectively, which suffices to support our theory.

From Table 2, using Algorithm 11, compared with Algorithm 10, we can get the same accurate results in the case of the almost same degrees of freedom, but the CPU time is significantly decreased.

Table 2: The 1st and 6th eigenvalues obtained by two algorithms on slit domain with .

1
5
5229
8.329697
0.43
5
5229
8.329697
0.25

1
10
17488
8.361012
1.33
10
17491
8.361012
0.79

1
15
64863
8.368891
5.14
15
64863
8.368891
2.90

1
20
249713
8.370756
23.4
20
249713
8.370756
12.9

1
23
569429
8.371082
61.5
23
569429
8.371082
31.2

1
24
—
—
—
24
748279
8.371148
44.0

6
5
8733
30.39428
0.69
5
8721
30.39422
0.41

6
10
35253
30.50367
3.21
10
35212
30.50363
1.68

6
15
141116
30.52867
16.2
15
140930
30.52866
8.20

6
19
426746
30.53371
58.8
19
426194
30.53371
28.9

6
20
566328
30.53432
82.6
20
565526
30.53431
39.1

6
21
—
—
—
21
743261
30.53464
54.9

(a)

(b)

Figure 5: The adaptive meshes of 1st eigenvalue after 5th iteration by Algorithm 10 (a) and Algorithm 11 (b).

(a)

(b)

Figure 6: The adaptive meshes of 6th eigenvalue after 5th iteration by Algorithm 10 (a) and Algorithm 11 (b).

(a)

(b)

Figure 7: The error curves of two algorithms on slit domain.

Remark 13. Based on the work of [30], we would like to believe that and Rayleigh quotient of are the lower and upper bounds of the exact eigenvalue , respectively. To see this point, the numerical results of Tables 1 and 2 also illustrate that the C-R element eigenvalues approximate the exact ones of the Laplace operator from below. Thus, we can establish iterative control condition by computing and for the two algorithms.

Remark 14. For Algorithm 12, by calculating, in the case of the almost same number of degrees of freedom, we can get the same accurate results to Algorithm 11, and CPU time is almost the same; thus, we do not list the associated numerical results in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 11161012) and Science and Technology Foundation of Guizhou Province of China (no. 2111). The authors cordially thank the referees and the editor for their valuable comments and suggestions that led to the improvement of this paper.

References

W. Dahmen, T. Rohwedder, R. Schneider, and A. Zeiser, “Adaptive eigenvalue computation: complexity estimates,” Numerische Mathematik, vol. 110, no. 3, pp. 277–312, 2008.

X. Dai, J. Xu, and A. Zhou, “Convergence and optimal complexity of adaptive finite element eigenvalue computations,” Numerische Mathematik, vol. 110, no. 3, pp. 313–355, 2008.

V. Heuveline and R. Rannacher, “A posteriori error control for finite approximations of elliptic eigenvalue problems,” Advances in Computational Mathematics, vol. 15, no. 1–4, pp. 107–138, 2001.

J. Han and Y. Yang, “A class of spectral element methods and its a priori/a posteriori error estimates for 2nd-order elliptic eigenvalue problems,” Abstract and Applied Analysis, vol. 2013, Article ID 262010, 14 pages, 2013.

D. Mao, L. Shen, and A. Zhou, “Adaptive finite element algorithms for eigenvalue problems based on local averaging type a posteriori error estimates,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 135–160, 2006.

V. Mehrmann and A. Miedlar, “Adaptive computation of smallest eigenvalues of self-adjoint elliptic partial differential equations,” Numerical Linear Algebra with Applications, vol. 18, no. 3, pp. 387–409, 2011.

T. Rohwedder, R. Schneider, and A. Zeiser, “Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization,” Advances in Computational Mathematics, vol. 34, no. 1, pp. 43–66, 2011.

Y. Yang, Y. Zhang, and H. Bi, “Multigrid discretization and iterative algorithm for mixed variational formulation of the eigenvalue problem of electric field,” Abstract and Applied Analysis, vol. 2012, Article ID 190768, 25 pages, 2012.

H. Bi, Y. Yang, and H. Li, “Local and parallel finite element discretizations for eigenvalue problems,” SIAM Journal on Scientific Computing, vol. 35, no. 6, pp. 2575–2597, 2013.

H. Li and Y. Yang, “The adaptive finite element method based on multi-scale discretizations for eigenvalue problems,” Computers & Mathematics with Applications, vol. 65, no. 7, pp. 1086–1102, 2013.

Y. Yang and H. Bi, “Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems,” SIAM Journal on Numerical Analysis, vol. 49, no. 4, pp. 1602–1624, 2011.

M. Crouzeix and P.-A. Raviart, “Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I,” RAIRO—Analyse Numérique, vol. 7, no. 3, pp. 33–75, 1973.

R. G. Durán, L. Gastaldi, and C. Padra, “A posteriori error estimators for mixed approximations of eigenvalue problems,” Mathematical Models & Methods in Applied Sciences, vol. 9, no. 8, pp. 1165–1178, 1999.

Y. Li, “A posteriori error analysis of nonconforming methods for the eigenvalue problem,” Journal of Systems Science & Complexity, vol. 22, no. 3, pp. 495–502, 2009.

C. Lovadina, M. Lyly, and R. Stenberg, “A posteriori estimates for the Stokes eigenvalue problem,” Numerical Methods for Partial Differential Equations, vol. 25, no. 1, pp. 244–257, 2009.

A. D. Russo and A. E. Alonso, “A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4100–4117, 2011.

M. G. Armentano and R. G. Durán, “Asymptotic lower bounds for eigenvalues by nonconforming finite element methods,” Electronic Transactions on Numerical Analysis, vol. 17, pp. 93–101, 2004.

L. Chen, iFEM: An Innovation Finite Element Methods Package in MATLAB, 2008.

I. Babuška and J. Osborn, “Eigenvalue problems,” in Handbook of Numerical Analysis: Finite Element Methods (Part 1), P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 641–787, North-Holland, Amsterdam, The Netherlands, 1991.

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 2002.

F. Chatelin, Spectral Approximation of Linear Operators, Computer Science and Applied Mathematics, Academic Press, New York, NY, USA, 1983.

P. G. Ciarlet, “Basic error estimates for elliptic problems,” in Handbook of Numerical Analysis: Finite Element Methods (Part 1), P. G. Ciarlet and J. L. Lions, Eds., vol. 2, pp. 17–351, North-Holland, Amsterdam, The Netherlands, 1991.

Y. Yang, Z. Zhang, and F. Lin, “Eigenvalue approximation from below using non-conforming finite elements,” Science China. Mathematics, vol. 53, no. 1, pp. 137–150, 2010.

Y. Yang, H. Bi, and Y. Yu, “The convergence and error estimates of the shifted inverse iteration based on multigrid discretizations for eigenvalue problems,” submitted to SIAM Journal on Numerical Analysis

A. Agouzal, “A posteriori error estimator for finite element discretizations of quasi-Newtonian Stokes flows,” International Journal of Numerical Analysis and Modeling, vol. 2, no. 2, pp. 221–239, 2005.

C. Carstensen, J. Hu, and A. Orlando, “Framework for the a posteriori error analysis of nonconforming finite elements,” SIAM Journal on Numerical Analysis, vol. 45, no. 1, pp. 68–82, 2007.

C. Carstensen and J. Hu, “A unifying theory of a posteriori error control for nonconforming finite element methods,” Numerische Mathematik, vol. 107, no. 3, pp. 473–502, 2007.

L. Wang and X. Xu, Foundation of Mathematics in Finite Element Methods, Science Press, Beijing, China, 2004 (Chinese).

L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1997.

Y. Yang, J. Han, H. Bi, and Y. Yu, “The lower/upper bound property of the nonoconforming Crouzeix-Raviart element eigenvalues on adaptive meshes,” Journal of Scientific Computing, 2014.