Abstract and Applied Analysis

Volume 2012 (2012), Article ID 812914, 29 pages

doi:10.1155/2012/812914

Research Article

A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems

Yidu Yang,¹ Wei Jiang,² Yu Zhang,¹ Wenjun Wang,¹ and Hai Bi¹

¹School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China

²School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen 361005, China

Received 1 March 2012; Accepted 24 April 2012

Academic Editor: Ibrahim Sadek

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 efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid . Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking , and when using the - element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking . Finally, numerical experiments are presented to support the theoretical analysis.

1. Introduction

To improve the efficiency of finite element method, Xu introduced a two-scale discretization scheme and applied it to nonsymmetric and nonlinear elliptic problems (see [1–3]). Later on, this scheme attracted the attention of academic circles and has been successfully applied to Stokes equations (see [4–7]), semilinear eigenvalue problems (see [8]), and linear differential operator eigenvalue problems, and so forth.

Up to now, two kinds of finite element two-scale discretization schemes have been developed for linear differential operator eigenvalue problems. The first kind is established by Xu and Zhou [9] in 2001, whose idea correlates with the iterative Galerkin method which was established by Lin and Xie [10] and Sloan [11], but it bases on the finite element spaces on two different scale grids. The scheme of Xu and Zhou has been applied to the conforming finite element method for electric structure eigenvalue problem (see [12–14]), the conforming/nonconforming finite element method for non-self-adjoint eigenvalue problem (see [15, 16]), the conforming/nonconforming finite element method for Steklov eigenvalue problem (see [17, 18]), the mixed finite element method for Stokes eigenvalue problem, and biharmonic equations eigenvalue problem (see [19, 20]). Another group of two-scale discretization scheme is proposed by Yang and Bi [21]. They established the conforming/nonconforming finite element two-scale discretization scheme based on the shifted-inverse power method.

With two-scale discretization schemes, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid , and the resulting solution still maintains an asymptotically optimal accuracy. Thus, the computational efficiency is improved.

Influenced by the work mentioned above, in this paper we establish a new finite element two-scale discretization scheme for mixed variational formulation of eigenvalue problem and apply it to Stokes eigenvalue problem and eigenvalue problem of electric field. The research of this paper has the following features.

(1) Our two-scale discretization scheme is a combinative production of the mixed finite element method and the shifted-inverse power method (see [22, Algorithm 27.2]). Comparing with the scheme in [19, 20], the scheme in this paper is more efficient: the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking when solving Stokes eigenvalue problem and when solving eigenvalue problem of electric field; however, with the scheme in [19, 20] the resulted solution maintains an asymptotically optimal accuracy by taking .(2)The literatures of high-efficient numerical method for eigenvalue problem of electric field are not too many by now, thus they seem to be very valuable. Our two-scale discretization scheme is a new and highly efficient method for eigenvalue problem of electric field.

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 eigenvalue problem mixed variational formulation (2.3)-(2.4) in general form, the finite element two-scale discretization scheme based on the shifted-inverse power method is established, and the validity of this scheme is proved theoretically. In Sections 4 and 5, the scheme established in Section 3 is applied to Stokes eigenvalue problem and eigenvalue problem of electric field, respectively. Finally, numerical experiments are presented in Section 6.

2. Preliminaries

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

is a continuous bilinear form on , that is,

In scientific and engineering computations, many eigenvalue problems for differential equation have the following 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

As for the mixed finite element method for boundary value problems, Brezzi and Babuska, and others have established a systematic theory. Denote

Theorem 2.1 (Brezzi-Babuska Theorem). Suppose that

(1) and are continuous bilinear forms, that is,

(2) there exists , such that

(3)inf-sup condition: there exists , such that

then there exists a unique solution to the problem (2.7)-(2.8), and

 where constant just depends on and . Furthermore, suppose that(4) there exists a constant , such that

(5) discrete inf-sup condition: there exists a constant , such that

Then there exists a unique solution to the problem (2.9)-(2.10); moreover, the following error estimate is valid:

where just depends on and .

Since is a symmetric, continuous, and -elliptic bilinear form on , is a continuous bilinear form, then conditions (1), (2), and (4) of Brezzi-Babuska Theorem hold. Suppose inf-sup condition and discrete inf-sup condition hold. Then by Brezzi-Babuska Theorem, we know that (2.7)-(2.8) and (2.9)-(2.10) are uniquely solvable for each . Thus we can define the corresponding linear bounded operators:

for all

for all ,

It is obvious that (2.3)-(2.4) have an equivalent operator form:

Equations (2.5)-(2.6) have an equivalent operator form:

It is easy to prove that are self-adjoint operators. In fact, for all , taking in (2.7)-(2.8) we obtain

Exchanging and , we get

then

It shows that is self-adjoint in the sense of inner product . Analogously, it can be proved that is self-adjoint in the sense of inner product .

Assume that (compact imbedded), then it is easy to prove that the operator 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 .

It is obvious that is an inner product on and are equivalent norms. Let in (2.5); then

From (2.6), we get . Then

therefore, is a completely normal eigenvector system on in the sense of 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 the th eigenvalue, too. Denote , , and . Define

The convergence and error estimate about mixed element method of eigenvalue problem have been studied by [23–25]. From these literatures we easily know that the following results are valid.

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

Let ; then there exists such that

Proof. From the spectral approximation theory (see [23]) we have (2.39).

Let satisfy (2.39), and . Next we will prove that this eigenpair satisfies (2.40)-(2.41). From Brezzi-Babuska Theorem and (2.39), we get

Using the triangle inequality and (2.43), we deduce

that is, (2.40) is valid.

Using the triangle inequality and (2.44), we get

which together with yields (2.41).

Let the eigenfunctions be an orthonormal system of in the sense of inner product . Then, from (2.41) and Lemma 3.1 we know that there exists a basis of satisfying and the following result is valid:

For any , we write . By calculation, we get

From (2.47), when we have , and thus we get .

Denote , then . From (2.47), we deduce

that is, (2.42) is valid. The proof is completed.

For , define the Rayleigh quotient

The following lemma is an extension of [23, Lemma 9.1].

Lemma 2.3. Let be an eigenpair of (2.3)-(2.4); then for all , with its Rayleigh quotient satisfying

Proof. From (2.3)-(2.4), we deduce

By dividing by on both sides of the above identity, we obtain (2.51).

Taking in (2.51) and using (2.4) and (2.6), we derive

Lemma 2.4. Let and be the th eigenpair of (2.3)-(2.4) and (2.5)-(2.6), respectively; then

3. The Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems

This paper establishes the following finite element two-scale discretization scheme based on the shifted-inverse power method.

Scheme 1. One has the following.

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

Step 2. Solve a equation on a fine grid : find such that

Set .

Step 3. Compute the Rayleigh quotient

Next we will discuss the validity of Scheme 1.

Lemma 3.1. For any nonzero elements ,

Proof. See [21].

Denote .

Consider the eigenvalue problem (2.25) on the space .

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 [21].

Theorem 3.3. Suppose that the conditions of Brezzi-Babuska Theorem hold and . Let be the approximate eigenpair obtained by the two-scale discretization scheme and small properly. Then there exists such that

Proof. We use Lemma 3.2 in the proof.

Select and . Let such that satisfies (2.39) and (2.41). By calculation we deduce

thus, using Lemma 3.1, we get

Using the triangle inequality and (2.42), we derive

From Lemma 2.2 we know ; then

When is small enough, noting that , from (3.11) and (3.10) we get

Having in mind that we have

which together with (3.12), 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 2 of Scheme 1, from the definition of and we have

Hence, Step 2 of Scheme 1 is equivalent to ,

.

From (3.21) we obtain

Combining (3.22), (3.18), and (3.20), we get

By (3.24), taking in (3.23), we obtain

Thus

From (3.26) we know that the first term on the left-hand side of (3.23) is equal to ; thus

then, using discrete inf-sup condition, we obtain

Thus Step  2 of Scheme 1 is equivalent to (3.26), (3.28), and . Noting that differs from by only a constant and denoting , we have

By (3.13), (3.15), (3.16), and (3.29), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.11) and (3.12) into (3.6), we obtain

Let the eigenfunctions be an orthonormal system of (in the sense of ). Then

Let

and noting that , from (3.30) we deduce

By Lemma 2.2, there exists such that satisfies (2.41). Let

then . Using (2.41) we deduce

Combining (3.33) with the previous inequality, we have

Substituting (3.10) into (3.36), we get (3.7).

We know that from Step 2 of two-scale scheme; then

Select . From Lemma 2.3, we get

Noting that, for all , we have

Since (continuously imbedded), . Then from (3.39) we obtain (3.8).

4. Two-Scale Discretization Scheme for Stokes Eigenvalue Problem

Consider the Stokes eigenvalue problem:

where is a polygonal domain in denotes the fluid velocity, and denotes the pressure.

In this paper, we use the symbol to stand for vector function. For the function in , let

For vector function , define

Using Green's formula, we derive the mixed variational form associated with (4.1)–(4.3).

Find with such that

Let be two mixed finite element spaces. The mixed finite element form is as follows.

Seek with such that

Denote

Let . It is clear that is a norm. Then (4.6)-(4.7) and (4.8)-(4.9) can be written in the forms of (2.3)-(2.4) and (2.5)-(2.6), respectively (we need to add for the vector function, e.g., should be written in the forms of ).

We apply Scheme 1 to the Stokes eigenvalue problem (4.6)-(4.7). Adding the symbol for the vector function we get two-scale discretization scheme of mixed finite element for solving the Stokes eigenvalue problem (4.6)-(4.7), which is still called Scheme 1.

Consider the associated source and approximate source problems.

Find such that

Seek such that

From [26] we know that (4.11)-(4.12) satisfy conditions (1)–(3) in Brezzi-Babuska Theorem; therefore, there exists a unique solution to the problem (4.11)-(4.12) and the following estimate is valid:

Condition (4) in Brezzi-Babuska Theorem holds since and . Suppose that condition (5) in Brezzi-Babuska Theorem (discrete inf-sup condition) is valid, then there exists a unique solution to the problem (4.13)-(4.14), and the following error estimate is valid (see [27–29]):

We assume that the following a prior estimate holds: for any , and

where is a number determined by the maximal inner angle of . When , (see [30]).

Suppose that the following estimate holds: for any , and for any ,

From Section 2, we know that (4.6)-(4.7) and (4.8)-(4.9) have the following equivalent operator forms, respectively,

Moreover, and are all self-adjoint compact operators.

Theorem 4.1. Assume that discrete inf-sup condition, (4.17) and (4.18) hold; let be properly small; and an approximate eigenpair obtained by Scheme 1. Then there exists such that

Proof. From Brezzi-Babuska Theorem, (4.17), and (4.18), we deduce

By virtue of Nitsche technique (see [29]) and (4.17), we derive

Using (4.24), we have

Hence, the conditions in Lemma 2.2 and Theorem 3.3 hold.

From (2.23), we know that

From (4.23), (4.18), and (4.17), we have

Substituting (4.26) and (4.27) into (3.7), we obtain (4.21). Substituting (4.21) and (4.28) into (3.8), we obtain (4.22).

4.1. Mini Mixed Finite Element

Consider two-scale discretization scheme of Mini mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).

Mini element was established by Arnold et al. in 1984 (see [31]). Let be a regular triangulation of under the meaning of paper [32], and

For any , let , , and denote barycentric coordinates. Denote , and set

From [31], we know that Mini element satisfies discrete inf-sup condition. From the interpolation theory in Sobolev space, we conclude that (4.18) is valid. Hence, for satisfying (4.17), Scheme 1 for Mini mixed finite element is effective. Theorem 4.1 is valid.

4.2. - Mixed Finite Element

Consider two-scale discretization scheme of - mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).

Let be a regular triangulation of , and is the product of refining in the middle point. The - mixed finite element space is defined by

where and are piecewise continuous linear polynomial spaces defined on and , respectively.

From [33, Proposition 3.3], we know that - element satisfies discrete inf-sup condition. By the interpolation theory in Sobolev space, we conclude that (4.18) holds. Therefore, for satisfying (4.17), Scheme 1 for - mixed finite element is effective. Theorem 4.1 is valid.

5. Two-Scale Discretization Scheme for Eigenvalue Problem of Electric Field

Consider the eigenvalue problem of electric field:

where is a polyhedron in and is the outward normal unit vector on .

In physics, in the above eigenvalue problem of electric field denotes electric field, denotes the time frequency, and is the speed of the light. Usually we set which is called eigenvalue.

The spaces (curl, ), (curl, ) are defined in the usual way:

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

Denote

From [34, 35], we see that ; is a coercive bilinear form in , and is a norm.

On the other hand, when is nonconvex the maximal interior angle belongs to . In this situation the problem is relatively complicated. Let denote a set of reentrant edge with dihedral angles belonging to , and let denote the distance to the set : . We introduce a weight function which is a nonnegative smooth function with respect to . It can be represented by in reentrant edge and angular domain. We write . Define the weight function space:

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 [36, 37], we know that, for all , the seminorm is a norm in , and is dense in .

In the following discussion, we will use both for non-convex and convex domain. We take for non-convex domain; otherwise, we take .

By introducing Lagrange multiplier , [36, 38, 39] changed (5.1)–(5.3) 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 as follows:

Here we set . means that is zero on the tetrahedron where reentrant edge and angular point are adjacent.

Restricting (5.10)-(5.11) to the previous finite element space, we get discrete mixed variational form: find such that

Set

Then (5.10)-(5.11) and (5.13)-(5.14) can be written in the forms of (2.3)-(2.4) and (2.5)-(2.6), respectively (we need to add for the vector function, e.g., should be written in the forms of ).

We apply Scheme 1 to the eigenvalue problem of electric field (5.10)-(5.11). Adding the symbol for the vector function we get two-scale discretization scheme of mixed finite element for solving the eigenvalue problem of electric field (5.10)-(5.11) which is still called Scheme 1.

It is easy to know that and are continuous bilinear forms on and , respectively. is compact embedded in (when is convex, it is valid obviously; when is non-convex, see [36]).

Consider the source problem corresponding to (5.10)-(5.11).

Find such that

For the problem (5.16)-(5.17) and its - element approximation, people have already proved the conditions in Brezzi-Babuska Theorem hold (see [38, 40, 41]).

Therefore, we can define operators ; moreover, (5.10)-(5.11) and (5.13)-(5.14) can be written in the forms of (2.23)-(2.24) and (2.25)-(2.26), respectively.

Lemma 5.1 is cited from the literature [36, 38].

Lemma 5.1. (5.1)–(5.3) is equivalent to (5.10)-(5.11), and the solutions of (5.10)-(5.11), , satisfy and with .

Denote

For the - element approximation of (5.16)-(5.17), [38] proved that the condition of [24, Theorem 1] (i.e., [38, Theorem 4.3]) is valid, hence, there holds the following.

Lemma 5.2. For the - element approximation of (5.16)-(5.17), there exists ,   such that

This lemma is very important. It tells us that . Based on this lemma, [38] also proved the following conclusion.

Lemma 5.3. For the - element approximation of (5.10)-(5.11), the following estimate is valid:

Theorem 5.4. Let be the approximate - element eigenpair obtained by Scheme 1; then there exists such that

Proof. We use Theorem 3.3 to complete the proof. From (5.19) we see that the condition in Theorem 3.3 holds, and by Lemma 5.1 we know .

From (2.18), we deduce

From (5.20), we derive

Substituting (5.24) and (5.25) into (3.7), we get (5.22).

Since , (3.8) can be simplified to

Substituting (5.22) into (5.26), we obtain (5.23).

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

Corollary 5.5. Under the condition of Theorem 5.4, if is a convex polyhedron there holds

if is a non-convex polyhedron, then

Proof. When is a convex polyhedron, for any we have (see [35], or [42, equation (44)]); therefore

Substituting these two formulae into (5.22) and (5.23), we obtain (5.27) and (5.28), respectively. When is non-convex, for all , from [38, equation (36)] we know that

Substituting the above two formulae into (5.22) and (5.23), we obtain (5.29) and (5.30), respectively.

6. Numerical Experiments

In the following two examples, let be the first four eigenvalues computed by using mixed finite method directly on coarse mesh and the first four eigenvalues computed by using mixed finite method directly on fine mesh . Let denote the first four eigenvalues computed by Scheme 1 on the meshes and .

Example 6.1. Consider the Stokes eigenvalue problem (4.1)–(4.3), where is a unit square domain. The smallest eigenvalue is approximately equal to for this problem.

We adopt a uniform isosceles right triangulation for the domain (the edge in each element is along three fixed directions), and we give an initial mesh in Figure 1 and refine the initial mesh in a uniform way (each triangle is divided into four congruent triangles) repeatedly to get meshes and .

Figure 1

We solve this problem by Scheme 1 with Mini element. The numerical results are shown in Table 1.

Table 1: The results on the square by Scheme 1 (Mini element) for Stokes eigenvalue problem.

1

57.50602342019
52.42255785241
52.42147553267

1

55.61324143591
52.37622690147
52.37604993774

1

54.59828590012
52.35983048690
52.35978975974

2

103.5158846668
92.30714521284
92.29553836981

2

99.36963738012
92.19619098643
92.19429134425

2

97.12904494102
92.15848299868
92.15804753694

3

110.7048253409
92.43050152401
92.39531304531

3

103.8058661064
92.24032470162
92.23510195554

3

100.1402638604
92.17887213187
92.17771885852

4

159.7476076282
128.8864896622
128.7085892117

4

148.7986378758
128.4767873556
128.4133574197

4

142.5976379771
128.3089668708
128.3076834410

From Table 1, we conclude that Scheme 1 can efficiently solve Stokes eigenvalue problem.

Example 6.2. Consider the eigenvalue problem of electric field (5.1)–(5.3), where is a square domain or an L-shaped domain . For the square domain, the first four exact eigenvalues are for this problem. For the L-shaped domain, the first four approximate eigenvalues are .

We adopt a uniform isosceles right triangulation for (the edge in each element is along three fixed directions, see Figure 2 for the L-shaped domain, and see Figure 1 for the square domain ) to produce the meshes and .

Figure 2

We use - mixed finite element to solve this problem. The definition of - mixed finite element space is given by

We compute the first four approximate eigenvalues by using Scheme 1 with - element on the mesh and . The numerical results are listed in Tables 2, 3, and 4.

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

1

1.000489112
1.000000005
1.000000005

1

1.000099609
1.000000005
1.000000005

1

1.000031944
1.000000005
1.000000005

2

1.000490210
1.000000005
1.000000005

2

1.000099685
1.000000005
1.000000005

2

1.000031956
1.000000005
1.000000005

3

2.006497695
2.000000070
2.000000070

3

2.001365268
2.000000070
2.000000070

3

2.000442889
2.000000070
2.000000070

4

4.028293524
4.000000321
4.000000333

4

4.006078511
4.000000321
4.000000321

4

4.001988912
4.000000321
4.000000321

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

1

3.004954998
2.557217952
2.557241250

1

2.896205625
2.557217952
2.557225625

1

2.842087274
2.557217952
2.557221723

2

3.686002173
3.536967001
3.536967768

2

3.623341929
3.536967001
3.536967079

2

3.595097794
3.536967001
3.536967017

3

9.938579092
9.869605192
9.869605219

3

9.884458574
9.869605192
9.869605192

3

9.874470046
9.869605192
9.869605192

4

9.938761731
9.869605192
9.869605221

4

9.884463697
9.869605192
9.869605192

4

9.874472304
9.869605192
9.869605192

Table 4: The results on the L-shaped domain by Scheme 1 (- element) for eigenvalue problem of electric field ().

1

1.913411786
1.517146260
1.517159275

1

1.779172346
1.517146260
1.517148955

1

1.723917701
1.517146260
1.517147418

2

3.589241343
3.534187594
3.534187611

2

3.558323120
3.534187594
3.534187595

2

3.547766251
3.534187594
3.534187595

3

9.937421407
9.869605190
9.869605213

3

9.884270121
9.869605190
9.869605191

3

9.874420606
9.869605190
9.869605191

4

9.937916802
9.869605191
9.869605216

4

9.884282799
9.869605191
9.869605190

4

9.874424604
9.869605191
9.869605190

From Tables 2–4, we conclude that Scheme 1 can efficiently solve the eigenvalue problem of electric field.

In this paper, denotes a positive constant independent of , which may stand for different values at its different occurrences.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11161012) and Science and Technology Foundation of Guizhou Province of China (no. [2011]2111).

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