Highly Efficient Calculation Schemes of Finite-Element Filter Approach for the Eigenvalue Problem of Electric Field

This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear algebraic system on finer and finer grids. Theoretical analysis and numerical results show that the scheme has high efficiency. Besides, we use interpolation postprocessing technique to improve the accuracy of solutions, and numerical results show that the scheme is an efficient and significant method for eigenvalue problem of electric field.


Introduction
In recent years, eigenvalue problems of electric field has attracted increasing attention in the fields of physics and mathematics, and its numerical methods the filter approach, the parameterized approach, and the mixed approach are also developed further see 1-7 .Although the filter approach is an effective and important method for solving eigenvalue problems of electric field, its computation costs and accuracy of numerical solutions still need to be improved.
In fact, it is really a challenging job to reduce the computation costs without decreasing the accuracy of finite-element solutions.As we know, two-grid discretization and multigrid discretization are reliable and important methods satisfying the above requirements.Twogrid discretization was first introduced by Xu for nonsymmetric and nonlinear elliptic When Ω is a convex polyhedron, we define the following function space:

2.4
Let σ D Δ ∈ 3/2, 2 be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition:

2.5
Set γ min 2 − σ D Δ and γ ∈ γ min , 1 .When Ω is a nonconvex polyhedron, let E denote a set of edges of reentrant dihedral angles on ∂Ω, and let d d x denote the distance to the set E : d x dist x, ∪ e∈E e .We introduce a weight function ω γ which is a nonnegative smooth function corresponding to x.It can be represented by d γ in reentrant edge and angular domain.We shall write ω γ d γ .Define the weighted functional spaces:

2.7
Note that X γ X when Ω is a convex polyhedron, namely, in the case of γ 0. Consider the variational formulation: Find λ, u ∈ R × X γ with u X γ 1, such that a u, v λb u, v , ∀v ∈ X γ .

2.8
Let π h be a regular simplex partition, and let X h be a space of piecewise polynomial of degree less than or equal to k defined on π h : Then, X h ⊆ X γ .
The discrete variational form of 2.8 : The eigenpairs of 2.1 must be that of 2.8 .But the converse of this statement may not be true, namely, 2.8 has spurious pairs.Hence, 2.10 has spurious pairs.It is easy to prove that a •, • and b •, • are symmetric bilinear forms.Next we shall prove that a •, • is continuous and V -elliptic.
From the definition of a •, • , we have

2.11
Therefore, continuity of a •, • is valid.And

2.14
It is easy to prove that T : X γ → X γ , T h : X h → X h is self-adjoint completely continuous operator, respectively.Actually, for all f, g ∈ X γ , we have 15 which shows that T : X γ → X γ is self-adjoint in the sense of inner product a •, • .Similarly, we can prove that T h : X h → X h is self-adjoint in the sense of inner product a •, • .From 2, 4 , we get X γ → L 2 Ω 3 compactly imbedded .Hence, we derive that operator T : X γ → X γ is completely continuous.Obviously, T h : X h → X h is a finite-rank operator.
By 3, 26 , we know that 2.8 has the following equivalent operator form: T u μu.

2.16
Denote Then, the eigenvalues of 2.8 are sorted as We can construct a complete orthogonal system of X γ by using the eigenfunctions corresponding to {λ k }: Equation 2.10 has the following equivalent operator form: Then the eigenvalues of 2.10 are sorted as Let M μ k be the space spanned by all eigenfunctions corresponding to μ k of T , and let M h μ k be the space spanned by all eigenfunctions corresponding to all eigenvalues of T h that converge to

The Filter Approach
Let λ h , u h be an eigenpair of 2.10 , we know that some of these eigenvalues are "real," but some are spurious namely, not divergence free .We should filter out the spurious pairs to obtain "real" eigenpairs.Hence, ones designed a filter ratio: The corresponding value of filter ratio is small for "real" pairs since the divergence part of the eigenvector is small, whereas it is large for spurious ones since the curl part small.Noting Mathematical Problems in Engineering that when a multiple eigenvalue is dealt with, an additional step must be carried out see 3, 5 .
Next we introduce error estimates for the filter approach.

2.23
Let P h : X γ → X h be orthogonal projection, namely, Then, T h P h T .Using the spectral theory see 26 , 3 discussed error estimates for the filter approach and gave the following lemmas.

2.26
For any u k ∈ M λ k , there exists where C 1 , C 2 , and C 3 are constants independent of mesh diameter.
In this paper, we will use the following lemma.

2.28
Proof.The proof is completed by using the same proof steps as that of Lemma 9.1 in 26 .

Two-Grid Discretization Scheme and Multigrid Discretization Scheme
Consider 2.19 on X h inner product a •, • and norm • X γ .We will discuss the high efficiency of two-grid discretization scheme and multigrid discretization scheme next.
Lemma 3.2.Let μ 0 , u 0 be an approximation for μ k , u k , where μ 0 is not an eigenvalue of T h , and where ρ min μ j / μ k |μ j − μ k | is the separation constant of the eigenvalue μ k .

Two-Grid Discretization Scheme
Reference 16 established the two-grid discretization scheme based on shifted-inverse power method.Next, we will apply the scheme to eigenvalue problem of electric field.
Let π H and π h be regular meshes see 3 with diameters H and h, respectively.Let and δ be a properly small positive number.
Step 1. Solve 2.8 on a coarse grid π H : Find λ H , u H ∈ R × X H , such that u H X γ 1, and And obtain the "real" eigenpair λ k,H , u k,H by filtering process.

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Step 2. Solve a linear system on a fine grid π h : Find u ∈ X h , such that And set u h k u/ u X γ .
Step 3. Compute the Rayleigh quotient

3.6
We use λ h k , u h k as the approximate eigenpair of 2.1 .
Theorem 3.3.Suppose that H is properly small.Let λ h k , u h k be the approximate eigenpair obtained by Scheme 1. Then there exists eigenpair λ k , u k of 2.1 , such that where C 4 , C 5 , and C 6 are positive constants independent of mesh diameters, and these constants are decided by 3.11 , 3.13 , and 3.30 in the following proof.
Proof.We use Lemma 3.2 to complete the proof.Select

3.11
Mathematical Problems in Engineering 9 Combining the above two inequalities with 2.25 and noting that δ h λ k is a small quantity of higher order than δ H λ k , we obtain max From Lemma 2.1, we know that T h − T X γ → 0 h → 0 , then there exists a constant C 5 independent of h, such that Obviously, there exists Then, we derive

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Combining the triangle inequality, 2.27 and 3.16 , we deduce

3.17
Since H is small enough and δ h λ k δ H λ k t 2 , from 2.26 and 3.17 , we know For j / k, k 1, . . ., k q − 1, since H is small enough, ρ is the separation constant, we have

3.19
From the Step 2 in Scheme 1 and 2.14 , we get From the arguments of 3.9 , 3.12 , 3.18 , 3.19 , and 3.22 , we see that the conditions of Lemma 3.2 hold.Hence, substituting 3.11 and 3.17 into 3.3 , we obtain

3.24
Mathematical Problems in Engineering 11 Set From 3.23 , we directly get

3.26
From Lemma 2.2, we know that there exist {u 0 j }

3.28
Besides, by 2.26 , we easily know which together with 3.28 and 2.25 leads to 3.7 .
From the continuous embedding of X γ into L 2 Ω n , we get that there exists a constant C 6 independent of meshes, such that λ k .Therefore, when h is small enough, we have

3.31
The proof of Theorem 3.3 is completed.
Let σ N Δ be the smallest singular exponent in the Laplace problem with homogenous Neumann boundary condition, then Corollary 3.4.Suppose that H is properly small.Let λ h k , u h k be an approximate eigenpair obtained by Scheme 1. Then there exists an eigenpair λ k , u k of 2.1 , such that when Ω is a convex domain, when Ω is a nonconvex domain, where C and C are stated in the proof as follows.
Proof.From 1, 4 , we know that when Ω is a convex domain, there exists a constant C independent of h, such that

3.36
Substituting the above inequality into 3.7 , and noting that δ h λ k is an infinitesimal of lower order comparing with δ H λ k 3 , we know that 3.32 is valid.
And when Ω is a nonconvex domain, there exists a constant C independent of h, such that where μ ∈ 0, τ .Substituting the above inequality into 3.7 , we know that 3.34 is valid.

Multigrid Discretization Scheme
Next, we will discuss finite-element multigrid discretization scheme based on Rayleigh quotient iteration method.Assume that partition satisfies the following condition.
. ., and δ is a properly small positive number.
Let {X h i } l 1 be the finite-element spaces defined on Scheme 2. Multigrid Discretization.
Step 1. Solve 2.8 on a coarse grid π

3.38
And obtain the "real" eigenpair λ k,H , u k,H by filtering process.
Step 3. Solve a linear system on a fine grid π h i : Find u ∈ X hi , such that

3.39
Set Step 4. Compute the Rayleigh quotient

3.40
Step 5.If i l, then output λ h l k , u h l k , stop.Else, i ⇐ i 1, and return to Step 3.
We use λ h l k , u h l k obtained by Scheme 2 as the approximate eigenpair of 2.1 .Next, we will discuss the efficiency of Scheme 2.

Theorem 3.5. Suppose that H is properly small and Condition (A) holds. Let λ h l
k , u h l k be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair λ k , u k of 2.1 , such that

3.42
Proof.We use induction to complete the proof of 3.41 .For l 2, Scheme 2 is actually Scheme 1.Hence, 3.41 is easily obtained from 3.28 .Suppose that 3.41 holds for l 3, 4, . . ., l − 1. Next, we shall prove that 3.41 holds for l. Select . Using the proof method of Theorem 3.3, we deduce

3.43
Using the triangle inequality and 2.27 , we get and together with the induction assumption, yields

3.45
From Step 3 of Scheme 2, we know that u h l k satisfies 1

3.46
From the above arguments, we know that the conditions of Lemma 3.2 hold.Define u * and u k as those in Theorem 3.3 using u h l k instead of u h k , u j,h l instead of u j,h , then

3.47
where u j,h l − u 0 j satisfies 2.26 .We can derive by Lemma 3.2 and the proof of 3.11 that

3.48
Substituting 3.44 into the above inequality, we deduce

3.49
Like the proof method of 3.27 , we get

3.50
From the above two inequalities, we obtain

3.51
There exists a constant C 5 independent of h l such that

3.52
Like the proof method of 3.16 , we can derive

3.53
Combining 3.51 and 3.53 , we know that 3.41 is valid.Like the proof method of 3.8 , we get 3.42 , namely, Theorem 3.5 is valid.
Corollary 3.6.Suppose that Condition (A) holds and h 1 (namely, H) is properly small.Let λ h l k , u h l k be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair λ k , u k of 2.1 , such that the following error estimates hold: when Ω is a convex domain,

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when Ω is a nonconvex domain, where the C and C are the ones in Corollary 3.4.

Interpolation Postprocessing Technique
In this section, we apply interpolation postprocessing technique to the filter approach for eigenvalue problem of electric field.Let π 2h be a regular simplex mesh of Ω.When n 2, the mesh π h is obtained by dividing each element of the mesh π 2h into four congruent triangular elements; when n 3, the mesh π h is obtained by connecting the midpoints on each edge of the tetrahedral element, which divides each element of tetrahedralization π 2h into eight tetrahedral elements.
Let I h : C 0 Ω n → X h with k 1 be a piecewise linear node interpolation operator on π h .Let I 2 2h : C 0 Ω n → X 2h with k 2 be a piecewise quadratic node interpolation operator on π 2h by using the corners of the mesh π h as interpolation nodes.

Scheme 3. Interpolation Postprocessing Technique.
Step 1. Use linear finite-element filter approach to solve the problem 2.1 on the mesh π h , and obtain the "real" eigenpair λ k,h , u k,h .
Step 2. On π 2h , use the value of the function u k,h on the corners of the mesh π h as interpolation conditions to construct a piecewise quadratic interpolation I 2 2h u k,h .
Step 3. Compute the Rayleigh quotient: Here, λ r k,h , I 2 2h u k,h is the eigenpair corrected.
We develop the work in 18 to get the following theorem.
by triangle inequality and interpolation error estimate, we get In Section 5, we will verify this theorem by the numerical experiments.
Here the weight is ω d γ d x 2 1 x 2 2 .In the numerical experiments, when Ω is the L-shaped domain, let γ 0.5 or 0.95; when Ω is the square domain, we choose γ 0. And we use the numerical integral formula with accuracy of order 2 in our experiments.
From the following tables, we know that these three schemes are reliable for solving Maxwell eigenvalue problems.In addition, the accuracy of solutions is improved highly by these schemes.Example 5.1.Solve problem 2.1 on the L-shaped domain −1, 0 × −1, 0 ∪ −1, 1 × 0, 1 by using Scheme 1 with quadratic finite element.The eigenvalues obtained by Scheme 1 can be seen in 27 .
To the square domain, eigenfunctions are smooth enough.And from Table 4, we see that λ r k,h obtained by interpolation postprocessing technique achieve the accuracy order of quadratic finite element.To the L-shape domain, eigenfunctions are not smooth enough, generally.For example, the first eigenfunction has a strong singularity to L-shape domain  5 and 6 show that the accuracy of λ r 3,h is improved obviously, and the improvement of λ r 1,h is not obvious.
Remark 5.4.Wang established two-grid discretization scheme of finite-element parameterized approach for eigenvalue problem of electric field see 29 .And she also proved error estimates of the Scheme.It will still be meaningful to extend the multigrid discretization scheme and the interpolation postprocessing technique discussed in our paper to parameterized approach.

Theorem 4 . 1 .
Let λ r k,h , I 2 2h u k,h be an approximate eigenpair obtained by Scheme 3. Assume that M λ k ⊂ H 2 α Ω and there exists an

Table 1 :
The results on square by Scheme 2 for eigenvalue problem of electric field γ 0 : Set h 1 H

Table 2 :
The results on L-shape domain by Scheme 2 for eigenvalue problem of electric field γ 0.5 : Set

Table 3 :
The results on L-shape domain by Scheme 2 for eigenvalue problem of electric field γ 0.95 : Set 5 namely, 4.2 is valid.Combining 2.28 and 4.2 , we know that 4.3 is valid.Remark 4.2.Generally, to 2nd-order elliptic eigenvalue problems, condition I h u k − u k,h H 1 Ω ≤ Ch 1 α is valid see 18-25 .But to eigenvalue problems of electric field, it is very difficult to prove that

Table 4 :
The results on square by Scheme 3 for eigenvalue problem of electric field γ 0 .

Table 5 :
The results on L-shape domain by Scheme 3 for eigenvalue problem of electric field γ 0.5 .

Table 6 :
The results on L-shape domain by Scheme 3 for eigenvalue problem of electric field γ 0.95 .