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

Abstract

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.

1. 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. Two-grid discretization was first introduced by Xu for nonsymmetric and nonlinear elliptic problems, and so forth (see [8–10]). Later on, it was successfully applied to Stokes equations, semilinear eigenvalue problems and linear eigenvalue problems, and so forth (see [11–16]). Recently, Yang and Bi [16] established two-grid finite-element discretization and multigrid discretization Schemes based on shifted-inverse power method. Referecnces [6, 7] applied the two Schemes to the mixed approach for eigenvalue problem of electric field, and [17] applied them to conforming finite element for the Steklov eigenvalue problem. Based on the work mentioned above, this paper discusses two-grid discretization and multigrid discretization Schemes of the filter approach for eigenvalue problem of electric field and analyzes error estimates. They are extensions of Scheme  2 and Scheme  3 in [16], respectively.

From 1989 to 1991, Lin and Yang firstly pointed out and proved that the function, obtained by using nodes of lower-order element as interpolation nodes to make a higher order interpolation of lower order finite-element solutions, can have global gradient superconvergence. The technique used to obtain global superconvergence was called finite-element interpolation postprocessing or finite-element interpolation correction (see reviews paper [18] and the references cited therein). For over 20 years, finite element interpolation postprocessing technique has been developed greatly and was applied to a variety of partial differential equations (see [19–25]). It is applied to this paper too. We give Theorem 4.1, and our numerical results show that interpolation postprocessing is an efficient and significant method for solving eigenvalue problems of electric field.

The rest of this paper is organized as follows. In the next section, some preliminaries which are needed are provided. In Section 3, two kinds of finite-element discretization schemes for eigenvalue problem of electric field are given and the error estimates are established. In Section 4, we introduce interpolation postprocessing technique. Finally, numerical experiments are presented.

2. Preliminaries

Let Ω⊂𝑅^𝑛  (𝑛=2,3) be a bounded polyhedron domain with boundary 𝜕Ω. We denote by 𝐧 the unit outward normal vector to 𝜕Ω, by 𝐮 the electric field, and by 𝜔 the time frequency. Let 𝑐=3.0×10⁸ m/s be the light velocity in vacuum, 𝐜𝐮𝐫𝐥 curl operator, and div divergence operator.

Consider the following eigenvalue problem of electric field: 𝑐²𝐜𝐮𝐫𝐥𝐜𝐮𝐫𝐥𝐮=𝜔²𝐮,inΩ,div𝐮=0,inΩ,𝐮×𝐧=0,on∈𝜕Ω. (2.1) Let 𝜆=𝜔²/𝑐² named eigenvalue.

Define function spaces as follows: 𝐻(𝐜𝐮𝐫𝐥,Ω)=𝐯∈𝐿₂(Ω)^𝑛∶𝐜𝐮𝐫𝐥𝐯∈𝐿₂(Ω)^𝑛,𝐻₀(𝐜𝐮𝐫𝐥,Ω)=𝐯∈𝐻(𝐜𝐮𝐫𝐥,Ω)∶𝐯×𝐧|_𝜕Ω=0. (2.2) When Ω is a convex polyhedron, we define the following function space: 𝑋=𝐯∈𝐻₀(𝐜𝐮𝐫𝐥,Ω)∶div𝐯∈𝐿²(Ω). (2.3) Denote 𝑏(𝐮,𝐯)=(𝐮,𝐯)₀=_Ω𝐮⋅𝐯d𝑥,‖𝐮‖₀=(𝐮,𝐮)₀^1/2,𝑎(𝐮,𝐯)=(𝐮,𝐯)_𝑋=(𝐜𝐮𝐫𝐥𝐮,𝐜𝐮𝐫𝐥𝐯)₀+(div𝐮,div𝐯)₀,‖𝐮‖_𝑋=(𝐮,𝐮)_𝑋^1/2. (2.4) Let 𝜎^𝐷_Δ∈(3/2,2) be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition: 𝜙∈𝐻¹(Ω)∶Δ𝜙∈𝐿₂(Ω),𝜓_𝜕Ω=0⊂∩_𝑠<𝜎^𝐷_Δ𝐻^𝑠(Ω),𝜙∈𝐻¹(Ω)∶Δ𝜙∈𝐿₂(Ω),𝜓_𝜕Ω=0⊄𝐻^𝜎𝐷Δ(Ω). (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∈𝐸𝑒). 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: 𝐿²_𝛾(Ω)=𝑤∈𝐿²_loc(Ω)∶𝜔_𝛾𝑤∈𝐿₂(Ω,𝑋_𝛾)=𝐯∈𝐿₂(Ω)^𝑛∶𝐜𝐮𝐫𝐥𝐯∈𝐿₂(Ω)^𝑛,𝐯×𝐧|_𝜕Ω=0,div𝐯∈𝐿²_𝛾(Ω). (2.6) Denote (𝑤,𝑣)_𝐿²_𝛾=_Ω𝜔²_𝛾𝑤𝑣d𝑥,‖𝑤‖_0,𝛾=(𝑤,𝑤)_𝐿^1/2_2𝛾,𝑎(𝐮,𝐯)=(𝐮,𝐯)_𝑋𝛾=(𝐜𝐮𝐫𝐥𝐮,𝐜𝐮𝐫𝐥𝐯)₀+(div𝐮,div𝐯)_𝐿²_𝛾,‖𝐮‖_𝑋𝛾=(𝐮,𝐮)_𝑋^1/2_𝛾. (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‖_𝑋𝛾=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: 𝑋_ℎ=𝐯∈𝐶⁰Ω^𝑛∶𝐯×𝐧_𝜕Ω=0,𝐯_𝜅∈𝑃_𝑘(𝜅)^𝑛,∀𝜅∈𝜋_ℎ. (2.9) Then, 𝑋_ℎ⊆𝑋_𝛾.

The discrete variational form of (2.8): Find (𝜆_ℎ,𝐮_ℎ)∈𝑅⁺×𝑋_ℎ with ‖𝐮_ℎ‖_𝑋𝛾=1, such that 𝑎𝐮_ℎ,𝐯=𝜆_ℎ𝑏𝐮_ℎ,𝐯,∀𝐯∈𝑋_ℎ. (2.10) 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 |𝑎(𝐰,𝐯)|=_Ω𝐜𝐮𝐫𝐥𝐰⋅𝐜𝐮𝐫𝐥𝐯d𝑥+_Ω𝜔²_𝛾≤‖‖div𝐰div𝐯d𝑥𝐜𝐮𝐫𝐥𝐰₀‖𝐜𝐮𝐫𝐥𝐯‖₀+‖div𝐰‖_0,𝛾‖div𝐯‖_0,𝛾≤‖𝐜𝐮𝐫𝐥𝐰‖²₀+‖div𝐰‖²_0,𝛾‖𝐜𝐮𝐫𝐥𝐯‖²₀+‖div𝐯‖²_0,𝛾≤‖𝐰‖_𝑋𝛾‖𝐯‖_𝑋𝛾. (2.11) Therefore, continuity of 𝑎(⋅,⋅) is valid. And 𝑎(𝐰,𝐰)=‖𝐰‖²_𝑋𝛾, (2.12) which indicates that 𝑎(⋅,⋅) is 𝑉-elliptic.

Define operator 𝑇∶𝑋_𝛾→𝑋_𝛾 satisfying 𝑎(𝑇𝐟,𝐯)=𝑏(𝐟,𝐯),∀𝐯∈𝑋_𝛾. (2.13) Define operator 𝑇_ℎ∶𝑋_ℎ→𝑋_ℎ satisfying 𝑎𝑇_ℎ𝐟,𝐯=𝑏(𝐟,𝐯),∀𝐯∈𝑋_ℎ. (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 𝑎(𝐟,𝑇𝐠)=𝑎(𝑇𝐠,𝐟)=𝑏(𝐠,𝐟)=𝑏(𝐟,𝐠)=𝑎(𝑇𝐟,𝐠), (2.15) 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↪𝐿₂(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: 𝑇𝐮=𝜇𝐮. (2.16) Denote 𝜇_𝑘=1/𝜆_𝑘, 𝜇_𝑘,ℎ=1/𝜆_𝑘,ℎ.

Then, the eigenvalues of (2.8) are sorted as 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑘≤⋯↗+∞. (2.17) We can construct a complete orthogonal system of 𝑋_𝛾 by using the eigenfunctions corresponding to {𝜆_𝑘}: 𝐮₁,𝐮₂,…,𝐮_𝑘,…. (2.18) Equation (2.10) has the following equivalent operator form: 𝜆_ℎ𝑇_ℎ𝐮_ℎ=𝐮_ℎ. (2.19) Then the eigenvalues of (2.10) are sorted as 0<𝜆_1,ℎ≤𝜆_2,ℎ≤⋯≤𝜆_𝑘,ℎ≤⋯𝜆_𝑁ℎ,ℎ, (2.20) and the corresponding eigenfunctions are 𝐮_1,ℎ,𝐮_2,ℎ,…,𝐮_𝑘,ℎ,…,𝐮_𝑁ℎ,ℎ, (2.21) where 𝑁_ℎ=dim𝑋_ℎ.

In this paper, 𝜇_𝑘 and 𝜇_𝑘,ℎ, 𝜆_𝑘 and 𝜆_𝑘,ℎ are all called eigenvalues.

Suppose that the algebraic multiplicity of 𝜇_𝑘 is equal to 𝑞. 𝜇_𝑘=𝜇_𝑘+1=⋯=𝜇_{𝑘+𝑞−1}. 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 𝜇_𝑘. Let 𝑀(𝜇_𝑘)={𝑣∶𝑣∈𝑀(𝜇_𝑘),‖𝑣‖_𝑋𝛾=1}, 𝑀_ℎ(𝜇_𝑘)={𝑣∶𝑣∈𝑀_ℎ(𝜇_𝑘),‖𝑣‖_𝑋𝛾=1}. We also write 𝑀(𝜆_𝑘)=𝑀(𝜇_𝑘), 𝑀_ℎ(𝜆_𝑘)=𝑀_ℎ(𝜇_𝑘), 𝑀(𝜆_𝑘)=𝑀(𝜇_𝑘), and 𝑀_ℎ(𝜆_𝑘)=𝑀_ℎ(𝜇_𝑘).

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: div𝐮_ℎ0,𝛾curl𝐮_ℎ0. (2.22) 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 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.

Define ‖(𝑇−𝑇_ℎ)|_{𝑀(𝜆𝑘)}‖_𝑋𝛾=max_{𝐮∈𝑀(𝜆𝑘)}(‖(𝑇−𝑇_ℎ)𝐮‖_𝑋𝛾/‖𝐮‖_𝑋𝛾).

Denote 𝛿_ℎ𝜆_𝑘=sup_𝐮∈𝑀_(𝜆𝑘)inf_{𝐯∈𝑋ℎ}‖𝐮−𝐯‖_𝑋𝛾. (2.23) Let 𝑃_ℎ∶𝑋_𝛾→𝑋_ℎ be orthogonal projection, namely, 𝑎𝐮−𝑃_ℎ𝐮,𝐯=0,∀𝐯∈𝑋_ℎ. (2.24) Then, 𝑇_ℎ=𝑃_ℎ𝑇.

Using the spectral theory (see [26]), [3] discussed error estimates for the filter approach and gave the following lemmas.

Lemma 2.1. ‖𝑇_ℎ−𝑇‖_𝑋𝛾→0  (ℎ→0).

Lemma 2.2. Let (𝜆_𝑘,ℎ,𝐮_𝑘,ℎ) be the 𝑘th eigenpair of (2.10) with ‖𝐮_𝑘,ℎ‖_𝑋𝛾=1. Let 𝜆_𝑘 be the 𝑘th eigenvalue of (2.8). Then, there exists 𝐮_𝑘∈𝑀(𝜆_𝑘) such that 𝜆_𝑘,ℎ−𝜆_𝑘≤𝐶₁𝛿²_ℎ𝜆_𝑘,𝐮_𝑘,ℎ−𝐮_𝑘𝑋𝛾≤𝐶₂𝛿_ℎ𝜆_𝑘. (2.25)(2.26) For any 𝐮_𝑘∈𝑀(𝜆_𝑘), there exists 𝐮_ℎ∈𝑀_ℎ(𝜆_𝑘) such that 𝐮_ℎ−𝐮_𝑘𝑋𝛾≤𝐶₃𝛿_ℎ𝜆_𝑘, (2.27) where 𝐶₁, 𝐶₂, and 𝐶₃ are constants independent of mesh diameter.

In this paper, we will use the following lemma.

Lemma 2.3. Let (𝜆,𝐮) be an eigenpair of (2.8), then for any 𝐰∈𝑋_𝛾, ‖𝐰‖₀≠0, the Rayleigh quotient 𝑎(𝐰,𝐰)/‖𝐰‖²₀ satisfies 𝑎(𝐰,𝐰)‖𝐰‖²₀‖‖−𝜆=𝐰−𝐮²_𝑋𝛾‖𝐰‖²₀‖‖−𝜆𝐰−𝐮²₀‖𝐰‖²₀. (2.28)

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

3. Two-Grid Discretization Scheme and Multigrid Discretization Scheme

Consider (2.19) on 𝑋_ℎ (inner product 𝑎(⋅,⋅) and norm ‖⋅‖_𝑋𝛾). We will discuss the high efficiency of two-grid discretization scheme and multigrid discretization scheme next.

Lemma 3.1. For all nonzero 𝐮,𝐯∈𝑋_𝛾, 𝐮‖𝐮‖_𝑋𝛾−𝐯‖𝐯‖_{𝑋𝛾𝑋𝛾}‖‖≤2𝐮−𝐯_𝑋𝛾‖𝐮‖_𝑋𝛾,𝐮‖𝐮‖_𝑋𝛾−𝐯‖𝐯‖_{𝑋𝛾𝑋𝛾}‖‖≤2𝐮−𝐯_𝑋𝛾‖𝐯‖_𝑋𝛾. (3.1)

Proof. See [16].

Lemma 3.2. Let (𝜇₀,𝐮₀) be an approximation for (𝜇_𝑘,𝐮_𝑘), where 𝜇₀ is not an eigenvalue of 𝑇_ℎ, and 𝐮₀∈𝑋_ℎ with ‖𝐮₀‖_𝑋𝛾=1. Suppose that max_{𝑘≤𝑗≤𝑘+𝑞−1}|(𝜇_𝑗,ℎ−𝜇_𝑘,ℎ)/(𝜇₀−𝜇_𝑗,ℎ)| ≤ 1/2, dist(𝐮₀,𝑀_ℎ(𝜇_𝑘)) ≤ 1/2,  |𝜇₀−𝜇_𝑗,ℎ| ≥ 𝜌/2,  (𝑗≠𝑘,𝑘+1,…,𝑘+𝑞−1), and 𝐮∈𝑋_ℎ,  𝐮^ℎ_𝑘∈𝑋_ℎ satisfy 𝜇₀−𝑇_ℎ𝐮=𝐮₀,𝐮^ℎ_𝑘=𝐮‖𝐮‖_𝑋𝛾. (3.2) Then dist𝐮^ℎ_𝑘,𝑀_ℎ𝜇_𝑘≤16𝜌𝜇₀−𝜇_𝑘,ℎdist𝐮₀,𝑀_ℎ𝜇_𝑘, (3.3) where 𝜌=min_{𝜇𝑗≠𝜇𝑘}|𝜇_𝑗−𝜇_𝑘| is the separation constant of the eigenvalue 𝜇_𝑘.

Proof. See [16].

3.1. 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 𝛿_ℎ(𝜆_𝑘)=𝛿_𝐻(𝜆_𝑘)^𝑡2, 𝑡₂∈[1+𝛿,3−𝛿], and 𝛿 be a properly small positive number.

Scheme 1. Two-grid Discretization.

Step 1. Solve (2.8) on a coarse grid 𝜋_𝐻: Find (𝜆_𝐻,𝐮_𝐻)∈𝑅⁺×𝑋_𝐻, such that ‖𝐮_𝐻‖_𝑋𝛾=1, and 𝑎𝐮_𝐻,𝐯=𝜆_𝐻𝑏𝐮_𝐻,𝐯,∀𝐯∈𝑋_𝐻. (3.4) And obtain the “real” eigenpair (𝜆_𝑘,𝐻,𝐮_𝑘,𝐻) by filtering process.

Step 2. Solve a linear system on a fine grid 𝜋_ℎ: Find 𝐮∈𝑋_ℎ, such that 𝑎(𝐮,𝐯)−𝜆_𝑘,𝐻𝑏(𝐮,𝐯)=𝑏𝐮_𝑘,𝐻,𝐯,∀𝐯∈𝑋_ℎ. (3.5) And set 𝐮^ℎ_𝑘=𝐮/‖𝐮‖_𝑋𝛾.

Step 3. Compute the Rayleigh quotient 𝜆^ℎ_𝑘=𝑎𝐮^ℎ_𝑘,𝐮^ℎ_𝑘𝑏𝐮^ℎ_𝑘,𝐮^ℎ_𝑘. (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 𝐮^ℎ_𝑘−𝐮_𝑘𝑋𝛾≤48𝜌𝐶₁𝐶₂𝐶₄𝐶₅𝛿³_𝐻𝜆_𝑘+3𝐶₂𝑞𝛿_ℎ𝜆_𝑘,𝜆^ℎ_𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾, (3.7)(3.8) where 𝐶₄, 𝐶₅, and 𝐶₆ 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 𝜇₀=1/𝜆_𝑘,𝐻 and 𝐮₀ = 𝜆_𝑘,𝐻𝑇_ℎ𝐮_𝑘,𝐻/‖𝜆_𝑘,𝐻𝑇_ℎ𝐮_𝑘,𝐻‖_𝑋𝛾. Obviously, 𝐮_0𝑋𝛾=1. (3.9) Noting that 𝜆_𝑘=𝜆_𝑘+1=⋯𝜆_{𝑘+𝑞−1}, for 𝑗=𝑘,𝑘+1,…,𝑘+𝑞−1, we have 𝜇_𝑗,ℎ−𝜇_𝑘,ℎ=𝜆_𝑘,ℎ−𝜆_𝑗,ℎ𝜆_𝑘,ℎ𝜆_𝑗,ℎ=𝜆_𝑘,ℎ−𝜆_𝑘+𝜆_𝑗−𝜆_𝑗,ℎ𝜆_𝑘,ℎ𝜆_𝑗,ℎ≤𝜆_𝑘,ℎ−𝜆_𝑘+𝜆_𝑗−𝜆_𝑗,ℎ𝜆_𝑘,ℎ𝜆_𝑗,ℎ,𝜇₀−𝜇_𝑗,ℎ=𝜆_𝑗,ℎ−𝜆_𝑘,𝐻𝜆_𝑗,ℎ𝜆_𝑘,𝐻=𝜆_𝑗,ℎ−𝜆_𝑗+𝜆_𝑘−𝜆_𝑘,𝐻𝜆_𝑗,ℎ𝜆_𝑘,𝐻≤𝐶₄𝜆_𝑘,𝐻−𝜆_𝑘. (3.10)(3.11) 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_{𝑘≤𝑗≤𝑘+𝑞−1}𝜇_𝑗,ℎ−𝜇_𝑘,ℎ𝜇₀−𝜇_𝑗,ℎ≤12. (3.12) From Lemma 2.1, we know that ‖𝑇_ℎ−𝑇‖_𝑋𝛾→0(ℎ→0), then there exists a constant 𝐶₅ independent of ℎ, such that 𝑇_ℎ𝐟_𝑋𝛾≤𝐶₅‖𝐟‖_𝑋𝛾,∀𝐟∈𝑋_ℎ. (3.13) Obviously, there exists 𝐮^𝑘∈𝑀(𝜆_𝑘), such that 𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾=dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘. (3.14) Then, we derive 𝜆_𝑘,𝐻𝑇_ℎ𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾=𝜆_𝑘,𝐻𝑇_ℎ𝐮_𝑘,𝐻−𝜆_𝑘𝑇𝐮^𝑘_𝑋𝛾≤(𝜆_𝑘,𝐻−𝜆_𝑘)𝑇_ℎ𝐮_{𝑘,𝐻𝑋𝛾}+𝜆_𝑘𝑇_ℎ(𝐮_𝑘,𝐻−𝐮^𝑘)_𝑋𝛾+𝜆_𝑘(𝑇_ℎ−𝑇)𝐮^𝑘_𝑋𝛾≤𝐶₅𝜆_𝑘,𝐻−𝜆_𝑘+𝐶₅𝜆_𝑘𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾+(𝑃_ℎ−𝐼)𝐮^𝑘_𝑋𝛾≤𝐶₅𝜆_𝑘,𝐻−𝜆_𝑘+𝐶₅𝜆_𝑘𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾+𝛿_ℎ𝜆_𝑘. (3.15) Hence, by Lemma 3.1, (3.15) and (2.28), we have dist𝐮₀,𝑀𝜆_𝑘≤𝐮₀−𝐮^𝑘_𝑋𝛾≤2𝜆_𝑘,𝐻𝑇_ℎ𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾≤2𝐶₅𝜆_𝑘,𝐻−𝜆_𝑘+2𝐶₅𝜆_𝑘𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾+2𝛿_ℎ𝜆_𝑘≤3𝐶₅𝜆_𝑘𝐮_𝑘,𝐻−𝐮^𝑘_𝑋𝛾+2𝛿_ℎ𝜆_𝑘=3𝐶₅𝜆_𝑘dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘+2𝛿_ℎ𝜆_𝑘. (3.16) Combining the triangle inequality, (2.27) and (3.16), we deduce dist𝐮₀,𝑀_ℎ𝜆_𝑘≤dist𝐮₀,𝑀𝜆_𝑘+𝐶₃𝛿_ℎ𝜆_𝑘≤3𝐶₅𝜆_𝑘dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘+𝐶₃+2𝛿_ℎ𝜆_𝑘. (3.17) Since 𝐻 is small enough and 𝛿_ℎ(𝜆_𝑘)=𝛿_𝐻(𝜆_𝑘)^𝑡2, from (2.26) and (3.17), we know dist𝐮₀,𝑀_ℎ𝜆_𝑘≤12. (3.18) For 𝑗≠𝑘,  𝑘+1,…,𝑘+𝑞−1, since 𝐻 is small enough, 𝜌 is the separation constant, we have 𝜇₀−𝜇_𝑗,ℎ≥𝜌2. (3.19) From the Step 2 in Scheme 1 and (2.14), we get 𝑎(𝐮,𝐯)−𝜆_𝑘,𝐻𝑎𝑇_ℎ𝐮,𝐯=𝑎𝑇_ℎ𝐮_𝑘,𝐻,𝐯, (3.20) namely, 𝐮−𝜆_𝑘,𝐻𝑇_ℎ𝐮=𝑇_ℎ𝐮_𝑘,𝐻. (3.21) Thus (1/𝜆_𝑘,𝐻)𝐮−𝑇_ℎ𝐮=(1/𝜆_𝑘,𝐻)𝑇_ℎ𝐮_𝑘,𝐻 and 𝐮^ℎ_𝑘=𝐮/‖𝐮‖_𝑋𝛾. Note that (1/𝜆_𝑘,𝐻)𝑇_ℎ𝐮_𝑘,𝐻 = ‖(1/𝜆_𝑘,𝐻)𝑇_ℎ𝐮_𝑘,𝐻‖_𝑋𝛾𝐮₀ differs from 𝐮₀ by only a constant; then Step 2 is equivalent to 1𝜆_𝑘,𝐻−𝑇_ℎ𝐮=𝐮₀,𝐮^ℎ_𝑘=𝐮‖𝐮‖_𝑋𝛾. (3.22) 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 dist𝐮^ℎ_𝑘,𝑀_ℎ𝜆_𝑘≤16𝜌𝐶₄𝜆_𝑘,𝐻−𝜆_𝑘3𝐶₅𝜆_𝑘dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘+𝐶₃+2𝛿_ℎ𝜆_𝑘. (3.23) Let eigenvectors {𝐮_𝑗,ℎ}_𝑘^{𝑘+𝑞−1} be an orthogonal basis of 𝑀_ℎ(𝜆_𝑘) (in the sense of inner product 𝑎(⋅,⋅)), then dist𝐮^ℎ_𝑘,𝑀_ℎ𝜆_𝑘=𝐮^ℎ_𝑘−^{𝑘+𝑞−1}_𝑗=𝑘𝑎(𝐮^ℎ_𝑘,𝐮_𝑗,ℎ)𝐮_{𝑗,ℎ𝑋𝛾}. (3.24) Set 𝐮^∗=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝐮^ℎ_𝑘,𝐮_𝑗,ℎ𝐮_𝑗,ℎ. (3.25) From (3.23), we directly get 𝐮^ℎ_𝑘−𝐮^∗_𝑋𝛾=dist𝐮^ℎ_𝑘,𝑀_ℎ𝜆_𝑘≤16𝜌𝐶₄𝜆_𝑘,𝐻−𝜆_𝑘3𝐶₅𝜆_𝑘dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘+𝐶₃+2𝛿_ℎ𝜆_𝑘. (3.26) From Lemma 2.2, we know that there exist {𝐮⁰_𝑗}_𝑘^{𝑘+𝑞−1}⊂𝑀(𝜆_𝑘) making 𝐮_𝑗,ℎ−𝐮⁰_𝑗 satisfy (2.26). Let 𝐮_𝑘=∑^{𝑘+𝑞−1}_𝑗=𝑘𝑎(𝐮^ℎ_𝑘,𝐮_𝑗,ℎ)𝐮⁰_𝑗, then 𝐮^∗−𝐮_𝑘𝑋𝛾=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝐮^ℎ_𝑘,𝐮_𝑗,ℎ𝐮_𝑗,ℎ−𝐮⁰_𝑗𝑋𝛾≤^{𝑘+𝑞−1}_𝑗=𝑘𝐮_𝑗,ℎ−𝐮⁰_𝑗²_𝑋𝛾^1/2≤𝐶₂^{𝑘+𝑞−1}_𝑗=𝑘𝛿_ℎ𝜆_𝑗≤𝐶₂𝑞𝛿_ℎ𝜆_𝑘. (3.27) Combining (3.26) and (3.27), we obtain 𝐮^ℎ_𝑘−𝐮_𝑘𝑋𝛾≤𝐮^ℎ_𝑘−𝐮^∗_𝑋𝛾+𝐮^∗−𝐮_𝑘𝑋𝛾≤48𝜌𝐶₄𝐶₅𝜆_𝑘𝜆_𝑘,𝐻−𝜆_𝑘dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘+3𝐶₂𝑞𝛿_ℎ𝜆_𝑘. (3.28) Besides, by (2.26), we easily know dist𝐮_𝑘,𝐻,𝑀𝜆_𝑘≤𝐶₂𝛿_𝐻𝜆_𝑘, (3.29) which together with (3.28) and (2.25) leads to (3.7).
From the continuous embedding of 𝑋_𝛾 into 𝐿₂(Ω)^𝑛, we get that there exists a constant 𝐶₆ independent of meshes, such that ‖𝐮‖²₀≤𝐶₆‖𝐮‖²_𝑋𝛾. (3.30) Equation (3.7) indicates that 𝐮^ℎ_𝑘 converges to 𝐮_𝑘 in the sense of norm ‖⋅‖_𝑋𝛾, then 𝐮^ℎ_𝑘 converges to 𝐮_𝑘 in the sense to norm ‖⋅‖₀; thus, 1/‖𝐮^ℎ_𝑘‖²₀→1/‖𝐮_𝑘‖²₀=𝜆_𝑘. Therefore, when ℎ is small enough, we have 𝜆^ℎ_𝑘−𝜆_𝑘≤𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾𝐮^ℎ_𝑘²₀+𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²₀𝐮^ℎ_𝑘²₀≤2𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾+2𝜆²_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²₀≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾. (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 𝜎^𝑁_Δ∈(3/2,2). Denote 𝜏=min(𝛾−𝛾_min,𝜎^𝑁_Δ−1).

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, 𝐮^ℎ_𝑘−𝐮_𝑘𝑋𝛾≤6𝑞𝐶₂𝐶^′𝜆ℎ,_ℎ𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾, (3.32)(3.33) when Ω is a nonconvex domain, 𝐮^ℎ_𝑘−𝐮_𝑘𝑋𝛾≤6𝑞𝐶₂𝐶^′′ℎ^𝜇(),𝜆,∀𝜇∈0,𝜏_ℎ𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ_𝑘−𝐮_𝑘²_𝑋𝛾, (3.34)(3.35) 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 𝛿_ℎ𝜆_𝑘≤𝐶^′ℎ. (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 𝛿_ℎ𝜆_𝑘≤𝐶^′′ℎ^𝜇, (3.37) where 𝜇∈(0,𝜏).
Substituting the above inequality into (3.7), we know that (3.34) is valid.

3.2. 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.

Condition (A). {𝜋_ℎ𝑖}^𝑙₁ is a family of regular meshes (see [3]) with diameters ℎ_𝑖, 𝛿_ℎ𝑖+1(𝜆_𝑘)=𝛿_ℎ𝑖(𝜆_𝑘)^𝑡𝑖+1, 𝑡_𝑖+1∈[1+𝛿,3−𝛿], 𝑖=1,2,…, and 𝛿 is a properly small positive number.

Let {𝑋_ℎ𝑖}^𝑙₁ be the finite-element spaces defined on {𝜋_ℎ𝑖}^𝑙₁. Further, let 𝜋_𝐻=𝜋_ℎ1, 𝑋_𝐻=𝑋_ℎ1.

Scheme 2. Multigrid Discretization.

Step 1. Solve (2.8) on a coarse grid 𝜋_𝐻: Find (𝜆_𝐻,𝐮_𝐻)∈𝑅⁺×𝑋_𝐻, with ‖𝐮_𝐻‖_𝑋𝛾=1, such that 𝑎𝐮_𝐻,𝐯=𝜆_𝐻𝑏𝐮_𝐻,𝐯,∀𝐯∈𝑋_𝐻. (3.38) And obtain the “real” eigenpair (𝜆_𝑘,𝐻,𝐮_𝑘,𝐻) by filtering process.

Step 2. 𝐮^ℎ1_𝑘=𝐮_𝑘,𝐻,  𝜆^ℎ1_𝑘=𝜆_𝑘,𝐻,  𝑖⇐2.

Step 3. Solve a linear system on a fine grid 𝜋_ℎ𝑖: Find 𝐮∈𝑋_ℎ𝑖, such that 𝑎(𝐮,𝐯)−𝜆^ℎ𝑖−1_𝑘𝑏(𝐮,𝐯)=𝑏𝐮^ℎ𝑖−1_𝑘,𝐯,∀𝐯∈𝑋_ℎ𝑖. (3.39) Set 𝐮^ℎ𝑖_𝑘=𝐮/‖𝐮‖_𝑋𝛾.

Step 4. Compute the Rayleigh quotient 𝜆^ℎ𝑖_𝑘=𝑎𝐮^ℎ𝑖_𝑘,𝐮^ℎ𝑖_𝑘𝑏𝐮^ℎ𝑖_𝑘,𝐮^ℎ𝑖_𝑘, (3.40)

Step 5. If 𝑖=𝑙, then output (𝜆^ℎ𝑙_𝑘,𝐮^ℎ𝑙_𝑘), stop. Else, 𝑖⇐𝑖+1, and return to Step 3.

We use (𝜆^ℎ𝑙_𝑘,𝐮^ℎ𝑙_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^ℎ𝑙_𝑘,𝐮^ℎ𝑙_𝑘) be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair (𝜆_𝑘,𝐮_𝑘) of (2.1), such that 𝐮^ℎ𝑙_𝑘−𝐮_𝑘𝑋𝛾≤48𝜌𝐶₄𝐶₅𝜆_𝑘𝜆^ℎ𝑙−1_𝑘−𝜆_𝑘dist𝐮^ℎ𝑙−1_𝑘,𝑀𝜆_𝑘+3𝐶₂𝑞𝛿_ℎ𝑙𝜆_𝑘,𝜆^ℎ𝑙_𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ𝑙_𝑘−𝐮_𝑘𝑋𝛾,𝑙≥2. (3.41)(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 𝜇₀=1/𝜆^ℎ𝑙−1_𝑘, 𝜇_𝑘,ℎ𝑙=1/𝜆_𝑘,ℎ𝑙, and 𝐮₀=𝜆^ℎ𝑙−1_𝑘𝑇_ℎ𝑙𝐮^ℎ𝑙−1_𝑘/‖𝜆^ℎ𝑙−1_𝑘𝑇_ℎ𝑙𝐮^ℎ𝑙−1_𝑘‖_𝑋𝛾. Using the proof method of Theorem 3.3, we deduce 𝜇₀−𝜇_𝑘,ℎ𝑙≤𝐶₄𝜆^ℎ𝑙−1_𝑘−𝜆_𝑘,max_{𝑘≤𝑗≤𝑘+𝑞−1}𝜇_𝑗,ℎ𝑙−𝜇_𝑘,ℎ𝑙𝜇₀−𝜇_𝑗,ℎ𝑙≤12. (3.43) Using the triangle inequality and (2.27), we get dist𝐮₀,𝑀_ℎ𝑙𝜆_𝑘≤dist𝐮₀,𝑀𝜆_𝑘+𝐶₃𝛿_ℎ𝑙𝜆_𝑘, (3.44) and together with the induction assumption, yields dist𝐮₀,𝑀_ℎ𝑙𝜆_𝑘≤12. (3.45) From Step 3 of Scheme 2, we know that 𝐮^ℎ𝑙_𝑘 satisfies 1𝜆^ℎ𝑙−1_𝑘−𝑇_ℎ𝐮=𝐮₀,𝐮^ℎ𝑙_𝑘=𝐮‖𝐮‖_𝑋𝛾. (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^ℎ𝑙_𝑘 instead of 𝐮^ℎ_𝑘, 𝐮_𝑗,ℎ𝑙 instead of 𝐮_𝑗,ℎ), then 𝐮^∗=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝐮^ℎ𝑙_𝑘,𝐮_𝑗,ℎ𝑙𝐮_𝑗,ℎ𝑙,𝐮_𝑘=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝐮^ℎ𝑙_𝑘,𝐮_𝑗,ℎ𝑙𝐮⁰_𝑗, (3.47) where 𝐮_𝑗,ℎ𝑙−𝐮⁰_𝑗 satisfies (2.26). We can derive by Lemma 3.2 and the proof of (3.11) that 𝐮^ℎ𝑙_𝑘−𝐮^∗_𝑋𝛾=dist𝐮^ℎ𝑙_𝑘,𝑀_ℎ𝑙𝜆_𝑘≤16𝜌𝜇₀−𝜇_𝑘,ℎ𝑙dist𝐮₀,𝑀_ℎ𝑙𝜆_𝑘≤16𝜌𝐶₄𝜆^ℎ𝑙−1_𝑘−𝜆_𝑘dist𝐮₀,𝑀_ℎ𝑙𝜆_𝑘. (3.48) Substituting (3.44) into the above inequality, we deduce 𝐮^ℎ𝑙_𝑘−𝐮^∗_𝑋𝛾≤16𝜌𝐶₄𝜆^ℎ𝑙−1_𝑘−𝜆_𝑘dist𝐮₀,𝑀𝜆_𝑘+𝐶₃𝛿_ℎ𝑙𝜆_𝑘. (3.49) Like the proof method of (3.27), we get 𝐮^∗−𝐮_𝑘𝑋𝛾≤𝐶₂𝑞𝛿_ℎ𝑙𝜆_𝑘. (3.50) From the above two inequalities, we obtain 𝐮^ℎ𝑙_𝑘−𝐮_𝑘𝑋𝛾≤16𝜌𝐶₄𝜆^ℎ𝑙−1_𝑘−𝜆_𝑘dist𝐮₀,𝑀𝜆_𝑘+2𝐶₂𝑞𝛿_ℎ𝑙𝜆_𝑘. (3.51) There exists a constant 𝐶₅ independent of ℎ_𝑙 such that 𝑇_ℎ𝑙𝐟_𝑋𝛾≤𝐶₅‖𝐟‖_𝑋𝛾,∀𝐟∈𝑋_ℎ𝑙. (3.52) Like the proof method of (3.16), we can derive dist𝐮₀,𝑀𝜆_𝑘≤3𝐶₅𝜆_𝑘dist𝐮^ℎ𝑙−1_𝑘,𝑀𝜆_𝑘+2𝛿_ℎ𝑙𝜆_𝑘. (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 ℎ₁ (namely, 𝐻) is properly small. Let (𝜆^ℎ𝑙_𝑘,𝐮^ℎ𝑙_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, 𝐮^ℎ𝑙_𝑘−𝐮_𝑘𝑋𝛾≤6𝑞𝐶₂𝐶^′ℎ_𝑙,𝜆^ℎ𝑙_𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ𝑙_𝑘−𝐮_𝑘²_𝑋𝛾; (3.54) when Ω is a nonconvex domain, 𝐮^ℎ𝑙_𝑘−𝐮_𝑘𝑋𝛾≤6𝑞𝐶₂𝐶^′′ℎ^𝜇_𝑙(),𝜆,∀𝜇∈0,𝜏_ℎ𝑙𝑘−𝜆_𝑘≤2𝜆_𝑘1+𝐶₆𝜆_𝑘𝐮^ℎ𝑙_𝑘−𝐮_𝑘²_𝑋𝛾, (3.55) where the 𝐶^′ and 𝐶^′′ are the ones in Corollary 3.4.

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∶𝐶⁰(Ω)^𝑛→𝑋_ℎ with 𝑘=1 be a piecewise linear node interpolation operator on 𝜋_ℎ. Let 𝐼⁽²⁾_2ℎ∶𝐶⁰(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⁾_2ℎ𝐮_𝑘,ℎ.

Step 3. Compute the Rayleigh quotient: 𝜆^𝑟_𝑘,ℎ=𝑎𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ,𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ𝑏𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ,𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ. (4.1)
Here, (𝜆^𝑟_𝑘,ℎ,𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ) is the eigenpair corrected.

We develop the work in [18] to get the following theorem.

Theorem 4.1. Let (𝜆^𝑟_𝑘,ℎ,𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ) be an approximate eigenpair obtained by Scheme 3. Assume that 𝑀(𝜆_𝑘)⊂𝐻^2+𝛼(Ω) and there exists an 𝐮_𝑘∈𝑀(𝜆_𝑘) such that ‖𝐼_ℎ𝐮_𝑘−𝐮_𝑘,ℎ‖_𝑋𝛾≤𝐶ℎ^1+𝛼, for some 𝛼>0. Then 𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ−𝐮_𝑘𝑋𝛾≤𝐶ℎ^1+𝛼,𝜆^𝑟_𝑘,ℎ−𝜆_𝑘≤𝐶ℎ^2+2𝛼. (4.2)(4.3)

Proof. From boundedness of interpolation 𝐼⁽²⁾_2ℎ, we have 𝐼⁽²⁾_2ℎ𝐮_𝑘−𝐼⁽²⁾_2ℎ𝐮_{𝑘,ℎ𝑋𝛾}=𝐼⁽²⁾_2ℎ𝐼_ℎ𝐮_𝑘−𝐼⁽²⁾_2ℎ𝐮_{𝑘,ℎ𝑋𝛾}≤𝐶𝐼_ℎ𝐮_𝑘−𝐮_{𝑘,ℎ𝑋𝛾}≤𝐶ℎ^1+𝛼, (4.4) by triangle inequality and interpolation error estimate, we get 𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ−𝐮_𝑘𝑋𝛾≤𝐼⁽²⁾_2ℎ𝐮_𝑘,ℎ−𝐼⁽²⁾_2ℎ𝐮_𝑘𝑋𝛾+𝐼⁽²⁾_2ℎ𝐮_𝑘−𝐮_𝑘𝑋𝛾≤𝐶ℎ^1+𝛼, (4.5) 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_ℎ𝐮_𝑘−𝐮_𝑘,ℎ‖_𝐻¹_(Ω)≤𝐶ℎ^1+𝛼 is valid (see [18–25]). But to eigenvalue problems of electric field, it is very difficult to prove that ‖𝐼_ℎ𝐮_𝑘−𝐮_𝑘,ℎ‖_𝑋𝛾≤𝐶ℎ^1+𝛼. In Section 5, we will verify this theorem by the numerical experiments.

5. Numerical Experiments

In this section, we consider numerical solutions of problem (2.1) on the L-shaped domain [−1,0]×[−1,0]∪[−1,1]×[0,1] and on the square domain [0,𝜋]×[0,𝜋]. The smallest five exact eigenvalues are 𝜆₁≈1.47562182408,𝜆₂≈3.53403136678,𝜆₃=𝜆₄≈9.86960440109,𝜆₅≈11.3894793979, and 𝜆₁ = 𝜆₂ = 1, 𝜆₃ = 2, 𝜆₄=𝜆₅=4, respectively.

We adopt a uniform isosceles right triangulation for Ω (the edge in each element is along three fixed directions, see Figures 1 and 2) to produce the meshes 𝜋_ℎ𝑙 with mesh diameter ℎ_𝑙.

Figure 1 

The L-shaped domain Ω.

Figure 2 

The square domain Ω.

Here the weight is 𝜔=𝑑^𝛾(𝑑=𝑥²₁+𝑥²₂). 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].

Example 5.2. Solve problem (2.1) on the square domain [0,𝜋]×[0,𝜋] and the L-shaped domain [−1,0]×[−1,0]∪[−1,1]×[0,1] by using Scheme 2 with quadratic finite element.

We compute the first five approximate eigenvalues by using Scheme 2. The numerical results are listed in Tables 1–3; here 𝜆^ℎ_𝑘,(𝑘=1,2,…,5) denote the first five “real” eigenvalues obtained by Scheme 2.

Table 1 

The results on square by Scheme 2 for eigenvalue problem of electric field (𝛾=0): Set ℎ₁=𝐻=√2/8,ℎ₂=ℎ₁/4,ℎ_𝑖=ℎ_𝑖−1/2,𝑖=3,4,….

Table 2 

The results on L-shape domain by Scheme 2 for eigenvalue problem of electric field (𝛾=0.5): Set ℎ₁=𝐻=√2/8,ℎ₂=ℎ₁/4,ℎ_𝑖=ℎ_𝑖−1/2,𝑖=3,4,….

Table 3 

The results on L-shape domain by Scheme 2 for eigenvalue problem of electric field (𝛾=0.95): Set ℎ₁=𝐻=√2/8,ℎ₂=ℎ₁/4,ℎ_𝑖=ℎ_𝑖−1/2,𝑖=3,4,….

From Tables 1–3, we see that Scheme 2 is highly efficient for solving eigenvalue problem of electric field.

Example 5.3. Solve problem (2.1) on the square domain [0,𝜋]×[0,𝜋] and the L-shaped domain [−1,0]×[−1,0]∪[−1,1]×[0,1] by using Scheme 3.

In Tables 4–6, 𝜆_𝑘,ℎ(𝑘=1,2,…,5) denote the first five “real” eigenvalues obtained by linear element filter approach directly, 𝜆^𝑟_𝑘,ℎ(𝑘=1,2,…,5) denote the first five “real” eigenvalues obtained by Scheme 3, 𝜆⁽²⁾_𝑘,2ℎ(𝑘=1,2,…,5) denote the eigenvalues obtained by quadratic element filter approach directly.

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).

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 (see [28]). Tables 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.

Acknowledgments

The authors cordially thank the editor and the referees for their valuable comments and suggestions that led to the improvement of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11161012) and Science and Technology Foundation of Guizhou Province of China (no. [2011]2111).