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

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 𝐾^ℎ is reduced to the solution of an eigenvalue problem on a much coarser grid 𝐾^𝐻 and the solution of a linear algebraic system on the fine grid 𝐾^ℎ. 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 𝐻=𝑂(√⁴ℎ), and when using the 𝑃_𝑘+1-𝑃_𝑘 element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking 𝐻=𝑂(√³ℎ). 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 𝐾^ℎ is reduced to the solution of an eigenvalue problem on a much coarser grid 𝐾^𝐻 and the solution of a linear algebraic system on the fine grid 𝐾^ℎ, 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 𝐻=𝑂(√⁴ℎ) when solving Stokes eigenvalue problem and 𝐻=𝑂(√³ℎ) when solving eigenvalue problem of electric field; however, with the scheme in [19, 20] the resulted solution maintains an asymptotically optimal accuracy by taking 𝐻=𝑂(√²ℎ).(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 𝑉, 𝑊, and 𝐷 be three real Hilbert spaces with inner products and norms (⋅,⋅)_𝑉, ‖⋅‖_𝑉, (⋅,⋅)_𝑊, ‖⋅‖_𝑊, and (⋅,⋅)_𝐷, ‖⋅‖_𝐷, respectively. We suppose that 𝑉↪𝐷 (continuously imbedded), 𝑎(⋅,⋅) is a symmetric, continuous, and 𝑉-elliptic bilinear form on 𝑉×𝑉, that is, 𝑎𝑞,𝜓≤𝑀₁𝑞_𝑉𝜓_𝑉𝑎𝑞,∀𝑞,𝜓∈𝑉,𝑞,𝑞≥𝜈²_𝑉,∀0≠𝑞∈𝑉, (2.1)𝑏(⋅,⋅) is a continuous bilinear form on 𝑉×𝑊, that is, 𝑏𝜓,𝑣≤𝑀₂𝜓_𝑉‖𝑣‖_𝑊,∀𝜓∈𝑉,𝑣∈𝑊. (2.2)

In scientific and engineering computations, many eigenvalue problems for differential equation have the following mixed variational formulation.

Find (𝜆,𝑢,𝜎)∈𝐑×𝑉×𝑊,  (𝑢,𝜎)≠(0,0), such that 𝑎𝑢,𝜓+𝑏𝜓,𝜎=𝜆𝑢,𝜓_𝐷𝑏(),∀𝜓∈𝑉,𝑢,𝑣=0,∀𝑣∈𝑊. (2.3)(2.4)

In order to solve problem (2.3)-(2.4), one should construct finite element spaces 𝑉_ℎ⊂𝑉 and 𝑊_ℎ⊂𝑊. Restricting (2.3)-(2.4) on 𝑉_ℎ×𝑊_ℎ we get the conforming mixed finite element approximation as follows. Find (𝜆_ℎ,𝑢_ℎ,𝜎_ℎ)∈𝐑×𝑉_ℎ×𝑊_ℎ,  (𝑢_ℎ,𝜎_ℎ)≠(0,0), such that 𝑎𝑢_ℎ,𝜓+𝑏𝜓,𝜎_ℎ=𝜆_ℎ𝑢_ℎ,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑢_ℎ,𝑣=0,∀𝑣∈𝑊_ℎ. (2.5)(2.6)

Consider the associated source and approximate source problems.

Given 𝑓∈𝐷, find (𝑤,𝑝)∈𝑉×𝑊 satisfying 𝑎𝑤,𝜓+𝑏𝜓,𝑝=𝑓,𝜓_𝐷𝑏(),∀𝜓∈𝑉,𝑤,𝑣=0,∀𝑣∈𝑊. (2.7)(2.8)

Given 𝑓∈𝐷, find (𝑤_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝑊_ℎ satisfying 𝑎𝑤_ℎ,𝜓+𝑏𝜓,𝑝_ℎ=𝑓,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑤_ℎ,𝑣=0,∀𝑣∈𝑊_ℎ. (2.9)(2.10)

As for the mixed finite element method for boundary value problems, Brezzi and Babuska, and others have established a systematic theory. Denote 𝑉₀={()},𝑉𝑢∈𝑉∶𝑏𝑢,𝑣=0,∀𝑣∈𝑊_ℎ0=𝑢∈𝑉_ℎ∶𝑏(𝑢,𝑣)=0,∀𝑣∈𝑊_ℎ. (2.11)

Theorem 2.1 (Brezzi-Babuska Theorem). Suppose that(1)𝑎(⋅,⋅) and 𝑏(⋅,⋅) are continuous bilinear forms, that is, 𝑎𝑞,𝜓≤𝑀₁𝑞_𝑉𝜓_𝑉𝑏,∀𝑞,𝜓∈𝑉,𝜓,𝑣≤𝑀₂𝜓_𝑉‖𝑣‖_𝑊,∀𝜓∈𝑉,𝑣∈𝑊; (2.12)(2) there exists 𝜈₁>0, such that 𝑎𝑞,𝑞≥𝜈₁𝑞²_𝑉,∀0≠𝑞∈𝑉₀; (2.13)(3)inf-sup condition: there exists 𝜈₂>0, such that sup_{𝜓∈𝑉,𝜓≠0}𝑏𝜓,𝑣𝜓_𝑉≥𝜈₂‖𝑣‖_𝑊,∀𝑣∈𝑊; (2.14) then there exists a unique solution to the problem (2.7)-(2.8), and ‖𝑤‖_𝑉+𝑝_𝑊≤𝐶𝑓_𝐷, (2.15) where constant 𝐶 just depends on 𝜈₁,𝜈₂ and 𝑀₁,𝑀₂. Furthermore, suppose that(4) there exists a constant 𝜇₁>0, such that 𝑎𝑞,𝑞≥𝜇₁𝑞²_𝑉,∀0≠𝑞∈𝑉_ℎ0; (2.16)(5) discrete inf-sup condition: there exists a constant 𝜇₂>0, such that sup_{𝜓∈𝑉ℎ,𝜓≠0}𝑏𝜓,𝑣𝜓_𝑉≥𝜇₂‖𝑣‖_𝑊,∀𝑣∈𝑊_ℎ. (2.17)Then there exists a unique solution (𝑤_ℎ,𝑝_ℎ) to the problem (2.9)-(2.10); moreover, the following error estimate is valid: 𝑤−𝑤_ℎ𝑉+𝑝−𝑝_ℎ𝑊≤𝐶inf_{𝑞∈𝑉ℎ}𝑤−𝑞_𝑉+inf_{𝑣∈𝑊ℎ}𝑝−𝑣_𝑊, (2.18) where 𝐶 just depends on 𝜇₁,𝜇₂ and 𝑀₁,𝑀₂.

Since 𝑎(⋅,⋅) is a symmetric, continuous, and 𝑉-elliptic bilinear form on 𝑉×𝑉, 𝑏(⋅,⋅) 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 𝑓∈𝐷. Thus we can define the corresponding linear bounded operators:

𝑇∶𝐷→𝑉,𝑆∶𝐷→𝑊∶ for all 𝑓∈𝐷𝑎𝑇𝑓,𝜓+𝑏𝜓,𝑆𝑓=𝑓,𝜓_𝐷𝑏,∀𝜓∈𝑉,𝑇𝑓,𝑣=0,∀𝑣∈𝑊, (2.19)(2.20)𝑇_ℎ∶𝐷→𝑉_ℎ⊂𝑉,𝑆_ℎ∶𝐷→𝑊_ℎ⊂𝑊∶ for all 𝑓∈𝐷, 𝑎𝑇_ℎ𝑓,𝜓+𝑏𝜓,𝑆_ℎ𝑓=𝑓,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑇_ℎ𝑓,𝑣=0,∀𝑣∈𝑊_ℎ. (2.21)(2.22) It is obvious that (2.3)-(2.4) have an equivalent operator form: ().𝜆𝑇𝑢=𝑢,𝜎=𝑆𝜆𝑢 (2.23)(2.24) Equations (2.5)-(2.6) have an equivalent operator form: 𝜆_ℎ𝑇_ℎ𝑢_ℎ=𝑢_ℎ,𝜎_ℎ=𝑆_ℎ𝜆_ℎ𝑢_ℎ. (2.25)(2.26) It is easy to prove that 𝑇∶𝐷→𝐷,𝑇_ℎ∶𝐷→𝐷 are self-adjoint operators. In fact, for all 𝑓,𝑔∈𝐷, taking 𝜓=𝑇𝑔,𝑣=𝑆𝑔 in (2.7)-(2.8) we obtain 𝑎𝑇𝑓,𝑇𝑔+𝑏𝑇𝑔,𝑆𝑓=𝑓,𝑇𝑔_𝐷,𝑏𝑇𝑓,𝑆𝑔=0. (2.27) Exchanging 𝑓 and 𝑔, we get 𝑎𝑇𝑔,𝑇𝑓+𝑏𝑇𝑓,𝑆𝑔=𝑔,𝑇𝑓_𝐷,𝑏𝑇𝑔,𝑆𝑓=0, (2.28)(2.29) then 𝑓,𝑇𝑔_𝐷==𝑎𝑇𝑓,𝑇𝑔+𝑏𝑇𝑔,𝑆𝑓+𝑏𝑇𝑓,𝑆𝑔=𝑎𝑇𝑔,𝑇𝑓+𝑏𝑇𝑓,𝑆𝑔+𝑏𝑇𝑔,𝑆𝑓𝑔,𝑇𝑓_𝐷+0=𝑇𝑓,𝑔_𝐷. (2.30) It shows that 𝑇∶𝐷→𝐷 is self-adjoint in the sense of inner product (⋅,⋅)_𝐷. Analogously, it can be proved that 𝑇_ℎ∶𝐷→𝐷 is self-adjoint in the sense of inner product (⋅,⋅)_𝐷.

Assume that 𝑉↪𝐷 (compact imbedded), then it is easy to prove that the operator 𝑇∶𝐷→𝐷 is completely continuous, 𝑇∶𝑉→𝑉 is completely continuous, and 𝑇_ℎ is a finite rank operator. Combining (2.3)-(2.4), (2.5)-(2.6), and the 𝑉-ellipticity of 𝑎(⋅,⋅), we deduce 𝑎()𝜆=𝑢,𝑢(𝑢,𝑢)_𝐷>0,𝜆_ℎ=𝑎𝑢_ℎ,𝑢_ℎ𝑢_ℎ,𝑢_ℎ𝐷>0. (2.31) 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 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑘≤⋯↗+∞, (2.32) and the corresponding eigenfunctions are 𝑢₁,𝜎₁,𝑢₂,𝜎₂,…,𝑢_𝑘,𝜎_𝑘,…, (2.33) where (𝑢_𝑖,𝑢_𝑗)_𝐷=𝛿_𝑖𝑗.

The eigenvalues of (2.5)-(2.6) can be sorted as 0<𝜆_1,ℎ≤𝜆_2,ℎ≤⋯≤𝜆_𝑘,ℎ≤⋯≤𝜆_𝑁ℎ,ℎ, (2.34) and the corresponding eigenfunctions are 𝑢_1,ℎ,𝜎_1,ℎ,𝑢_2,ℎ,𝜎_2,ℎ,…,𝑢_𝑘,ℎ,𝜎_𝑘,ℎ,…,𝑢_𝑁ℎ,ℎ,𝜎_𝑁ℎ,ℎ, (2.35) where (𝑢_𝑖,ℎ,𝑢_𝑗,ℎ)_𝐷=𝛿_𝑖𝑗.

It is obvious that 𝑎(⋅,⋅) is an inner product on 𝑉,‖⋅‖_𝑎=𝑎(⋅,⋅) and ‖⋅‖_𝑉 are equivalent norms. Let 𝑢_ℎ=𝑢_𝑖,ℎ in (2.5); then 𝑎𝑢_𝑖,ℎ,𝑢_𝑗,ℎ+𝑏𝑢_𝑗,ℎ,𝜎_ℎ=𝜆_𝑖,ℎ𝑢_𝑖,ℎ,𝑢_𝑗,ℎ𝐷. (2.36) From (2.6), we get 𝑏(𝑢_𝑗,ℎ,𝜎_ℎ)=0. Then 𝑎𝑢_𝑖,ℎ,𝑢_𝑗,ℎ=𝜆_𝑖,ℎ𝑢_𝑖,ℎ,𝑢_𝑗,ℎ𝐷=𝜆_𝑖,ℎ𝛿_𝑖𝑗; (2.37) therefore, {𝑢_𝑖,ℎ/‖𝑢_𝑖,ℎ‖_𝑎} is a completely normal eigenvector system on 𝑉_ℎ in the sense of inner product 𝑎(⋅,⋅).

Denote 𝜆_𝑘=1/𝜇_𝑘,𝜆_𝑘,ℎ=1/𝜇_𝑘,ℎ. In this paper, 𝜇_𝑘 and 𝜇_𝑘,ℎ, 𝜆_𝑘, and 𝜆_𝑘,ℎ are all called eigenvalues. Let 𝜇 be the 𝑘th eigenvalue with algebraic multiplicity 𝑞,𝜇=𝜇_𝑘=𝜇_𝑘+1=⋯=𝜇_{𝑘+𝑞−1}. 𝑀(𝜇) is the space spanned by all eigenfunctions {𝑢_𝑗}_𝑘^{𝑘+𝑞−1} corresponding to 𝜇 of 𝑇. 𝑀_ℎ(𝜇) is the space spanned by all eigenfunctions {𝑢_𝑗,ℎ}_𝑘^{𝑘+𝑞−1} corresponding to all eigenvalues of 𝑇_ℎ that converge to 𝜇. Let 𝑀(𝜇)={𝑣∶𝑣∈𝑀(𝜇),‖𝑣‖_𝑎=1}, 𝑀_ℎ(𝜇)={𝑣∶𝑣∈𝑀_ℎ(𝜇),‖𝑣‖_𝑎=1}. We call 𝜆=1/𝜇 the 𝑘th eigenvalue, too. Denote 𝑀(𝜆)=𝑀(𝜇), 𝑀_ℎ(𝜆)=𝑀_ℎ(𝜇), and 𝑀(𝜆)=𝑀(𝜇). Define 𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷}=max_{𝑢∈𝑀(𝜆),𝑢≠0}𝑇−𝑇_ℎ𝑢_𝐷‖𝑢‖_𝐷,𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}=max_{𝑢∈𝑀(𝜆),𝑢≠0}𝑇−𝑇_ℎ𝑢_𝑎‖𝑢‖_𝑎. (2.38)

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 ‖𝑇−𝑇_ℎ‖_𝐷→0(ℎ→0). Let (𝜆_ℎ,𝑢_ℎ,𝜎_ℎ) be the 𝑘th eigenpair of (2.5)-(2.6), ‖𝑢_ℎ‖_𝑎=1, and 𝜆 the 𝑘th eigenvalue of (2.3)-(2.4). Then 𝜆_ℎ→𝜆(ℎ→0), and there exists an eigenfunction (𝑢,𝜎) corresponding to 𝜆 such that 𝜆_ℎ−𝜆+𝑢_ℎ−𝑢_𝐷≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷},𝜎−𝜎_ℎ𝑊≤𝑆_ℎ(𝜆𝑢)−𝑆(𝜆𝑢)_𝑊+𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷},𝑢−𝑢_ℎ𝑎≤𝐶𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}. (2.39)(2.40)(2.41) Let 𝑢∈𝑀(𝜆); then there exists 𝑢_ℎ∈𝑀_ℎ(𝜆) such that 𝑢−𝑢_ℎ𝑎≤𝐶𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}. (2.42)

Proof. From the spectral approximation theory (see [23]) we have (2.39).
Let 𝑢 satisfy (2.39), and 𝜎=𝑆(𝜆𝑢). Next we will prove that this eigenpair satisfies (2.40)-(2.41). From Brezzi-Babuska Theorem and (2.39), we get 𝑆𝜆𝑢−𝜆_ℎ𝑢_ℎ𝑊≤𝐶𝜆𝑢−𝜆_ℎ𝑢_ℎ𝐷≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷},𝑇𝜆𝑢−𝜆_ℎ𝑢_ℎ𝑉≤𝐶𝜆𝑢−𝜆_ℎ𝑢_ℎ𝐷≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷}. (2.43)(2.44) Using the triangle inequality and (2.43), we deduce 𝜎−𝜎_ℎ𝑊−𝑆_ℎ(𝜆𝑢)−𝑆(𝜆𝑢)_𝑊=𝑆(𝜆𝑢)−𝑆_ℎ𝜆_ℎ𝑢_ℎ𝑊−𝑆_ℎ(𝜆𝑢)−𝑆(𝜆𝑢)_𝑊≤𝑆_ℎ𝜆𝑢−𝜆_ℎ𝑢_ℎ𝑊≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷}, (2.45) that is, (2.40) is valid.
Using the triangle inequality and (2.44), we get 𝑢−𝑢_ℎ𝑎−𝑇_ℎ(𝜆𝑢)−𝑇(𝜆𝑢)_𝑎=𝑇(𝜆𝑢)−𝑇_ℎ𝜆_ℎ𝑢_ℎ𝑎−𝑇_ℎ(𝜆𝑢)−𝑇(𝜆𝑢)_𝑎≤𝑇_ℎ𝜆𝑢−𝜆_ℎ𝑢_ℎ𝑎≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷}≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}, (2.46) which together with ‖𝑇_ℎ(𝜆𝑢)−𝑇(𝜆𝑢)‖_𝑎≤𝐶‖(𝑇−𝑇_ℎ)|_𝑀(𝜆)‖_𝑎 yields (2.41).
Let the eigenfunctions {𝑢_𝑗,ℎ} be an orthonormal system of 𝑀_ℎ(𝜆) in the sense of inner product 𝑎(⋅,⋅). Then, from (2.41) and Lemma 3.1 we know that there exists a basis {𝑢⁰_𝑗} of 𝑀(𝜆) satisfying ‖𝑢⁰_𝑗‖_𝑎=1 and the following result is valid: 𝑢⁰_𝑗−𝑢_𝑗,ℎ𝑎≤𝐶𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}. (2.47) For any 𝑢∈𝑀(𝜆), we write 𝑢=∑^{𝑘+𝑞−1}_𝑗=𝑘𝛼_𝑗𝑢⁰_𝑗. By calculation, we get 1=‖𝑢‖²_𝑎=^{𝑘+𝑞−1}_𝑗=𝑘𝛼²_𝑗+^{𝑘+𝑞−1}_{𝑖≠𝑗,𝑖,𝑗=𝑘}𝛼_𝑖𝛼_𝑗𝑎𝑢⁰_𝑖,𝑢⁰_𝑗. (2.48) From (2.47), when 𝑖≠𝑗 we have 𝑎(𝑢⁰_𝑖,𝑢⁰_𝑗)=𝑎(𝑢⁰_𝑖,𝑢⁰_𝑗)−𝑎(𝑢_𝑖,ℎ,𝑢_𝑗,ℎ)→0(ℎ→0), and thus we get ∑^{𝑘+𝑞−1}_𝑗=𝑘𝛼²_𝑗→1(ℎ→0).
Denote 𝑢_ℎ=∑^{𝑘+𝑞−1}_𝑗=𝑘𝛼_𝑗𝑢_𝑗,ℎ, then 𝑢_ℎ∈𝑀_ℎ(𝜆). From (2.47), we deduce 𝑢−𝑢_ℎ𝑎≤𝐶^{𝑘+𝑞−1}_𝑗=𝑘𝑢_𝑗,ℎ−𝑢⁰_𝑗𝑎≤𝐶𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}, (2.49) that is, (2.42) is valid. The proof is completed.

For (𝑢^∗,𝜎^∗)∈𝑉×𝑊,𝑢^∗≠0, define the Rayleigh quotient 𝜆^𝑟=𝑎𝑢^∗,𝑢^∗+2𝑏𝑢^∗,𝜎^∗(𝑢^∗,𝑢^∗)_𝐷. (2.50)

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

Lemma 2.3. Let (𝜆,𝑢,𝜎) be an eigenpair of (2.3)-(2.4); then for all (𝑢^∗,𝜎^∗)∈𝑉×𝑊, (𝑢^∗,𝜎^∗)≠(0,0) with its Rayleigh quotient satisfying 𝜆^𝑟𝑎𝑢−𝜆=^∗−𝑢,𝑢^∗−𝑢+2𝑏𝑢^∗−𝑢,𝜎^∗−𝜎(𝑢^∗,𝑢^∗)_𝐷𝑢−𝜆^∗−𝑢,𝑢^∗−𝑢_𝐷(𝑢^∗,𝑢^∗)_𝐷. (2.51)

Proof. From (2.3)-(2.4), we deduce 𝑎𝑢^∗−𝑢,𝑢^∗−𝑢+2𝑏𝑢^∗−𝑢,𝜎^∗−𝜎−𝜆𝑢^∗−𝑢,𝑢^∗−𝑢_𝐷=𝑎𝑢^∗,𝑢^∗−2𝑎𝑢^∗,𝑢+𝑎(𝑢,𝑢)+2𝑏𝑢^∗,𝜎^∗−2𝑏𝑢^∗,𝜎−2𝑏𝑢,𝜎^∗+2𝑏(𝑢,𝜎)−𝜆𝑢^∗,𝑢^∗_𝐷+2𝜆𝑢^∗,𝑢_𝐷−𝜆(𝑢,𝑢)_𝐷.=𝑎𝑢^∗,𝑢^∗+2𝑏𝑢^∗,𝜎^∗−𝜆𝑢^∗,𝑢^∗_𝐷−2𝑎𝑢^∗,𝑢+𝑏𝑢^∗,𝜎−𝜆𝑢^∗,𝑢_𝐷−2𝑏𝑢,𝜎^∗+𝑎(𝑢,𝑢)+𝑏(𝑢,𝜎)−𝜆(𝑢,𝑢)_𝐷().𝑢+𝑏𝑢,𝜎=𝑎^∗,𝑢^∗+2𝑏𝑢^∗,𝜎^∗−𝜆𝑢^∗,𝑢^∗_𝐷−0−0+0+0. (2.52) By dividing by (𝑢^∗,𝑢^∗)_𝐷 on both sides of the above identity, we obtain (2.51).

Taking (𝑢^∗,𝜎^∗)=(𝑢_ℎ,𝜎_ℎ) in (2.51) and using (2.4) and (2.6), we derive

Lemma 2.4. Let (𝜆,𝑢,𝜎) and (𝜆_ℎ,𝑢_ℎ,𝜎_ℎ) be the 𝑘th eigenpair of (2.3)-(2.4) and (2.5)-(2.6), respectively; then 𝜆_ℎ𝑎𝑢−𝜆=^ℎ−𝑢,𝑢_ℎ−𝑢+2𝑏𝑢_ℎ−𝑢,𝑣−𝜎𝑢_ℎ,𝑢_ℎ𝐷𝑢−𝜆^ℎ−𝑢,𝑢_ℎ−𝑢_𝐷𝑢_ℎ,𝑢_ℎ𝐷,∀𝑣∈𝑊_ℎ𝜆_ℎ,−𝜆≤𝐶𝑢_ℎ−𝑢²_𝑎+𝑢_ℎ−𝑢_𝑎‖𝜎−𝑣‖_𝑊,∀𝑣∈𝑊_ℎ. (2.53)(2.54)

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 𝐾^𝐻: find (𝜆_𝐻,𝑢_𝐻,𝜎_𝐻)∈𝐑×𝑉_𝐻×𝑊_𝐻,  ‖𝑢_𝐻‖_𝑎=1 such that 𝑎𝑢_𝐻,𝜓+𝑏𝜓,𝜎_𝐻=𝜆_𝐻𝑢_𝐻,𝜓_𝐷,∀𝜓∈𝑉_𝐻,𝑏𝑢_𝐻,𝑣=0,∀𝑣∈𝑊_𝐻. (3.1)
Step 2. Solve a equation on a fine grid 𝐾^ℎ: find (𝑢′,𝜎′)∈𝑉_ℎ×𝑊_ℎ such that 𝑎𝑢^′,𝜓+𝑏𝜓,𝜎^′−𝜆_𝐻𝑢^′,𝜓_𝐷=𝑢_𝐻,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑢^′,𝑣=0,∀𝑣∈𝑊_ℎ. (3.2) Set 𝑢^ℎ=𝑢^′/‖𝑢^′‖_𝑎,𝜎^ℎ=𝜎^′/‖𝑢^′‖_𝑎.
Step 3. Compute the Rayleigh quotient 𝜆^ℎ=𝑎𝑢^ℎ,𝑢^ℎ𝑢^ℎ,𝑢^ℎ_𝐷. (3.3)

Next we will discuss the validity of Scheme 1.

Lemma 3.1. For any nonzero elements 𝑢,𝑣∈𝑉, 𝑢‖𝑢‖_𝑎−𝑣‖𝑣‖_𝑎𝑎‖‖≤2𝑢−𝑣_𝑎‖𝑢‖_𝑎,𝑢‖𝑢‖_𝑎−𝑣‖𝑣‖_𝑎𝑎‖‖≤2𝑢−𝑣_𝑎‖𝑣‖_𝑎. (3.4)

Proof. See [21].

Denote dist(𝑢,𝑉)=inf_𝑣∈𝑉‖𝑢−𝑣‖_𝑎.

Consider the eigenvalue problem (2.25) on the space 𝑉_ℎ.

Lemma 3.2. Suppose that 𝜇 and 𝜇_ℎ are the 𝑘th eigenvalue of 𝑇 and 𝑇_ℎ, respectively, and (𝜇₀,𝑢₀) is an approximate eigenpair where 𝜇₀ is not an eigenvalue of 𝑇_ℎ, 𝑢₀∈𝑉_ℎ, ‖𝑢₀‖_𝑎=1, dist(𝑢₀,𝑀_ℎ(𝜇))≤1/2, max_{𝑘≤𝑗≤𝑘+𝑞−1}|(𝜇_𝑗,ℎ−𝜇_ℎ)/(𝜇₀−𝜇_𝑗,ℎ)|≤1/2, |𝜇₀−𝜇_𝑗,ℎ|≥(𝜌/2)(𝑗≠𝑘,𝑘+1,…,𝑘+𝑞−1), and 𝑢^𝑠∈𝑉_ℎ, 𝑢^ℎ∈𝑉_ℎ satisfy 𝜇₀−𝑇_ℎ𝑢^𝑠=𝑢₀,𝑢^ℎ=𝑢^𝑠‖𝑢^𝑠‖_𝑎. (3.5) Then dist𝑢^ℎ,𝑀_ℎ𝜇≤16𝜌𝜇₀−𝜇_ℎdist𝑢₀,𝑀_ℎ𝜇, (3.6) where 𝜌=min_{𝜇𝑗≠𝜇}|𝜇_𝑗−𝜇| is the separation constant of the eigenvalue 𝜇.

Proof. See [21].

Theorem 3.3. Suppose that the conditions of Brezzi-Babuska Theorem hold and ‖𝑇_ℎ−𝑇‖_𝐷→0(ℎ→0). Let (𝜆^ℎ,𝑢^ℎ,𝜎^ℎ) be the approximate eigenpair obtained by the two-scale discretization scheme and 𝐻 small properly. Then there exists 𝑢∈𝑀(𝜆) such that 𝑢^ℎ−𝑢_𝑎≤𝐶𝜆_𝐻−𝜆²+𝜆_𝐻−𝜆𝑇−𝑇_𝐻|_{𝑀(𝜆)𝐷}+𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎},𝜆^ℎ−𝜆≤𝐶𝑢^ℎ−𝑢²_𝑎+‖𝑢^ℎ−𝑢‖_𝑎inf_{𝑣∈𝑊ℎ}‖𝜎−𝑣‖_𝑊. (3.7)(3.8)

Proof. We use Lemma 3.2 in the proof.
Select 𝜇₀=1/𝜆_𝐻 and 𝑢₀=𝜆_𝐻𝑇_ℎ𝑢_𝐻/‖𝜆_𝐻𝑇_ℎ𝑢_𝐻‖_𝑎. Let 𝑢⁰∈𝑀(𝜆) such that 𝑢_𝐻−𝑢⁰ satisfies (2.39) and (2.41). By calculation we deduce 𝜆_𝐻𝑇_ℎ𝑢_𝐻−𝑢⁰_𝑎=𝜆_𝐻𝑇_ℎ𝑢_𝐻−𝜆𝑇𝑢⁰_𝑎≤𝐶𝜆_𝐻−𝜆+𝑢_𝐻−𝑢⁰_𝐷+𝑇−𝑇_ℎ𝑢⁰_𝑎≤𝐶𝜆_𝐻−𝜆+𝑇−𝑇_𝐻|_{𝑀(𝜆)𝐷}+𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}; (3.9) thus, using Lemma 3.1, we get dist𝑢₀,𝑀(𝜆)≤𝑢₀−𝑢⁰𝑢⁰_𝑎𝑎≤𝐶𝜆_𝐻𝑇_ℎ𝑢_𝐻−𝑢⁰_𝑎≤𝐶𝜆_𝐻−𝜆+𝑇−𝑇_𝐻|_{𝑀(𝜆)𝐷}+𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}. (3.10) Using the triangle inequality and (2.42), we derive dist𝑢₀,𝑀_ℎ(𝜆)≤dist𝑢₀,𝑀(𝜆)+𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}. (3.11) From Lemma 2.2 we know 𝜆_𝐻→𝜆,𝜆_𝑗,ℎ→𝜆; then 𝜇₀−𝜇_𝑗,ℎ=𝜆_𝐻−𝜆+𝜆−𝜆_𝑗,ℎ𝜆_𝑗,ℎ𝜆_𝐻. (3.12) When 𝐻 is small enough, noting that ℎ≪𝐻, from (3.11) and (3.10) we get dist𝑢₀,𝑀_ℎ(𝜆≤1)2. (3.13) Having in mind that 𝜆=𝜆_𝑘+1=⋯=𝜆_{𝑘+𝑞−1} we have 𝜇_𝑗,ℎ−𝜇_ℎ=𝜆_ℎ−𝜆_𝑗,ℎ𝜆_ℎ𝜆_𝑗,ℎ=𝜆_ℎ−𝜆+𝜆_𝑗−𝜆_𝑗,ℎ𝜆_ℎ𝜆_𝑗,ℎ, (3.14) which together with (3.12), noting that 𝜆_𝑗,ℎ−𝜆 is an infinitesimal of higher order comparing with 𝜆_𝐻−𝜆, yields max_{𝑘≤𝑗≤𝑘+𝑞−1}𝜇_𝑗,ℎ−𝜇_ℎ𝜇₀−𝜇_𝑗,ℎ≤12. (3.15) Since 𝜌 is the separation constant, 𝐻 is small enough, and ℎ≪𝐻, there holds 𝜇₀−𝜇_𝑗,ℎ≥𝜌2,𝑗≠𝑘,𝑘+1,…,𝑘+𝑞−1. (3.16) For 𝑢′ in Step 2 of Scheme 1, from the definition of 𝑇_ℎ and 𝑆_ℎ we have 𝑎𝑇_ℎ𝜆_𝐻𝑢^′,𝜓+𝑏𝜓,𝑆_ℎ𝜆_𝐻𝑢^′=𝜆_𝐻𝑢^′,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑇_ℎ𝑢^′,𝑣=0,∀𝑣∈𝑊_ℎ𝑎𝑇_ℎ.𝑢_𝐻,𝜓+𝑏𝜓,𝑆_ℎ𝑢_𝐻=𝑢_𝐻,𝜓_𝐷,∀𝜓∈𝑉_ℎ,𝑏𝑇_ℎ𝑢_𝐻,𝑣=0,∀𝑣∈𝑊_ℎ. (3.17)(3.18)(3.19)(3.20) Hence, Step 2 of Scheme 1 is equivalent to (𝑢′,𝜎′)∈𝑉_ℎ×𝑊_ℎ, 𝑎𝑢^′,𝜓+𝑏𝜓,𝜎^′−𝜆_𝐻𝑎𝑇_ℎ𝑢^′,𝜓−𝜆_𝐻𝑏𝜓,𝑆_ℎ𝑢^′=𝑎𝑇_ℎ𝑢_𝐻,𝜓+𝑏𝜓,𝑆_ℎ𝑢_𝐻,∀𝜓∈𝑉_ℎ,𝑏𝑢^′,𝑣=0,∀𝑣∈𝑊_ℎ, (3.21)(3.22)𝑢^ℎ=𝑢^′/‖𝑢^′‖_𝑎,𝜎^ℎ=𝜎^′/‖𝑢^′‖_𝑎.
From (3.21) we obtain 𝑎𝑢^′−𝜆_𝐻𝑇_ℎ𝑢^′−𝑇_ℎ𝑢_𝐻,𝜓+𝑏𝜓,𝜎^′−𝜆_𝐻𝑆_ℎ𝑢^′−𝑆_ℎ𝑢_𝐻=0,∀𝜓∈𝑉_ℎ. (3.23) Combining (3.22), (3.18), and (3.20), we get 𝑏𝑢^′−𝜆_𝐻𝑇_ℎ𝑢^′−𝑇_ℎ𝑢_𝐻,𝑣=0,∀𝑣∈𝑊_ℎ. (3.24) By (3.24), taking 𝜓=𝑢′−𝜆_𝐻𝑇_ℎ𝑢′−𝑇_ℎ𝑢_𝐻 in (3.23), we obtain 𝑎𝑢^′−𝜆_𝐻𝑇_ℎ𝑢^′−𝑇_ℎ𝑢_𝐻,𝑢^′−𝜆_𝐻𝑇_ℎ𝑢^′−𝑇_ℎ𝑢_𝐻=0. (3.25) Thus 1𝜆_𝐻−𝑇_ℎ1𝑢′=𝜆_𝐻𝑇_ℎ𝑢_𝐻,𝑢^ℎ=𝑢′𝑢^′_𝑎. (3.26) From (3.26) we know that the first term on the left-hand side of (3.23) is equal to 0; thus 𝑏𝜓,𝜎^′−𝜆_𝐻𝑆_ℎ𝑢^′−𝑆_ℎ𝑢_𝐻=0,∀𝜓∈𝑉_ℎ; (3.27) then, using discrete inf-sup condition, we obtain 𝜎′=𝜆_𝐻𝑆_ℎ𝑢′+𝑆_ℎ𝑢_𝐻. (3.28) Thus Step  2 of Scheme 1 is equivalent to (3.26), (3.28), and 𝑢^ℎ=𝑢^′/‖𝑢^′‖_𝑎,𝜎^ℎ=𝜎^′/‖𝑢^′‖_𝑎. Noting that 𝜆_𝐻⁻¹𝑇_ℎ𝑢_𝐻=‖𝜆_𝐻⁻¹𝑇_ℎ𝑢_𝐻‖_𝑎𝑢₀ differs from 𝑢₀ by only a constant and denoting 𝑢^𝑠=𝑢′/‖𝜆_𝐻⁻¹𝑇_ℎ𝑢_𝐻‖_𝑎, we have 1𝜆_𝐻−𝑇_ℎ𝑢^𝑠=𝑢₀,𝑢^ℎ=𝑢^𝑠‖𝑢^𝑠‖_𝑎. (3.29) 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 dist𝑢^ℎ,𝑀_ℎ(𝜆)≤𝐶𝜆_𝐻−𝜆dist𝑢₀,𝑀(𝜆)+𝑇𝑢−𝑇_ℎ𝑢_𝑎. (3.30) Let the eigenfunctions {𝑢_𝑗,ℎ}_𝑘^{𝑘+𝑞−1} be an orthonormal system of 𝑀_ℎ(𝜆) (in the sense of 𝑎(⋅,⋅)). Then dist𝑢^ℎ,𝑀_ℎ(𝜆)=𝑢^ℎ−^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝑢^ℎ,𝑢_𝑗,ℎ𝑢_𝑗,ℎ𝑎. (3.31) Let 𝑢^∗=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝑢^ℎ,𝑢_𝑗,ℎ𝑢_𝑗,ℎ, (3.32) and noting that ‖𝑢^ℎ−𝑢^∗‖_𝑎≤dist(𝑢^ℎ,𝑀_ℎ(𝜆)), from (3.30) we deduce 𝑢^ℎ−𝑢^∗_𝑎≤𝐶𝜆_𝐻−𝜆dist𝑢₀,𝑀(𝜆)+𝑇𝑢−𝑇_ℎ𝑢_𝑎. (3.33) By Lemma 2.2, there exists {𝑢⁰_𝑗}_𝑘^{𝑘+𝑞−1}⊂𝑀(𝜆) such that 𝑢_𝑗,ℎ−𝑢⁰_𝑗 satisfies (2.41). Let 𝑢=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝑢^ℎ,𝑢_𝑗,ℎ𝑢⁰_𝑗; (3.34) then 𝑢∈𝑀(𝜆). Using (2.41) we deduce 𝑢^∗−𝑢_𝑎=^{𝑘+𝑞−1}_𝑗=𝑘𝑎𝑢^ℎ,𝑢_𝑗,ℎ𝑢_𝑗,ℎ−𝑢⁰_𝑗𝑎≤𝐶^{𝑘+𝑞−1}_𝑗=𝑘𝑢_𝑗,ℎ−𝑢⁰_𝑗²_𝑎^1/2≤𝐶𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}. (3.35) Combining (3.33) with the previous inequality, we have 𝑢^ℎ−𝑢_𝑎≤𝐶𝜆_𝐻−𝜆dist𝑢₀,𝑀(𝜆)+𝑇_ℎ−𝑇|_{𝑀(𝜆)𝑎}. (3.36) Substituting (3.10) into (3.36), we get (3.7).
We know that 𝑏(𝑢^ℎ,𝜎^ℎ)=0 from Step 2 of two-scale scheme; then 𝜆^ℎ=𝑎𝑢^ℎ,𝑢^ℎ𝑢^ℎ,𝑢^ℎ_𝐷=𝑎𝑢^ℎ,𝑢^ℎ+2𝑏𝑢^ℎ,𝜎^ℎ𝑢^ℎ,𝑢^ℎ_𝐷. (3.37) Select 𝜆^𝑟=𝜆^ℎ,𝑢^∗=𝑢^ℎ,𝜎^∗=𝜎^ℎ. From Lemma 2.3, we get 𝜆^ℎ𝑎𝑢−𝜆=^ℎ−𝑢,𝑢^ℎ−𝑢+2𝑏𝑢^ℎ−𝑢,𝜎^ℎ−𝜎𝑢^ℎ,𝑢^ℎ_𝐷𝑢−𝜆^ℎ−𝑢,𝑢^ℎ−𝑢_𝐷𝑢^ℎ,𝑢^ℎ_𝐷. (3.38) Noting that, for all 𝑣∈𝑊_ℎ,𝑏(𝑢^ℎ−𝑢,𝑣)=0, we have 𝜆^ℎ𝑎𝑢−𝜆=^ℎ−𝑢,𝑢^ℎ−𝑢+2𝑏𝑢^ℎ−𝑢,𝑣−𝜎𝑢^ℎ,𝑢^ℎ_𝐷𝑢−𝜆^ℎ−𝑢,𝑢^ℎ−𝑢_𝐷𝑢^ℎ,𝑢^ℎ_𝐷,∀𝑣∈𝑊_ℎ. (3.39) Since 𝑉↪𝐷 (continuously imbedded), ‖𝑢^ℎ−𝑢‖_𝐷≤𝐶‖𝑢^ℎ−𝑢‖_𝑎. Then from (3.39) we obtain (3.8).

4. Two-Scale Discretization Scheme for Stokes Eigenvalue Problem

Consider the Stokes eigenvalue problem: −Δ^→𝑢+∇𝜎=𝜆^→𝑢,inΩ,div^→𝑢=0,inΩ,_→𝑢=0,on𝜕Ω, (4.1)(4.2)(4.3) where Ω is a polygonal domain in 𝐑²,^→𝑢=(𝑢₁,𝑢₂) denotes the fluid velocity, and 𝜎 denotes the pressure.

In this paper, we use the symbol → to stand for vector function. For the function 𝜎 in 𝐻^𝑚(Ω), let ‖𝜎‖_𝑚=_{|𝛼|≤𝑚Ω}𝜕^𝛼𝜎^21/2,𝛼=𝛼₁,𝛼₂,|𝛼|=𝛼₁+𝛼₂. (4.4) For vector function ^→𝑢=(𝑢₁,𝑢₂), define ^→𝑢_𝑚=𝑢₁²_𝑚+𝑢₂²_𝑚^1/2,^→𝑢_𝑚=𝑢₁²_𝑚+𝑢₂²_𝑚^1/2. (4.5)

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

Find (𝜆,^→𝑢,𝜎)∈𝐑×𝐻¹₀(Ω)²×𝐿²₀(Ω) with |^→𝑢|₁=1 such that _Ω²_𝑖=1▿𝑢_𝑖⋅▿𝜓_𝑖−_Ωdiv^→𝜓𝜎=𝜆_Ω^→𝑢⋅^→𝜓,∀^→𝜓∈𝐻¹₀(Ω)²,−_Ωdiv^→𝑢𝑣=0,∀𝑣∈𝐿²₀(Ω). (4.6)(4.7)

Let 𝑉_ℎ⊂𝐻¹₀(Ω)²,𝑊_ℎ⊂𝐿²₀(Ω) be two mixed finite element spaces. The mixed finite element form is as follows.

Seek (𝜆_ℎ,^→𝑢_ℎ,𝜎_ℎ)∈𝐑×𝑉_ℎ×𝑊_ℎ with |^→𝑢_ℎ|₁=1 such that _Ω²_𝑖=1▿𝑢_ℎ𝑖⋅▿𝜓_𝑖−_Ωdiv^→𝜓_ℎ𝜎_ℎ=𝜆_Ω^→𝑢_ℎ⋅^→𝜓,∀^→𝜓∈𝑉_ℎ,−_Ωdiv^→𝑢_ℎ𝑣=0,∀𝑣∈𝑊_ℎ. (4.8)(4.9)

Denote 𝑉=𝐻¹₀(Ω)×𝐻¹₀(Ω),𝑊=𝐿²₀(Ω)=𝑣∈𝐿²(Ω)∶_Ω𝑣=0,𝐷=𝐿²(Ω)×𝐿²(Ω),𝑎^→𝑢,^→𝜓=_Ω²_𝑖=1▿𝑢_𝑖⋅▿𝜓_𝑖,𝑏^→𝑢,𝑣=−_Ωdiv^→𝑢𝑣,^→𝑢,^→𝜓_𝐷=_Ω^→𝑢⋅^→𝜓. (4.10) Let ‖^→𝑢‖_𝑎=𝑎(^→𝑢,^→𝑢). It is clear that ‖^→𝑢‖_𝑎=|^→𝑢|₁ 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., 𝑢,𝜓 should be written in the forms of ^→𝑢,^→𝜓).

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 (𝑤^→,𝑝)∈𝑉×𝑊 such that 𝑎𝑤^→,^→𝜓+𝑏^→𝜓,𝑝=^→𝑓,^→𝜓_𝐷,∀^→𝑏𝜓∈𝑉,_→𝑤,𝑣=0,∀𝑣∈𝑊. (4.11)(4.12)

Seek (𝑤^→_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝑊_ℎ such that 𝑎𝑤^→_ℎ,^→𝜓+𝑏^→𝜓,𝑝_ℎ=^→𝑓,^→𝜓_𝐷,∀^→𝜓∈𝑉_ℎ,𝑏𝑤^→_ℎ,𝑣=0,∀𝑣∈𝑊_ℎ. (4.13)(4.14)

From [26] we know that (4.11)-(4.12) satisfy conditions (1)–(3) in Brezzi-Babuska Theorem; therefore, there exists a unique solution (𝑤^→,𝑝)∈𝑉×𝑊 to the problem (4.11)-(4.12) and the following estimate is valid: 𝑤^→₁+𝑝₀≤𝐶_𝑝^→𝑓₀. (4.15) Condition (4) in Brezzi-Babuska Theorem holds since 𝑉_ℎ⊂𝑉 and 𝑊_ℎ⊂𝑊. Suppose that condition (5) in Brezzi-Babuska Theorem (discrete inf-sup condition) is valid, then there exists a unique solution (𝑤^→_ℎ,𝑝_ℎ)∈𝑉_ℎ×𝑊_ℎ to the problem (4.13)-(4.14), and the following error estimate is valid (see [27–29]): 𝑤^→−𝑤^→_ℎ1+𝑝−𝑝_ℎ0≤𝐶inf𝑤_{→𝜓∈𝑉ℎ}^→−^→𝜓₁+inf_{𝑣∈𝑊ℎ}𝑝−𝑣₀. (4.16)

We assume that the following a prior estimate holds: for any ^→𝑓∈𝐷,𝑤^→∈𝐻^1+𝑟(Ω)×𝐻^1+𝑟(Ω),𝑝∈𝐻^𝑟(Ω), and 𝑤^→_1+𝑟+𝑝_𝑟≤𝐶^→𝑓₀, (4.17) where 0<𝑟≤1 is a number determined by the maximal inner angle 𝜔 of Ω. When 𝜔<𝜋, 𝑟=1 (see [30]).

Suppose that the following estimate holds: for any 𝑤^→∈𝐻^1+𝑟(Ω)×𝐻^1+𝑟(Ω), and for any 𝑝∈𝐻^𝑟(Ω), inf𝑤_{→𝜓∈𝑉ℎ}^→−^→𝜓_𝑉+inf_{𝑣∈𝑊ℎ}𝑝−𝑣_𝑊≤𝐶ℎ𝑤^𝑟→_1+𝑟+𝑝_𝑟. (4.18)

From Section 2, we know that (4.6)-(4.7) and (4.8)-(4.9) have the following equivalent operator forms, respectively, 𝜆𝑇^→𝑢=^→𝑢,𝜎=𝑆𝜆^→𝑢,𝜆_ℎ𝑇_ℎ^→𝑢_ℎ=^→𝑢_ℎ,𝜎_ℎ=𝑆_ℎ𝜆_ℎ^→𝑢_ℎ. (4.19)(4.20) Moreover, 𝑇 and 𝑇_ℎ are all self-adjoint compact operators.

Theorem 4.1. Assume that discrete inf-sup condition, (4.17) and (4.18) hold; let 𝐻 be properly small; and (𝜆^ℎ,^→𝑢^ℎ,𝜎^ℎ) an approximate eigenpair obtained by Scheme 1. Then there exists ^→𝑢∈𝑀(𝜆) such that ^→𝑢^ℎ−^→𝑢_𝑎≤𝐶𝐻^4𝑟+ℎ^𝑟,𝜆^ℎ−𝜆≤𝐶𝐻^8𝑟+ℎ^2𝑟. (4.21)(4.22)

Proof. From Brezzi-Babuska Theorem, (4.17), and (4.18), we deduce 𝑇^→𝑓−𝑇_ℎ^→𝑓_𝑉+𝑆^→𝑓−𝑆_ℎ^→𝑓_𝑊≤𝐶ℎ^𝑟→𝑓_𝐷,∀^→𝑓∈𝐷. (4.23) By virtue of Nitsche technique (see [29]) and (4.17), we derive 𝑇^→𝑓−𝑇_ℎ^→𝑓_𝐷≤𝐶ℎ^𝑟inf_{𝑞∈𝑉ℎ}𝑇^→𝑓−𝑞_𝑉+inf_{𝑣∈𝑊ℎ}𝑆^→𝑓−𝑣_𝑊≤𝐶ℎ^2𝑟→𝑓_𝐷,∀𝑓∈𝐷. (4.24) Using (4.24), we have 𝑇−𝑇_ℎ𝐷=sup_{→𝑓∈𝐷,‖→𝑓‖𝐷=1}𝑇^→𝑓−𝑇_ℎ^→𝑓_𝐷≤𝐶ℎ^2𝑟⟶0(ℎ⟶0). (4.25) Hence, the conditions in Lemma 2.2 and Theorem 3.3 hold.
From (2.23), we know that 𝜆_ℎ−𝜆+^→𝑢_ℎ−^→𝑢_𝐷≤𝐶𝑇−𝑇_ℎ|_{𝑀(𝜆)𝐷}≤𝐶ℎ^2𝑟. (4.26) From (4.23), (4.18), and (4.17), we have 𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}=sup_{→𝑢∈𝑀(𝜆),→𝑢≠0}𝑇^→𝑢−𝑇_ℎ^→𝑢_𝑎→𝑢_𝑎≤𝐶ℎ^𝑟,inf_{𝑣∈𝑊ℎ}‖𝜎−𝑣‖_𝑊≤𝐶ℎ^𝑟‖𝜎‖_𝑟≤𝐶ℎ^𝑟→𝑢_𝐷. (4.27)(4.28) 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 𝐾^ℎ be a regular triangulation of Ω under the meaning of paper [32], and 𝑆^ℎ=𝑣∈𝐶Ω∶𝑣|_𝜅∈𝑃₁,𝜅∈𝐾^ℎ,𝑆^ℎ₀=𝑆^ℎ𝐻¹₀(Ω). (4.29) For any 𝜅∈𝐾^ℎ, let 𝑁₁, 𝑁₂, and 𝑁₃ denote barycentric coordinates. Denote 𝐵_ℎ={𝑣∶𝑣|_𝜅∈span{𝑁₁𝑁₂𝑁₃},𝜅∈𝐾^ℎ}, and set 𝑉_ℎ=𝑆^ℎ₀𝐵_ℎ²,𝑊_ℎ=𝑆^ℎ𝐿²₀(Ω). (4.30)

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 𝑟 satisfying (4.17), Scheme 1 for Mini mixed finite element is effective. Theorem 4.1 is valid.

4.2. 𝑃₁-𝑃₁ Mixed Finite Element

Consider two-scale discretization scheme of 𝑃₁-𝑃₁ mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).

Let 𝐾^2ℎ be a regular triangulation of Ω, and 𝐾^ℎ is the product of refining 𝐾^2ℎ in the middle point. The 𝑃₁-𝑃₁ mixed finite element space is defined by 𝑉_ℎ=𝑆^ℎ𝐻¹₀(Ω)²,𝑊_ℎ=𝑆^2ℎ𝐿²₀(Ω), (4.31) where 𝑆^2ℎ and 𝑆^ℎ are piecewise continuous linear polynomial spaces defined on 𝐾^2ℎ and 𝐾^ℎ, respectively.

From [33, Proposition 3.3], we know that 𝑃₁-𝑃₁ element satisfies discrete inf-sup condition. By the interpolation theory in Sobolev space, we conclude that (4.18) holds. Therefore, for 𝑟 satisfying (4.17), Scheme 1 for 𝑃₁-𝑃₁ 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: 𝑐²curlcurl^→𝑢=𝜔^2→𝑢,inΩ,div^→𝑢=0,inΩ,_→𝑢×^→𝛾=0,on∈𝜕Ω, (5.1)(5.2)(5.3) where Ω is a polyhedron in 𝐑^𝟑 and ^→𝛾 is the outward normal unit vector on 𝜕Ω.

In physics, ^→𝑢 in the above eigenvalue problem of electric field denotes electric field, 𝜔 denotes the time frequency, and 𝑐 is the speed of the light. Usually we set 𝜆=𝜔²/𝑐² which is called eigenvalue.

The spaces 𝐻(curl, Ω), 𝐻₀(curl, Ω) are defined in the usual way: 𝐻(curl,Ω)=^→𝑞∈𝐿₂(Ω)³∶curl^→𝑞∈𝐿₂(Ω)³,𝐻₀(curl,Ω)=^→𝑞∈𝐻(curl,Ω)∶^→𝑞×^→𝛾|_𝜕Ω=0. (5.4)

When Ω is convex polyhedron, we define the following function space: 𝜒=^→𝑞∈𝐻₀(curl,Ω)∶div^→𝑞∈𝐿²(Ω). (5.5) Denote ^→𝑞,^→𝜓₀=_Ω^→𝑞⋅^→𝜓𝑑𝑥,^→𝑞₀=^→𝑞,^→𝑞₀^1/2._→𝑞,^→𝜓_𝜒=curl^→𝑞,curl^→𝜓₀+div^→𝑞,div^→𝜓₀,^→𝑞_𝜒=^→𝑞,^→𝑞_𝜒^1/2. (5.6) From [34, 35], we see that 𝜒⊂𝐻¹(Ω)³; (^→𝑞,^→𝜓)_𝜒 is a coercive bilinear form in 𝜒, and ‖^→𝑞‖_𝜒 is a norm.

On the other hand, when Ω is nonconvex the maximal interior angle belongs to (𝜋,2𝜋). In this situation the problem is relatively complicated. Let 𝐸 denote a set of reentrant edge with dihedral angles belonging to (𝜋,2𝜋), and let 𝑑 denote the distance to the set 𝐸: 𝑑(𝑥)=dist(𝑥,∪_𝑒∈𝐸𝑒). We introduce a weight function 𝜔_𝑟 which is a nonnegative smooth function with respect to 𝑥. It can be represented by 𝑑^𝑟 in reentrant edge and angular domain. We write 𝜔_𝑟≃𝑑^𝑟. Define the weight function space: 𝐿²_𝑟(Ω)=𝑣∈𝐿²_loc(Ω)∶𝜔_𝑟𝑣∈𝐿₂(Ω,𝜒_𝑟)=^→𝑞∈𝐻₀(curl,Ω)∶div^→𝑞∈𝐿²_𝑟(Ω). (5.7) Denote ^→𝑞,^→𝜓_𝐿2𝑟=_Ω𝜔²_𝑟^→𝑞⋅^→𝜓𝑑𝑥,^→𝑞_𝐿2𝑟=^→𝑞,^→𝑞_𝐿^1/2_2𝑟,_→𝑞,^→𝜓_𝜒𝑟=curl^→𝑞,curl^→𝜓₀+div^→𝑞,div^→𝜓_𝐿2𝑟,^→𝑞_𝜒𝑟=^→𝑞,^→𝑞_𝜒^1/2_𝑟. (5.8) Let 𝜎^𝑁_Δ be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition: 𝜙∈𝐻¹₀(Ω)∶Δ𝜙∈𝐿₂(Ω)⊂_{𝑠<𝜎𝐷Δ}𝐻^𝑠(Ω),𝜙∈𝐻¹₀(Ω)∶Δ𝜙∈𝐿₂(Ω)⊄𝐻^𝜎𝐷Δ(Ω). (5.9) From the regularity estimate we know 𝜎^𝐷_Δ∈(3/2,2). Let 𝑟_min=2−𝜎^𝐷_Δ.

From [36, 37], we know that, for all 𝑟∈(𝑟_min,1), the seminorm |^→𝑞|_𝜒𝑟 is a norm in 𝜒_𝑟, and 𝜒_𝑟∩𝐻¹(Ω)³ is dense in 𝜒_𝑟.

In the following discussion, we will use 𝜒_𝑟,𝐿²_𝑟(Ω) both for non-convex and convex domain. We take 𝑟∈(𝑟_min,1) for non-convex domain; otherwise, we take 𝜒_𝑟=𝜒,𝐿²_𝑟(Ω)=𝐿²(Ω).

By introducing Lagrange multiplier 𝜎, [36, 38, 39] changed (5.1)–(5.3) into the mixed variational formulation: find (𝜆,^→𝑢,𝜎)∈𝐑⁺×𝜒_𝑟×𝐿²_𝑟(Ω) such that ^→𝑢,^→𝜓_𝜒𝑟+div^→𝜓,𝜎_𝐿2𝑟=𝜆^→𝑢,^→𝜓₀,∀^→𝜓∈𝜒_𝑟,div^→𝑢,𝑣_𝐿2𝑟=0,∀𝑣∈𝐿²_𝑟(Ω). (5.10)(5.11)

Let 𝐾^ℎ be a regular simplex partition (tetrahedral partition) of Ω with the mesh diameter ℎ. Define the 𝑃_𝑘+1-𝑃_𝑘 finite element space as follows: 𝑉_ℎ=^→𝑞∈𝐶⁰Ω³∶^→𝑞×^→𝛾|_𝜕Ω=0,^→𝑞|_𝜅∈𝑃_𝑘+1(𝜅)³,∀𝜅∈𝐾^ℎ,𝑊_ℎ=𝑣∈𝐶⁰Ω∶𝑣|_𝜅∈𝑃_𝑘(𝜅),∀𝜅∈𝐾^ℎ,𝑣|_𝐸ℎ=0. (5.12) Here we set 𝐸_ℎ=∪_{𝜅∈𝐾ℎ,𝜕𝜅∩𝐸≠𝜙}𝜅. 𝑣|_𝐸ℎ=0 means that 𝑣 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 (𝜆_ℎ,^→𝑢_ℎ,𝜎_ℎ)∈𝐑⁺×𝜒_ℎ×𝑀_ℎ such that ^→𝑢_ℎ,^→𝜓_𝜒𝑟+div^→𝜓,𝜎_ℎ𝐿2𝑟=𝜆_ℎ^→𝑢_ℎ,^→𝜓₀,∀^→𝜓∈𝜒_ℎ,div^→𝑢_ℎ,𝑣_𝐿2𝑟=0,∀𝑣∈𝑀_ℎ. (5.13)(5.14)

Set 𝑉=𝜒_𝑟,‖⋅‖_𝑉=‖⋅‖_𝜒𝑟,𝑊=𝐿²_𝑟(Ω),‖⋅‖_𝑊=‖⋅‖_𝐿2𝑟,𝐷=𝐿₂(Ω)³,‖⋅‖_𝐷=‖⋅‖₀,𝑎^→𝑞,^→𝜓=^→𝑞,^→𝜓_𝜒𝑟,𝑏^→𝜓,𝑣=div^→𝜓,𝑣_𝐿2𝑟. (5.15) 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., 𝑢,𝜓 should be written in the forms of ^→𝑢,^→𝜓).

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 𝑎(⋅,⋅) and 𝑏(⋅,⋅) are continuous bilinear forms on 𝑉×𝑉 and 𝑉×𝑊, respectively. 𝑉 is compact embedded in 𝐷 (when Ω is convex, it is valid obviously; when Ω is non-convex, see [36]).

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

Find (𝑤^→,𝑝)∈𝜒_𝑟×𝐿²_𝑟(Ω) such that 𝑤^→,^→𝜓_𝜒𝑟+div^→𝜓,𝑝_𝐿2𝑟=^→𝑓,^→𝜓₀,∀^→𝜓∈𝜒_𝑟,div𝑤^→,𝑣_𝐿2𝑟=0,∀𝑣∈𝐿²_𝑟(Ω). (5.16)(5.17)

For the problem (5.16)-(5.17) and its 𝑃_𝑘+1-𝑃_𝑘 element approximation, people have already proved the conditions in Brezzi-Babuska Theorem hold (see [38, 40, 41]).

Therefore, we can define operators 𝑇,𝑆,𝑇_ℎ,𝑆_ℎ; 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), (^→𝑢,𝜎), satisfy 𝜎=𝑆(𝜆^→𝑢)=0 and ^→𝑢∈𝜒_𝑟 with div^→𝑢=0.

Denote 𝜀_𝜆(ℎ)=sup_→𝑢∈𝑀_(𝜆)inf_{→𝜓∈𝑉ℎ}𝑇^→𝑢−^→𝜓_𝑎. (5.18)

For the 𝑃_𝑘+1-𝑃_𝑘 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 𝑃_𝑘+1-𝑃_𝑘 element approximation of (5.16)-(5.17), there exists 𝑟(ℎ)>0,  𝑟(ℎ)→0(ℎ→0) such that sup_{→𝑓∈𝐷,→𝑓≠0}𝑇−𝑇_ℎ^→𝑓_𝑉→𝑓_𝐷≤𝑟(ℎ). (5.19)

This lemma is very important. It tells us that ‖𝑇−𝑇_ℎ‖_𝐷→0(ℎ→0). Based on this lemma, [38] also proved the following conclusion.

Lemma 5.3. For the 𝑃_𝑘+1-𝑃_𝑘 element approximation of (5.10)-(5.11), the following estimate is valid: 𝜆−𝜆_ℎ≤𝐶𝜀_𝜆(ℎ)²,_→𝑢_ℎ−^→𝑢_𝑎≤𝐶𝜀_𝜆(ℎ). (5.20)(5.21)

Theorem 5.4. Let (𝜆^ℎ,^→𝑢^ℎ) be the approximate 𝑃_𝑘+1-𝑃_𝑘 element eigenpair obtained by Scheme 1; then there exists ^→𝑢∈𝑀(𝜆) such that ^→𝑢^ℎ−^→𝑢_𝑎≤𝐶𝜀_𝜆(𝐻)³+𝜀_𝜆(ℎ),𝜆^ℎ−𝜆≤𝐶𝜀_𝜆(𝐻)⁶+𝜀_𝜆(ℎ)². (5.22)(5.23)

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 𝜎=0.
From (2.18), we deduce 𝑇−𝑇_ℎ|_{𝑀(𝜆)𝑎}=sup_→𝑢∈𝑀_(𝜆)𝑇^→𝑢−𝑇_ℎ^→𝑢_𝑎≤𝐶sup_→𝑢∈𝑀₍^𝜆₎inf_{→𝜓∈𝑉ℎ}𝑇^→𝑢−^→𝜓_𝑎≤𝐶𝜀_𝜆(ℎ). (5.24) From (5.20), we derive 𝜆_𝐻−𝜆≤𝐶𝜀_𝜆(𝐻)². (5.25) Substituting (5.24) and (5.25) into (3.7), we get (5.22).
Since 𝜎=0, (3.8) can be simplified to 𝜆^ℎ−𝜆≤𝐶^→𝑢^ℎ−^→𝑢²_𝑎. (5.26) Substituting (5.22) into (5.26), we obtain (5.23).

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(𝑟−𝑟_min,𝜎^𝑁_Δ−1).

Corollary 5.5. Under the condition of Theorem 5.4, if Ω is a convex polyhedron there holds ^→𝑢^ℎ−^→𝑢_𝑎≤𝐶𝐻³,𝜆+ℎ_ℎ−𝜆≤𝐶𝐻⁶+ℎ²; (5.27)(5.28) if Ω is a non-convex polyhedron, then ^→𝑢^ℎ−^→𝑢_𝑎≤𝐶𝐻^3𝜇+ℎ^𝜇(),𝜆,∀𝜇∈0,𝜏_ℎ−𝜆≤𝐶𝐻^6𝜇+ℎ^2𝜇,∀𝜇∈(0,𝜏). (5.29)(5.30)

Proof. When Ω is a convex polyhedron, for any ^→𝑢∈𝑀(𝜆) we have ^→𝑢=𝑇(𝜆^→𝑢)∈𝐻²(Ω) (see [35], or [42, equation (44)]); therefore 𝜀_𝜆(𝐻)≤𝐶𝐻,𝜀_𝜆(ℎ)≤𝐶ℎ. (5.31) Substituting these two formulae into (5.22) and (5.23), we obtain (5.27) and (5.28), respectively. When Ω is non-convex, for all ^→𝑢∈𝑀(𝜆), from [38, equation (36)] we know that 𝜀_𝜆(𝐻)≤𝐶𝐻^𝜇,𝜀_𝜆(ℎ)≤𝐶ℎ^𝜇. (5.32) 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 𝜆_1,𝐻,𝜆_2,𝐻,…,𝜆_4,𝐻 be the first four eigenvalues computed by using mixed finite method directly on coarse mesh 𝐾^𝐻 and 𝜆_1,ℎ,𝜆_2,ℎ,…,𝜆_4,ℎ the first four eigenvalues computed by using mixed finite method directly on fine mesh 𝐾^ℎ. Let 𝜆^ℎ₁,𝜆^ℎ₂,…,𝜆^ℎ₄ denote the first four eigenvalues computed by Scheme 1 on the meshes 𝐾^𝐻 and 𝐾^ℎ.

Example 6.1. Consider the Stokes eigenvalue problem (4.1)–(4.3), where Ω⊂𝐑² is a unit square domain. The smallest eigenvalue 𝜆₁ is approximately equal to 52.3446911 for this problem.

We adopt a uniform isosceles right triangulation for the domain Ω (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 𝐾^𝐻 and 𝐾^ℎ.

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.

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 Ω is a square domain [0,𝜋]×[0,𝜋] or an L-shaped domain [−1,0]×[−1,0]∪[−1,1]×[0,1]. For the square domain, the first four exact eigenvalues are 𝜆₁=1,𝜆₂=1,𝜆₃=2,𝜆₄=4 for this problem. For the L-shaped domain, the first four approximate eigenvalues are 𝜆₁≈1.475622,𝜆₂≈3.534031,𝜆₃≈9.869604,𝜆₄≈9.869604.

We adopt a uniform isosceles right triangulation for Ω (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 [0,𝜋]×[0,𝜋]) to produce the meshes 𝐾^𝐻 and 𝐾^ℎ.

Figure 2 

We use 𝑃₂-𝑃₁ mixed finite element to solve this problem. The definition of 𝑃₂-𝑃₁ mixed finite element space is given by 𝑉_ℎ=^→𝑞∈𝐶⁰Ω²∶^→𝑞×^→𝛾|_𝜕Ω=0,^→𝑞|_𝜅∈𝑃₂(𝜅)²,∀𝜅∈𝐾^ℎ,𝑊_ℎ=𝑣∈𝐶⁰Ω∶𝑣|_𝜅∈𝑃₁(𝜅),∀𝜅∈𝐾^ℎ,𝑣|_𝐸ℎ=0. (6.1)

We compute the first four approximate eigenvalues by using Scheme 1 with 𝑃₂-𝑃₁ element on the mesh 𝐾^𝐻 and 𝐾^ℎ. The numerical results are listed in Tables 2, 3, and 4.

Table 2 

The results on the square by Scheme 1 (𝑃₂-𝑃₁ element) for eigenvalue problem of electric field (𝑟=0).

Table 3 

The results on the L-shaped domain by Scheme 1 (𝑃₂-𝑃₁ element) for eigenvalue problem of electric field (𝑟=0.5).

Table 4 

The results on the L-shaped domain by Scheme 1 (𝑃₂-𝑃₁ element) for eigenvalue problem of electric field (𝑟=0.95).

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

In this paper, 𝐶 denotes a positive constant independent of ℎ, 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).