Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods

A short survey of lower bounds of eigenvalue problems by nonconforming finite element methods 
is given. The class of eigenvalue problems considered covers Laplace, Steklov, biharmonic and 
Stokes eigenvalue problems.


1.
Introduction. It is well known that the eigenvalue problems are very important, which appear in many fields, such as fluid mechanics, quantum mechanics, stochastic process and etc.. Thus, a fundamental task of computational mathematics is to find the eigenvalues of partial differential equations. Since last century, abundant works have been dedicated to this topic.
In the famous paper [18], Feng cites the pioneer work of Pólya in computing the upper bound of Laplace eigenvalue problem. It is known that based on the minimum-maximum principle discovered by Rayleigh, Poincaré, Courant and Fischer etc., any conforming finite element method will give the upper bound of the eigenvalue (see [40]). Nevertheless, for the lower bound, until 1979, Rannacher [39] first gives some numerical results for the biharmonic eigenvalue problem.
There is few work on analysis of the lower bound for a long time. Inspired by the minimum-maximum principle, people try to find the lower bound with the nonconforming element methods. Recently, a series of works make progress in this aspect, e.g. Lin and Lin [29] use the integral-identity technique to obtain the asymptotic expansion of the eigenvalue approximations by nonconforming finite element method; also see the numerical reports of Liu and Liu [34], Liu and Yan [35], Lin, Huang and Li [28], and the work of Yao and Qiao [47]. Another way by Armentano and Durán [3] is to use a general kind of expansion method to get the lower bound, which is of less restriction on the partition compared with the integral-identity skill. Then this idea has been extended and generalized to more and more nonconforming finite element methods and eigenvalue problems: Zhang, Yang and Chen [49], Yang and Bi [42], Li [26] considered the Wilson's element, Li [25] analyzed EQ rot 1 element, etc.. For more details, please read papers: Zhang, Yang and Chen [49], Yang, Zhang and Lin [46], Hu, Huang and Lin [23], Lin, Xie, etc. [31], Luo, Lin and Xie [36]. The paper [49] surveys the topic of obtaining lower bound for the eigenvalues of second order elliptic and biharmonic operators by the corresponding nonconforming finite elements. In [23], the lower bounds of eigenvalue approximations have been analyzed for the Laplace, biharmonic and general 2morder eigenvalue problems in a general framework. The lower bounds of eigenvalues of the Stokes, Steklov eigenvalue problems have been analyzed in [32] and [27], respectively. The lower-bound result of Laplace eigenvalue problem without the convergence-order assumption is given in Luo, Lin and Xie [36]. Hu, Huang and Shen [22] gets the lower bound of Laplace eigenvalue problem by conforming linear and bilinear elements together with the mass lumping method.
The aim of this paper is to introduce the current art of the lower bounds of the eigenvalue problems by nonconforming finite element methods. Furthermore, we only concentrate on the class of the nonconforming finite element methods which can produce the lower bounds without any additional regularity assumption of the eigenfunctions. For simplicity, we only discuss the problem in R 2 , but the methods and results here can be extended to the case R 3 . In this paper, we will use the standard notation of Sobolev spaces (see, e.g., [14,16,40]).
The outline of the paper will go as follows. In Section 2, some abstract results for the eigenvalue problem by nonconforming finite element methods are introduced. In Section 3, 4 and 5, we will give lower-bound results for the Laplace, Steklov, biharmonic eigenvalue problems, respectively. Section 6 is devoted to analyzing the lower-bound results of the Stokes eigenvalue problem by mixed finite element methods. Some concluding remarks are given in the last section.
2. The eigenvalue problem and finite element methods. In this section, we introduce some notation and error estimates of the nonconforming finite element approximation for eigenvalue problems. The letter C (with or without subscripts) denotes a generic positive constant which may be different at its different occurrences through the paper.
Let (V, · ) be a real Hilbert space with inner product (·, ·) and norm · , respectively. Let a(·, ·) be a symmetric bilinear form on V × V satisfying From (1) and (2), we know that · a := a(·, ·) 1/2 and · are two equivalent norms on V and we shall use a(·, ·) and · a as the inner product and norm on V in the rest of this paper. Let W denote another Hilbert space and b(·, ·) be a symmetric bilinear form on W × W satisfying 0 < b(w, w), ∀w ∈ W and w = 0.
We assume b(·, ·) is an inner product on W and the norm · is relatively compact with respect to the norm · b = b(·, ·) 1/2 in the sense that any sequence in V which is bounded in · , one can extract a subsequence which is Cauchy in W with respect to · b .
In this paper we are concerned with the abstract eigenvalue problem: From [5], we know that the eigenvalue problem (4) has a positive eigenvalue sequence {λ j } with 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ k ≤ · · · , lim k→∞ λ k = ∞, and the corresponding eigenfunction sequence {u j } From the result in [5], the Rayleigh quotient is defined by Let T h be a quasi-uniform decomposition of the polygonal domain Ω into triangles or rectangles (c.f. [14,16]). The diameter of a cell K ∈ T h is denoted by h K . The mesh diameter h describes the maximum diameter of all cells denotes the edge set lying on the boundary ∂Ω. The finite element space V h is the corresponding finite element space on the partition, i.e. V h V as a nonconforming space and V h ⊂ V a conforming space.
In the rest of this paper, we only concentrate on the nonconforming finite element case and assume the bilinear form b(·, ·) can also be defined on V h × V h . The finite element approximation of (4) is defined as follows: where the bilinear form a h (·, ·) coincides with an elementwise representation of a(·, ·) in the nonconforming situation, e.g.
We assume the bilinear form a h (·, ·) is V h -elliptic on V + V h . Thus we define the For the eigenvalue problem (5), the Rayleigh quotient holds for the eigenvalue Similarly, the discrete eigenvalue problem (5) has also an eigenvalue sequence {λ j,h } with 0 < λ 1,h ≤ λ 2,h ≤ · · · ≤ λ N h ,h , and the corresponding discrete eigenfunction sequence {u j,h } There is a classical result about eigenvalue: minimum-maximum principle. Let λ j be the j-th eigenvalue of (4) and λ j,h be the j-th eigenvalue of (5), respectively. Arranging them by increasing order, then we have (see e.g., [5]) From (6), it is obvious that the conforming element methods (V h ⊂ V ) can only obtain the upper bounds of eigenvalue problems.
In order to give the error estimates of the eigenpair approximation by finite element methods, we define the operator T : for any f ∈ W . As we know, the operator T is compact. Then the eigenvalue problem (4) can be written as We also define the corresponding discrete operator T h : for any f ∈ W . Similarly the discrete eigenvalue problem (5) can be written as Let M (λ j ) denote the eigenfunction set corresponding to the eigenvalue λ j which is defined by w is an eigenfunction of (4) corresponding to λ j and w b = 1 .
Now we state the convergence result of the eigenvalue problems by nonconforming finite element methods. For this aim, we define the following notation be the j-th nonconforming finite element eigenpair approximation satisfying (5). Then λ j,h → λ j and there exist u j ∈ M (λ j ) such that For the eigenvalue problem, we have the following basic expansion which was introduced in [3,49] and has been extensively used in [23,31,41,42,46]. For the convenience of reading, we give a simple proof.
is the corresponding j-th eigenpair of the discrete problem (5). We have the following expansion Then (17) can be obtained and we complete the proof.
3. Laplace eigenvalue problem. In this section, we consider the lower-bound results of the Laplace eigenvalue problem by nonconforming finite element methods. The concerned Laplace eigenvalue problem can be stated as follows: where Ω is a bounded POLYGONAL domain in R 2 with continuous Lipschitz boundary ∂Ω. In order to give the error estimates of the finite element method, we assume the eigenfunction has the regularity where 0 < γ ≤ 1 depends on the maximum angle of the boundary ∂Ω and γ = 1 when the domain Ω is convex (c.f. [20]).
• ECR element is defined on the triangular partition and where K, K 1 , K 2 ∈ T h . • EQ rot 1 element is defined on the rectangular partition and where K, K 1 , K 2 ∈ T h . Based on the ECR and EQ rot 1 elements, the discrete eigenvalue problem (5) for the eigenvalue problem (18) can be define with The interpolation operator Π h : V −→ V h corresponding to ECR and EQ rot 1 elements can be defined in the same way (c.f. [23,30,32]): Lemma 3.1. ( [36]) For any u ∈ V , the interpolation defined in (22)-(23) has the following results Furthermore, the interpolation operator has error estimates for any u ∈ H 1+γ (Ω). Now we state a lower bound of the convergence rate for the eigenfunction approximation by finite element methods which will be used in the analysis of the lower bound of eigenvalue approximation.
For analyzing the lower bounds of the eigenvalue problems, we need to show that ) Assume the nonconforming finite element owns a type of interpolation operator Π h satisfying the orthogonal property (24). We have the following error estimate where ϕ ψ = T ψ, ϕ h = T h ψ and First for the first term in the right hand side of (31), we have the estimate From the standard error estimate theory of the nonconforming finite element method, the following estimate holds From the orthogonal property (24), we have the estimate for the third term in the right hand side of (31) Combining (31), (32), (33), (34) and (35), we have The desired inequality (30) can be obtained by combining (29), (36) and f b = 1 and we complete the proof.
From [4,5,23,31,41], the following basic error estimates for the two nonconforming finite elements hold The following lower-bound result of the eigenvalue approximation by ECR or EQ rot 1 element has been discussed in [23,25,31,36]. Here we state the result and give a simple proof.
Theorem 3.4. ( [36]) Let λ j and λ j,h be the j-th exact eigenvalue and its corresponding numerical approximation by ECR or EQ rot 1 element. When h is small enough, we have Proof. The result for ECR and EQ rot 1 elements can be proved in the uniform way. We choose v h = Π h u j in (17). For the second term in (17), from (39) and Lemma 3.1, we have 1+γ . For the third term in (17), from (27) Here and hereafter in this paper Π 0 denotes the piecewise constant interpolation. Together with (24) and Lemma 3.2, the first positive term is dominant in (17) and then (40) can be derived with (37). 4. Steklov eigenvalue problem. In this section, we are concerned with the lowerbound results of the Steklov eigenvalue problem. Steklov eigenvalue problem arises in a number of applications such as surface waves (see [9]), stability of mechanical oscillators immersed in a viscous fluid (see [17]), the vibration modes of a structure in contact with an incompressible fluid (see [10]), the antiplane shearing on a system of collinear faults under slip-dependent friction law (see [13]), vibrations of a pendulum (see [1]), eigen oscillations of mechanical systems with boundary conditions containing the frequency (see [21]).
The analysis of the conforming finite element methods for the Steklov eigenvalue problems has been given by Bramble and Osborn [11], Andreev and Todorov [2]. Recently, the nonconforming finite element methods for the Steklov eigenvalue problems have also been analyzed by Yang, Li and Li [44] on the convex domain and by Li, Lin and Xie [27] on both convex and concave domains.
In this section, we are concerned with the following Steklov eigenvalue problem where Ω ⊂ R 2 is a bounded polygonal domain and ∂ ν denote the outward normal derivative on ∂Ω.
The corresponding weak form (4) for the eigenvalue problem (41) can be defined with V = H 1 (Ω), · a = · 1 , W = L 2 (∂Ω) and In this section, we consider the lower bounds of the Steklov eigenvalue problem by ECR and EQ rot 1 elements defined in (20) and (21), respectively. Then the nonconforming finite element approximation of (41) can be defined by (5) with  (41), the corresponding operator T defined by (7) has the regularity results where 0 < γ ≤ 1 depends on the maximum angle of the boundary ∂Ω.
where C is a constant independent of h and λ j .
Proof. The estimates (44) and (46) can be obtained from Theorem 3.1 in [27]. The estimate (45) can be proved by the same process in the proof of Theorem 3.3.
In order to derivative the lower bounds of the Steklov eigenvalue problem, we need the following estimates.
for ECR and EQ rot 1 elements. Proof. From (24), we have With the help of (23), the following estimate holds This is the desired result (47) and we complete the proof.
Lemma 4.4. Let u ∈ H 1+γ (Ω) be an eigenfunction of (41). Then the following estimate holds Proof. The interpolation error in the · b can be derived from the following trace inequality v 0,∂K ≤ Ch where |R| ≤ Ch 1+2γ .
Proof. First from (43), we have u j ∈ H 1+γ (Ω). Taking v h = Π h u j in (17), we estimate the second, third and fourth terms on the right-hand side of (17). From (45) and (49), we have In addition, we introduce the piecewise constant interpolation operator J 0 on ∂Ω. Then, from (49), we have Corollary 1. let λ j and λ j,h be the j-th exact eigenvalue and its corresponding numerical approximation by ECR or EQ rot 1 element. Under the condition of γ > 1/2, we have when h is small enough.
Proof. From Lemma 3.2, (47) and |R| ≤ Ch 1+2γ , we know that the second and the third terms on the right-hand side of (50) are infinitesimals of higher order than the first term u j − u j,h 2 a,h . So the sign of the right-hand side of (50) is determined by the first positive term. Thus (53) holds.

Biharmonic eigenvalue problem.
In this section, we analyze the lower-bound results of the biharmonic eigenvalue problem: Find (λ, u) such that on ∂Ω, where Ω ⊂ R 2 is a bounded polygonal domain with Lipschitz continuous boundary ∂Ω, ∂ ν denotes the outward normal derivative on ∂Ω.
The corresponding weak form (4) for the eigenvalue problem (54) can be defined with V = H 2 0 (Ω), W = L 2 (Ω) and Evidently the bilinear form a(·, ·) is symmetric, continuous and coercive over the product space V × V .
Here we analyze the triangular Morley element which is defined by where P 2 = span 1, x, y, x 2 , xy, y 2 .
Based on the nonconforming finite element space V h defined in (55), the discrete eigenvalue problem (5) for the eigenvalue problem (54) can be defined with The eigenfunction of (54) has the following regularity where 0 < γ ≤ 1 depends on the maximum angle of the boundary ∂Ω. It is well known that γ = 1 when Ω is convex.
Here, we give the analysis of the lower-bound results of eigenvalues by the Morley element method. First, we define the corresponding interpolation operator Π h : From [4,5,31,39,41], the eigenpair approximation (λ j,h , u j,h ) obtained by the Morley element method defined (55) has the following error estimates Lemma 5.1. The interpolation operator Π h defined by (57) has the following error estimates where u ∈ H 2+γ (Ω). Furthermore, the following orthogonal property holds (c.f. [23,31,45]) Similarly, we have the following lower-bound result of the convergence rate by the Morley element method.  [23,31,45]) Assume λ j and λ j,h are the j-th eigenvalues of (54) and its approximation by Morley element, respectively. We have the following lowerbound result when h is small enough.
Proof. Let v h = Π h u j . We estimate the terms of (17).
First, we have the following estimates for the second and third terms Combining (62), (63), (65), (66) and Lemma 2.2, we know that the first positive term is dominant. This means we obtain the desired result (64) and complete the proof. 6. Stokes eigenvalue problem. In this section, we are concerned with the lowerbound results of the Stokes eigenvalue problems by nonconforming mixed finite element methods. The study of Stokes eigenmodes is required when the dynamics behaviors governed by the Navier-Stokes equations resulting from the way this nonlinear dynamics is controlled by diffusion. For the other reasons to study the Stokes eigenmodes, please read the papers [8,24].
Osborn [38], Mercier, Rappaz and Raviart [37] give an abstract analysis for the eigenpair approximations by mixed/hybrid finite element methods based on the general theory of compact operators (c.f. [15]). Recently, Xie, Yin and Gao in [48] obtains asymptotic error expansions of the Stokes eigenvalue approximations and gives extrapolation schemes to improve the convergence order for the eigenvalue approximations. Their numerical results show the lower-bound phenomena.
The corresponding weak form of (67) is: For the eigenvalue, there exists the following Rayleigh quotient expression The eigenfunction of (67) has the following regularity (c.f. [6,7]) where 0 < γ ≤ 1 depends on the maximum angle of the boundary ∂Ω. It is well known that γ = 1 when Ω is convex.
Associated with the partition T h , we define the nonconforming finite element spaces V h ⊂ V and W h ⊂ W (c.f. [12,19]). The nonconforming finite element (20) or (21) for ECR or EQ rot 1 , respectively. The finite element space W h is defined as Now, let us define the approximation of the eigenpair (λ, u, p) by the nonconforming mixed finite element method: where the bilinear forms a h (·, ·) and b(·, ·) are defined by Since the bilinear a h (·, ·) is symmetric and elliptic on V + V h , we can define the following norm on From (72), we can know the following Rayleigh quotient for λ h also holds We know from [5] the Stokes eigenvalue problem (72) has an eigenvalue sequence λ j,h with 0 < λ 1,h ≤ λ 2,h ≤ · · · ≤ λ k,h ≤ · · · ≤ λ N h ,h , and the corresponding eigenfunction sequence (u j,h , p j,h ) The well-posedness (no spurious eigenvalues) of the discrete eigenvalue problem (72) can be guaranteed by the fact that the corresponding approximation spaces V h and W h satisfy the Babuška-Brezzi condition (c.f. [12,19]) The eigenpair approximation (λ j,h , u j,h , p j,h ) by the mixed finite element space V h × W h has the following error estimates (c.f. [12,19,37,38]): Similarly, we have the following lower-bound result of the convergence rate for the Stokes eigenvalue problem.
The following lemma is similar to Lemma 2.2 but with an additional term. Lemma 6.2. ( [32]) Assume that (λ j , u j , p j ) ∈ R × V × W is an eigenpair of (67) and (λ j,h , u j,h , p j,h ) ∈ R × V h × W h is the corresponding eigenpair approximation and W h ⊂ W . We have the following expansion Proof. Since s(u j , u j ) = 1, s(u j,h , u j,h ) = 1, a h (u j , u j ) = λ j and a h (u j,h , u j,h ) = λ j,h , we have From (72), we have Adding −2b h (v h , p j,h ) to both sides of (80) and combining the fact that b h (u j , p j,h ) = 0 and (81) leads to ). This is the desired result (79) and we complete the proof.
Applying Lemma 6.2 to ECR and EQ rot 1 elements, we have the following lowerbound result. Theorem 6.3. ( [32]) Let λ j and λ j,h be the j-th exact eigenvalue and its corresponding numerical approximation by ECR/P 0 or EQ rot 1 /P 0 , respectively. Then, when h is small enough, we have Proof. Setting v h = Π h u j , we estimate all the five terms in the eigenvalue expansion (79).
First, we have the following estimates for the second and third terms Similarly, the following orthogonal property holds Combining the integration by parts and (22) Combining (78), Lemma 6.2, (83), (84), (85) and (86), we know the first positive term in the right hand side of (79) is dominant. Then we obtain the desired result (82) and complete the proof. 7. Concluding remarks. In this paper, lower-bound results of the eigenvalue problems by nonconforming finite element methods are introduced. Here, we only concentrate on the class of nonconforming elements which can produce the lower bounds of eigenvalues without any additional eigenfunction regularity assumption. From the analysis given in this paper, we find that the main ingredients of producing the lower-bound results is the eigenvalue expansion (17), the lower bound of the convergence rate (29) by finite element methods and the properties of the associated nonconforming finite element methods. For the numerical experiments, please read the corresponding references. Of course, the idea and results can be extended to more general eigenvalue problems and other nonconforming finite element methods.