# A new finite element approach for the Dirichlet eigenvalue problem

In this paper, we propose a new finite element approach, which is different than the classic Babuska-Osborn theory, to approximate Dirichlet eigenvalues. The Dirichlet eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Using conforming finite elements, the convergence is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.

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## 1 Introduction

Finite element methods for eigenvalue problems have been studied extensively [1, 2, 9]. In this paper, we propose a new finite element approach for the Dirichlet eigenvalue problem. The problem is formulated as an eigenvalue problem of a holomorphic Fredholm operator function [3]. Using Lagrange finite elements, the convergence is proved by the abstract approximation theory for holomorphic operator functions [7, 8].

The new approach has the following characteristics: 1) it provides a new finite element methodology which is different than the classic Babuška-Osborn theory; 2) it can be applied to a large class of nonlinear eigenvalue problems [4]; and 3) combined with the spectral indicator method [5, 6, 4], it can be parallelized to compute many eigenvalues effectively.

The rest of the paper is arranged as follows. In Section 2, preliminaries of holomorphic Fredholm operator functions and the associated abstract approximation are presented. In Section 3, we reformulate the Dirichlet eigenvalue problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The linear Lagrange finite element is used for discretization and the convergence is proved using the abstract approximation results of Karma [7, 8]. In Section 4, the spectral indicator method [5, 6, 4] is applied to compute the eigenvalues of the unit square.

## 2 Preliminaries.

We present some preliminaries on the eigenvalue approximation theory of holomorphic Fredholm operator functions following [7, 8]. Let be complex Banach spaces, be compact. Denote by the space of bounded linear operators and the set of holomorphic Fredholm operator functions of index zero [3]. Assume that . We consider the problem of finding , such that

 F(λ)u=0. (2.1)

The resolvent set and the spectrum of are defined as

 ρ(F)={λ∈Ω: F(λ)−1∈L(Y,X)},σ(F)=Ω∖ρ(F).

Throughout the paper, we assume that . Then the spectrum has no cluster points in , and every is an eigenvalue [7].

To approximate the eigenvalues of , we consider a sequence of operator functions . Assume the following properties hold.

• There exist Banach spaces , and linear bounded mappings with the property

 limn→∞∥pnv∥Xn=∥v∥X,v∈X,limn→∞∥qnv∥Yn=∥v∥Y,v∈Y. (2.2)
• is equibounded on , i.e., there exists a constant such that

 ∥Fn(λ)∥≤c∀λ∈Ω,n∈N.
• approximates for every , i.e.,

 limn→∞∥Fn(λ)pnu−qnF(λ)u∥Yn=0∀u∈X.
• is regular for every , i.e., if and is compact in the sense of Karma [7], then

## 3 Finite Element Approximation

Let be a bounded Lipschitz domain. The Dirichlet eigenvalue problem is to find and such that

 −△u=λu  in Dandu=0  % on ∂D. (3.3)

The associated source problem is, given , to find such that

 −△u=f  in Dandu=0  on ∂D. (3.4)

For , the weak formulation of (3.4) is finding such that

 a(u,v)=(f,v)∀ v∈H10(D), (3.5)

where

 a(u,v)=∫D∇u⋅∇v dx,(f,v)=∫Dfv dx.

Due to the wellposeness of (3.5) (see, e.g., [9]), there exists a linear compact solution operator such that . The Dirichelt eigenvalue problem (3.3) is equivalent to the operator eigenvalue problem: .

Assume that . Define a nonlinear operator function by

 F(λ):=T−1λI,λ∈Ω, (3.6)

where is the identity operator. Clearly, is a Dirichlet eigenvalue of (3.3) if and only if is an eigenvalue of .

###### Lemma 3.1

Let be a compact set. Then is a holomorphic Fredholm operator function of index zero.

Proof. Since is compact and is the identity operator, is a Fredholm operator of index zero. Clearly, is holomorphic in .

In the rest of the paper, stands for and is a generic constant. Let be a regular triangular mesh for with mesh size and is the linear Lagrange element space associated with . Then the discrete formulation of (3.5) is to find , such that

 a(uh,vh)=(f,vh)=(phf,vh)∀ vh∈Vh, (3.7)

where is the -projection operator. Obviously, is bounded and as , for any . Let . The well-posedness of (3.7) implies that .

Let and be the solutions of (3.5) and (3.7), respectively. Then there exists ( if is convex) such that (see Sec. 3.2 in [9])

 ∥u−uh∥≤Ch2β∥f∥. (3.8)

Let , be the solution operator of (3.7). Define an operator function

 Fh(λ):=Th−1λI. (3.9)

The error estimate (3.8) indicates that

. This implies that

 ∥F(λ)|Vh−Fh(λ)∥⩽Ch2β, (3.10)

which is due to for all .

###### Lemma 3.2

There exists small enough such that, for every compact set

 suphsupλ∈Ω∥Fh(λ)∥<∞,h

Proof. Let be sufficiently small and . Then

 ∥Fh(λ)fh∥=∥∥∥uh−1λfh∥∥∥≤∥uh∥+1|λ|∥fh∥≤(C+1|λ|)∥fh∥.

Since is compact and , (3.11) holds.

###### Lemma 3.3

Let . Then

 limh→0∥Fh(λ)phf−phF(λ)f∥=0. (3.12)

Proof. From (3.8), we have that

 ∥Fh(λ)phf−phF(λ)f∥ =∥∥∥Th(λ)phf−1λphf−phT(λ)f+1λphf∥∥∥ =∥uh−u+u−phu∥≤∥u−uh∥+∥u−phu∥≤Ch2β∥f∥.

Now we are ready to present the main convergence theorem.

###### Theorem 3.4

Let . Assume that is small enough. Then there exists such that as . For any sequence the following estimate hold

 |λh−λ0|≤Ch2βr0, (3.13)

where

is the maximum rank of eigenvectors.

Proof. Let be a sequence of sufficiently small positive numbers with as and , and . Clearly, (b1) holds with , , and . (b2) and (b3) hold due to Lemma 3.2 and 3.3.

To verify (b4), assume that , with and

 limn→∞∥Fn(λ)vn−pny∥=0, (3.14)

for some . We estimate as follows by considering and separately.

If , then exists and is bounded. Let . We have

 vn−pnv=F(λ)−1((F(λ)−Fn(λ))(vn−pnv)+Fn(λ)vn−pnF(λ)v+pnF(λ)v−Fn(λ)pnv).

Recalling from (3.10) it holds

 ∥vn−pnv∥⩽C(h2βn∥vn−pnv∥+∥Fn(λ)vn−pnF(λ)v∥+∥pnF(λ)v−Fn(λ)pnv∥). (3.15)

Using (3.14) and Lemma 3.3 we have as .

Assume that . Let

be the finite dimensional generalized eigenspace of

[7]. We denote by the projection from to , by the inverse of from to . Due (3.14), we have that

 ∥F(λ)vn−y∥⩽∥F(λ)vn−Fn(λ)vn∥+∥Fn(λ)vn−pny∥+∥pny−y∥→0,n→∞.

Since is closed, . Let and . Since

 ∥Fn(λ)pnPG(λ)vn∥⩽∥Fn(λ)pn−pnF(λ)∥∥PG(λ)vn∥→0,n→∞,

by Lemma 3.3, similar to (3.15), we deduce that

 ∥v′n−pnv′∥⩽(1−Ch2βn)−1C(∥Fn(λ)v′n−pnF(λ)v′∥+∥pnF(λ)v′−Fn(λ)pnv′∥)→0.

On the other hand, since is finite dimensional, there is a subsequence and such that as . Therefore we have

 ∥vn−pnv∥⩽∥v′n−pnv′∥+∥pnPG(λ)vn−pnv′′∥→0,asN′′∋n→∞,

where . Now, we have verified (b1)-(b4) which are the conditions for Theorem 2 of [8]. Then (3.13) follows readily.

###### Remark 3.5

Under the same conditions of the above theorem, it is possible to obtain error estimates for generalized eigenvectors [8], which is not included in this paper due to simplicity.

## 4 Numerical Results

Let be the unit square . The smallest eigenvalue is and its rank . We use the linear Lagrange element on a series of uniformly refined meshes for discretization. The spectral indicator method [5, 6, 4] is employed to compute the smallest eigenvalue of (3.9). The results are shown in Table 1, which confirms the second order convergence.

## Aknowledgement

The research of B. Gong is supported partially by China Postdoctoral Science Foundation Grant 2019M650460. The research of J. Sun is supported partially by MTU REF. The research of Z. Zhang is supported partially by the National Natural Science Foundation of China grants NSFC 11871092, NSAF U1930402, and NSF 11926356.

## References

• [1] I. Babuška and J. Osborn, Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, Elsevier Science Publishers, North-Holland, 1991.
• [2] D. Boffi, Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010), 1-120.
• [3] I. Gohberg and J. Leiterer, Holomorphic operator functions of one variable and applications. 192. Birkhäuser Verlag, Basel, 2009.
• [4] B. Gong, J. Sun, T. Turner, C. Zheng, Finite element approximation of the nonlinear transmission eigenvalue problem for anisotropic media. preprint, 2019.
• [5] R. Huang, A. Struthers, J. Sun, and R. Zhang, Recursive integral method for transmission eigenvalues. J. Comput. Phys. 327 (2016), 830-840.
• [6] R. Huang, J. Sun and C. Yang, Recursive integral method with Cayley transformation. Numer. Linear Algebra Appl., 25 (2018), no. 6, e2199.
• [7] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 365-387.
• [8] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II. (Convergence rate). Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 389-408.
• [9] J. Sun and A. Zhou, Finite element methods for eigenvalue problem. CRC Press, Taylor Francis Group, Boca Raton, London, New York, 2016.