A new finite element approach for the Dirichlet eigenvalue problem

Journal Pre-proof A new finite element approach for the Dirichlet eigenvalue problem Wenqiang Xiao, Bo Gong, Jiguang Sun, Zhimin Zhang

PII: DOI: Reference:

S0893-9659(20)30088-4 https://doi.org/10.1016/j.aml.2020.106295 AML 106295

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Applied Mathematics Letters

Received date : 29 December 2019 Revised date : 12 February 2020 Accepted date : 12 February 2020 Please cite this article as: W. Xiao, B. Gong, J. Sun et al., A new finite element approach for the Dirichlet eigenvalue problem, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2020.106295. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

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A new finite element approach for the Dirichlet eigenvalue problem Wenqiang Xiaoa , Bo Gonga, Jiguang Sunb,∗, Zhimin Zhanga,c a Beijing

Computational Science Research Center, Beijing 100193, China. of Mathematical Sciences, Michigan Technological University, USA c Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

b Department

Abstract

We propose a new finite element approach, which is different than the classic Babuška-Osborn theory, to approximate Dirichlet eigenvalues. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for conforming finite elements 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.

1. Introduction

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Keywords: Dirichlet eigenvalue; finite element; holomorphic Fredholm operator function; spectral indicator method.

Finite element methods for eigenvalue problems have been studied extensively [1, 2, 3]. In this paper, we propose a new finite element approach for the Dirichlet eigenvalue problem. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function [4]. Using Lagrange finite elements, the convergence is proved by the abstract approximation theory for holomorphic operator functions [5, 6].

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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 [7]; and 3) combined with the spectral indicator method [8, 9], 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 [5, 6]. In Section 4, the spectral indicator method [8, 9, 7] is employed to compute the eigenvalues of the unit square.

∗ Corresponding

author. Email addresses: [email protected] (Wenqiang Xiao), [email protected] (Bo Gong), [email protected] (Jiguang Sun), [email protected] (Zhimin Zhang)

Preprint submitted to Journal of LATEX Templates

February 12, 2020

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2. Preliminaries. We present some preliminaries on the eigenvalue approximation theory of holomorphic Fredholm operator functions following [5, 6]. Let X, Y be complex Banach spaces and Ω ⊂ C be compact. Denote by L(X, Y ) the space of

bounded linear operators and Φ0 (Ω, L(X, Y )) the set of holomorphic Fredholm operator functions of index zero [4]. Assume that F ∈ Φ0 (Ω, L(X, Y )). We consider the problem of finding (λ, u) ∈ Ω × X, u 6= 0, such that F (λ)u = 0.

(2.1)

The resolvent set ρ(F ) and the spectrum σ(F ) of F are respectively defined as ρ(F ) = {λ ∈ Ω : F (λ)−1 ∈ L(Y, X)},

σ(F ) = Ω\ρ(F ).

Throughout the paper, we assume that ρ(F ) 6= ∅. The spectrum σ(F ) has no cluster points in Ω, and every λ ∈ σ(F ) is an eigenvalue [5].

To approximate the eigenvalues of F , we consider a sequence of operator functions Fn ∈ Φ0 (Ω, L(Xn , Yn )), n ∈

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N. Assume the following properties hold.

(b1) There exist Banach spaces Xn , Yn , n ∈ N and linear bounded mappings pn ∈ L(X, Xn ), qn ∈ L(Y, Yn ) with the property

lim kpn vkXn = kvkX , v ∈ X,

n→∞

lim kqn vkYn = kvkY , v ∈ Y.

n→∞

(2.2)

(b2) {Fn (·)}n∈N is equibounded on Ω, i.e., there exists a constant c such that

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kFn (λ)k ≤ c

∀λ ∈ Ω, n ∈ N.

(b3) {Fn (·)}n∈N approximates F (λ) for every λ ∈ Ω, i.e., lim kFn (λ)pn u − qn F (λ)ukYn = 0

n→∞

∀u ∈ X.

(b4) {Fn (·)}n∈N is regular for every λ ∈ Ω, i.e., if kxn k ≤ 1 (n ∈ N) and {Fn (λ)xn }n∈N is compact in the sense of Karma [5], then {xn }n∈N is compact.

3. Finite Element Approximation

Let D ⊂ R2 be a bounded Lipschitz domain. The Dirichlet eigenvalue problem is to find λ ∈ Ω and u 6= 0 such

that

−△u = λu in D

and u = 0 on ∂D.

(3.3)

The associated source problem is, given f , to find u such that −△u = f in D

and u = 0 on ∂D. 2

(3.4)

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For f ∈ L2 (D), the weak formulation of (3.4) is finding u ∈ H01 (D) such that ∀ v ∈ H01 (D),

a(u, v) = (f, v) where a(u, v) =

Z

D

∇u · ∇¯ v dx,

(f, v) =

Z

(3.5)

f v¯ dx.

D

Due to the wellposeness of (3.5) (see, e.g., [3]), there exists a linear compact solution operator T : L2 (D) →

H01 (D) ⊂ L2 (D) such that T f = u. The Dirichlet eigenvalue problem (3.3) is equivalent to the operator eigenvalue problem: T (λu) = u.

Assume that 0 ∈ / Ω. Define a nonlinear operator function F : Ω → L(L2 (D), L2 (D)) by F (λ) := T −

1 I, λ

λ ∈ Ω,

(3.6)

where I is the identity operator. Clearly, λ is a Dirichlet eigenvalue of (3.3) if and only if λ is an eigenvalue of F (λ). Lemma 3.1. Let Ω ⊂ C\{0} be a compact set. Then F (·) : Ω → L(L2 (D), L2 (D)) is a holomorphic Fredholm

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operator function of index zero.

Proof. It is clear that F (λ) is holomorphic in Ω. Since T is compact and I is the identity operator, F (λ) is a Fredholm operator of index zero.

In the rest of the paper, k · k stands for k · kL2 (D) and C > 0 is a generic constant. Let Th be a regular triangular

mesh for D with mesh size h and Vh ⊂ H01 (D) is the linear Lagrange element space associated with Th . Then the

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discrete formulation of (3.5) is to find uh ∈ Vh , such that

a(uh , vh ) = (f, vh ) = (ph f, vh )

∀ vh ∈ Vh ,

(3.7)

where ph : L2 (D) → Vh is the L2 -projection operator. Obviously, ph is bounded and kph f − f k → 0 as h → 0, for any f ∈ L2 (D). Let fh = ph f . The well-posedness of (3.7) implies that kuh k ≤ Ckfh k.

Let u and uh be the solutions of (3.5) and (3.7), respectively. Then there exists 1/2 < β ≤ 1 (β = 1 if D is

convex) such that (see Sec. 3.2 in [3])

ku − uh k ≤ Ch2β kf k.

(3.8)

Let Th : Vh → Vh , Th fh = uh be the solution operator of (3.7). Define an operator function Fh : Ω → L(Vh , Vh ) Fh (λ) := Th −

1 I. λ

(3.9)

The error estimate (3.8) indicates that kT − Th ph k 6 Ch2β . This implies that kF (λ)|Vh − Fh (λ)k 6 Ch2β , which is due to kF (λ)vh − Fh (λ)vh k = kT vh − Th vh k 6 kT − Th ph kkvh k for all vh ∈ Vh . 3

(3.10)

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Lemma 3.2. There exists h0 > 0 small enough such that, for every compact set Ω ⊂ C\{0}, sup sup kFh (λ)k < ∞,

h < h0 .

h λ∈Ω

(3.11)

Proof. Let h be sufficiently small and fh ∈ Vh . Then

 

1 1 1

kFh (λ)fh k = uh − fh ≤ kuh k + kfh k. kfh k ≤ C + λ |λ| |λ|

Since Ω is compact and 0 ∈ / Ω, (3.11) holds. Lemma 3.3. Let f ∈ L2 (D). Then

lim kFh (λ)ph f − ph F (λ)f k = 0.

h→0

Proof. From (3.8), we have that

(3.12)



1 1

kFh (λ)ph f − ph F (λ)f k = Th (λ)ph f − ph f − ph T (λ)f + ph f

λ λ

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= kuh − u + u − ph uk ≤ ku − uh k + ku − ph uk ≤ Ch2β kf k.

Now we are ready to present the main convergence theorem.

Theorem 3.4. Let λ0 ∈ σ(F ). Assume that h is small enough. Then there exists λh ∈ σ(Fh ) such that λh → λ0 as h → 0. For any sequence λh ∈ σ(Fh ) the following estimate holds

2β

|λh − λ0 | ≤ Ch r0 ,

(3.13)

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where r0 is the maximum rank of eigenvectors.

Proof. Let {hn } be a sequence of sufficiently small positive numbers with hn → 0 as n → ∞ and Fn (λ) := Fhn (λ),

Vn := Vhn and pn := phn . Clearly, (b1) holds with X = Y = L2 (D), Xn = Yn = Vn , and qn = pn . (b2) and (b3) hold due to Lemma 3.2 and Lemma 3.3.

To verify (b4), assume that vn ∈ Vn , n ∈ N′ ⊂ N with kvn k ≤ 1 and lim kFn (λ)vn − pn yk = 0,

n→∞

(3.14)

for some y ∈ L2 (D). We estimate kvn − pn vk as follows by considering λ ∈ ρ(F ) and λ ∈ σ(F ) separately. If λ ∈ ρ(F ), then F (λ)−1 exists and is bounded. Let v = F (λ)−1 y. We have


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

Recalling kF (λ)|Vn − Fn (λ)k 6 Ch2β n from (3.10) it holds


kvn − pn vk 6 C h2β n kvn − pn vk + kFn (λ)vn − pn F (λ)vk + kpn F (λ)v − Fn (λ)pn vk . 4

(3.15)

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Using (3.14) and Lemma 3.3 we have kvn − pn vk → 0 as n → ∞.

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Assume that λ ∈ σ(F ). Let G(λ) be the finite dimensional eigenspace of λ [5]. We denote by PG(λ) the projection

from L2 (D) to G(λ), by F (λ)−1 the inverse of F (λ)|L2 (D)/G(λ) from R(F (λ)) to L2 (D)/G(λ). Due to (3.14), we

have that

kF (λ)vn − yk 6 kF (λ)vn − Fn (λ)vn k + kFn (λ)vn − pn yk + kpn y − yk → 0, Since R(F (λ)) is closed, y ∈ R(F (λ)).

n → ∞.

Let v ′ := F (λ)−1 y and vn′ := (I − pn PG(λ) )vn . Since

kFn (λ)pn PG(λ) vn k 6 kFn (λ)pn − pn F (λ)kkPG(λ) vn k → 0,

n → ∞,

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


−1 C kFn (λ)vn′ − pn F (λ)v ′ k + kpn F (λ)v ′ − Fn (λ)pn v ′ k → 0. kvn′ − pn v ′ k 6 (1 − Ch2β n )

On the other hand, since G(λ) is finite dimensional, there is a subsequence N′′ and v ′′ ∈ G(λ) such that kPG(λ) vn −

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v ′′ k → 0 as N′′ ∋ n → ∞. Therefore we have

kvn − pn vk 6 kvn′ − pn v ′ k + kpn PG(λ) vn − pn v ′′ k → 0,

as N′′ ∋ n → ∞,

where v := v ′ + v ′′ . Now, we have verified (b1)-(b4) which are the conditions for Theorem 2 of [6]. 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

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eigenvectors [6].

4. Numerical Results

Let D be the unit square (0, 1) × (0, 1). The smallest eigenvalue of (3.3) is 2π 2 and its rank r0 = 1. We use the

linear Lagrange element on a series of uniformly refined meshes for discretization. The spectral indicator method in

Section 5 of [7] (see also [8, 9]) is employed to compute the smallest eigenvalue of (3.9) with Ω ⊂ C being a disc

centered at (20, 0) with radius 1. 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.

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|λ − λh |

convergence order

0.1889

-

19.7871

0.0479

1.9795

1/40

19.7512

0.0120

1.9970

1/80

19.7422

0.0029

2.0489

λh

1/10

19.9281

1/20

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h

Table 1: The smallest Dirichlet eigenvalue of the unit square (linear Lagrange element)

References

[1] I. Babuška, 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] J. Sun, A. Zhou, Finite element methods for eigenvalue problem. CRC Press, Taylor Francis Group, Boca Raton,

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London, New York, 2016.

[4] I. Gohberg, J. Leiterer, Holomorphic operator functions of one variable and applications. 192. Birkhäuser Verlag, Basel, 2009.

[5] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 365-387.

[6] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II. (Conver-

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gence rate). Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 389-408. [7] B. Gong, J. Sun, T. Turner, C. Zheng, Finite element approximation of the nonlinear transmission eigenvalue problem for anisotropic media. arXiv:2001.05340, 2019. [8] R. Huang, A. Struthers, J. Sun, R. Zhang, Recursive integral method for transmission eigenvalues. J. Comput. Phys. 327 (2016), 830-840.

[9] R. Huang, J. Sun, C. Yang, Recursive integral method with Cayley transformation. Numer. Linear Algebra Appl., 25 (2018), no. 6, e2199.

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Journal Pre-proof *Credit Author Statement

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CRediT Author Statement

• Wenqiang Xiao: Coding, Writing, Editing.

• Bo Gong: Analyzation, Discussion, Validation.

• Jiguang Sun: Methodology, Writing, Editing, Proof-reading, Supervision.

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• Zhimin Zhang: Supervision, Reviewing, Editing,

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