# Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field

Authors: Yidu Yang; Yu Zhang; Hai Bi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.


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## Body

## 1. Introduction

The finite element method for eigenvalue problem of electric field has become a hot topic in the field of mathematics and physics (see, e.g., [1–7]). This paper discusses high efficient mixed finite element calculation schemes for the eigenvalue problem of electric field.Kikuchi [6] introduced the first type of mixed variational formulation for the eigenvalue problem of electric field. Based on this formulation, in [3] Buffa et al.analyzed the approximation of nodal finite element. Boffi et al. [1] discussed the second type of mixed variational formulation for the eigenvalue problem of electric field and analyzed approximations of edge element and nodal element. Yang et al. [7] studied a two-grid discretization scheme of finite element for the first type of mixed variational formulation.Based on the work mentioned above, in this paper a new multi-grid discretization scheme and an adaptive algorithm are proposed for the first type of mixed variational formulation of eigenvalue problem and applied to the eigenvalue problem of electric field. The main features of the research in this paper are as follows.(1)
Our multi-grid discretization scheme and adaptive algorithm, which are the extension of conforming finite element multi-grid discretization scheme (see scheme  3 in [8] and scheme  1 in [9]), are a combination of the mixed finite element method and the Rayleigh quotient iteration method (see the algorithm  27.3 in [10]). With our algorithm one solves an eigenvalue problem on a coarse grid just at the first step and always solves a linear algebraic system on finer and finer grids at each following step. We derive the error estimates for the algorithm and prove that the constants appeared in the error estimates are independent of the iteration degrees. Thus we prove the convergence of iterations.(2)
The eigenvalue problem of electric field is so complicated that it is very difficult to obtain local a posteriori error estimates for the eigenfunctions of mixed finite element. As yet, there is no achievement reported in this field. Our adaptive algorithm substitutes the weight method established by Costabel and Dauge (see [3, 11]) for local refinement, which uses θ×|λhl-λhl-1|(θ∈(0,1]) as a posteriori error estimator of λhl instead of estimating local a posteriori error for the eigenfunction. And the results are satisfying.(3)
We analyze the mixed finite element error for the eigenvalue problem of electric field (see Theorem2.2 and Theorem 4.2). We refer to [12] to propose a new proof method which differs from the usual one in [13].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 Section3, for the first type of mixed variational formulation of eigenvalue problem, the finite element multi-grid discretization scheme and the adaptive algorithm are established and the validity of these schemes is proved theoretically. In Section 4, the multi-grid discretization scheme is applied to the eigenvalue problem of electric field. Finally, numerical experiments are presented in Section 5.


## 2. Preliminaries

LetV, W, and D be three real Hilbert spaces with inner products and norms (·,·)V, ∥·∥V, (·,·)W, ∥·∥W, (·,·)D, and ∥·∥D, respectively. Suppose that V↪D (continuously imbedded), a(·,·) is a symmetric, continuous, and V-elliptic bilinear form on V×V, that is,
(2.1)|a(q,ψ)|≤M1∥q∥V∥ψ∥V,∀q,ψ∈V,a(q,q)≥ν∥q∥V2,∀0≠q∈V.b(·,·) is a continuous bilinear form on V×W, that is,
(2.2)|b(ψ,v)|≤M2∥ψ∥V∥v∥W,∀ψ∈V,v∈W.
It is obvious that a(·,·) is an inner product on V and ∥·∥a=a(·,·) is an equivalent norm to ∥·∥V.In scientific and engineering computations, many eigenvalue problems have the following first type of mixed variational formulation: find(λ,u,σ)∈R×V×W, (u,σ)≠(0,0), such that
(2.3)a(u,ψ)+b(ψ,σ)=λ(u,ψ)D,∀ψ∈V,(2.4)b(u,v)=0,∀v∈W.In order to solve problem (2.3)-(2.4), one should construct finite element spaces Vh⊂V and Wh⊂W. Restricting (2.3)-(2.4) on Vh×Wh, we get the conforming mixed finite element approximation as follows: find (λh,uh,σh)∈R×Vh×Wh,(uh,σh)≠(0,0), such that
(2.5)a(uh,ψ)+b(ψ,σh)=λh(uh,ψ)D,∀ψ∈Vh,(2.6)b(uh,v)=0,∀v∈Wh.Consider the associated source and approximate source problems. Givenf∈D, find (w,p)∈V×W satisfying
(2.7)a(w,ψ)+b(ψ,p)=(f,ψ)D,∀ψ∈V,b(w,v)=0,∀v∈W.
Given f∈D, find (wh,ph)∈Vh×Wh satisfying
(2.8)a(wh,ψ)+b(ψ,ph)=(f,ψ)D,∀ψ∈Vh,b(wh,v)=0,∀v∈Wh.
Note that the source term f is independent of the solution.As for the mixed finite element method for boundary value problems, Brezzi and Fortinand so forth have established a systematic theory (see [14]). By Brezzi-Babuska Theorem, we have the following.Lemma 2.1 (Brezzi-Babuska).
Suppose that(C1)
(2.1)-(2.2) hold;(C2)
inf-sup condition holds, namely, there exists a constantν1>0, such that
(2.9)supψ∈V,ψ≠0b(ψ,v)∥ψ∥V≥ν1∥v∥W,∀v∈W,then there exists a unique solution(w,p) to the problem (2.7) and
(2.10)∥w∥a+∥p∥W≤Cr∥f∥D,whereCr just depends on ν,ν1, and M1. Moreover, suppose;(C3)
discrete inf-sup condition holds, namely, there exists a constantν2>0 independent of h, such that
(2.11)supψ∈Vh,ψ≠0b(ψ,v)∥ψ∥V≥ν2∥v∥W,∀v∈Wh,then there exists a unique solution(wh,ph) to the problem (2.8) and the following error estimate is valid:
(2.12)∥w-wh∥a+∥p-ph∥W≤Ce{infq∈Vh∥w-q∥a+infv∈Wh∥p-v∥W},whereCe just depends on ν,ν2 and M1,M2.Suppose conditions (C1)–(C3) hold in Lemma2.1. Then there exist unique solutions to the problem (2.7) and (2.8), respectively. Thus, we can define linear bounded operators as follows: T:D→V,S:D→W:∀f∈D,
(2.13)a(Tf,ψ)+b(ψ,Sf)=(f,ψ)D,∀ψ∈V,b(Tf,v)=0,∀v∈W.Th:D→Vh⊂V,Sh:D→Wh⊂W:∀f∈D,
(2.14)a(Thf,ψ)+b(ψ,Shf)=(f,ψ)D,∀ψ∈Vh;b(Thf,v)=0,∀v∈Wh.
Obviously, (2.3)-(2.4) has the following equivalent operator form
(2.15)λTu=u,σ=S(λu),
and (2.5)-(2.6) has the following equivalent operator form
(2.16)λhThuh=uh,(2.17)σh=Sh(λhuh).It is easy to verify thatT:D→D,Th:D→D are self-adjoint operators in the sense of inner product (·,·)D (see [7]).AssumeV↪cD (compactly embedded), then it’s easy to prove that T:D→D is completely continuous, T:V→V  is completely continuous, and Th is a finite rank operator. Combining (2.3)-(2.4), (2.5)-(2.6), and the V-ellipticity of a(·,·), we deduce
(2.18)λ=a(u,u)(u,u)D>0,λh=a(uh,uh)(uh,uh)D>0.
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
(2.19)0<λ1≤λ2≤⋯≤λk≤⋯↗+∞,
and the corresponding eigenfunctions are
(2.20)(u1,σ1),(u2,σ2),…,(uk,σk),…,
where (ui,uj)D=δij.The eigenvalues of (2.5)-(2.6) can be sorted as
(2.21)0<λ1,h≤λ2,h≤⋯≤λk,h≤⋯≤λNh,h,
and the corresponding eigenfunctions are
(2.22)(u1,h,σ1,h),(u2,h,σ2,h),…,(uk,h,σk,h),…,(uNh,h,σNh,h),
where Nh=dimVh, (ui,h,uj,h)D=δij.From (2.5), by taking uh=ui,h, ψ=uj,h, we have
(2.23)a(ui,h,uj,h)+b(uj,h,σh)=λi,h(ui,h,uj,h)D,
and from (2.6) we see that b(uj,h,σh)=0, then
(2.24)a(ui,h,uj,h)=λi,h(ui,h,uj,h)D=λi,hδij,
thus {ui,h/∥ui,h∥a} is a completely normal eigenvector system on Vh in the sense of the inner product a(·,·).Denoteλk=1/μk, λk,h=1/μk,h. In this paper, μk and μk,h, λk and λk,h are all called eigenvalues.Letμ be the kth eigenvalue with algebraic multiplicity q, μ=μk=μk+1=⋯=μk+q-1. M(μ) is the space spanned by all eigenfunctions {uj}kk+q-1 corresponding to μ of T. Mh(μ) is the space spanned by all eigenfunctions {uj,h}kk+q-1 corresponding to all eigenvalues of Th that converge to μ. Let M^(μ)={v:v∈M(μ),∥v∥a=1}, M^h(μ)={v:v∈Mh(μ),∥v∥a=1}. We call λ=1/μ as the kth eigenvalue, too. Denote M(λ)=M(μ), Mh(λ)=Mh(μ), and M^(λ)=M^(μ). Define
(2.25)∥(T-Th)|M(λ)∥D=maxu∈M(λ)∥(T-Th)u∥D∥u∥D,∥(T-Th)|M(λ)∥a=maxu∈M(λ)∥(T-Th)u∥a∥u∥a.The convergence and error estimates about the mixed element method of eigenvalue problem have been studied in [7, 12, 13, 15, 16]. From [12], we know that the following results are valid.Theorem 2.2.
SupposeV↪cD, a(u,v) is symmetric, and the conditions of Lemma 2.1 hold; moreover, for any f∈D,
(2.26)infq∈Vh∥Tf-q∥V→0(h→0),(2.27)infv∈Wh∥Sf-v∥W→0(h→0).
Then ∥Th-T∥D→0(h→0).Proof.
FromV↪cD, we derive that T:D→D is a completely continuous operator. It is obvious that Th:D→D is a finite rank operator. From (2.12), (2.26), and (2.27), we deduce
(2.28)∥Tf-Thf∥a≤Ce(infq∈Vh∥Tf-q∥a+infv∈Wh∥Sf-v∥W)→0(h→0).
It shows that Th:D→V pointwisely converges to T. From (2.10) and (2.12) we derive that both T:D→V and Th:D→V are linear bounded. Hence, from Banach-Steinhaus Theorem, we know that there exists a positive constant M independent of h, such that
(2.29)suph∥Th∥L(D,V)≤M.
Thus, ∪h>0(T-Th)B is a bounded set in V with respect to the unit ball B of D. From V↪cD, we know that ∪h>0(T-Th)B is a relatively compact set in D, which proves that {Th} is collectively compact. From (2.28), we know that T:D→D,Th:D→D, and Th pointwisely converge to T. From [7], T:D→D,Th:D→D are self-adjoint operators in the sense of inner product (·,·)D. Then by Lemma 3.7 or Table  3.1 in [17], we get ∥Th-T∥D→0(h→0). The proof is completed.Lemma 2.3.
Suppose that the conditions of Theorem2.2 hold. Let (λh,uh,σh) be the kth eigenpair of (2.5)-(2.6) with ∥uh∥a=1, let λ be the kth eigenvalue of (2.3)-(2.4). Then λh→λ(h→0), and there exists an eigenfunction (u,σ) corresponding to λ, such that
(2.30)|λh-λ|+∥uh-u∥D≤C1∥(T-Th)|M(λ)∥D,(2.31)∥σ-σh∥W≤∥Sh(λu)-S(λu)∥W+C∥(T-Th)|M(λ)∥D,(2.32)∥u-uh∥a≤C2∥(Th-T)|M(λ)∥a,
let u∈M^(λ), then there exists uh∈Mh(λ) such that
(2.33)∥u-uh∥a≤C3∥(Th-T)|M(λ)∥a,
where u depends on h in general, and C1, C2, and C3 are constants independent of h.Proof.
By Theorem2.2, we know ∥Th-T∥D→0(h→0). Thus from Theorem  2.2 in [7], we see that the desired results are valid. The proof is completed.For(u*,σ*)∈V×W, u*≠0, define the Rayleigh quotient
(2.34)λr=a(u*,u*)+2b(u*,σ*)(u*,u*)D.Lemma 2.4.
Suppose(λ,u,σ) is an eigenpair of (2.3)-(2.4), then for any (u*,σ*)∈V×W, u*≠0, the Rayleigh quotient λr satisfies
(2.35)λr-λ=a(u*-u,u*-u)+2b(u*-u,σ*-σ)(u*,u*)D-λ(u*-u,u*-u)D(u*,u*)D.Proof.
The proof is completed by using the same argument as that of Lemma 9.1 (see [7, 18]).SinceV↪D (continuously imbedded), there exists a constant C4 independent of h such that
(2.36)∥v∥D≤C4∥v∥a,∀v∈V.
Taking (u*,σ*)=(uh,σh) in (2.35) and using (2.4) and (2.6), we deduce the following.Lemma 2.5.
Suppose(λh,uh,σh) is an approximation of (λ,u,σ)  and ∥uh∥a=1, then
(2.37)λh-λ=a(uh-u,uh-u)+2b(uh-u,v-σ)(uh,uh)D-λ(uh-u,uh-u)D(uh,uh)D,∀v∈Wh,(2.38)|λh-λ|≤(λh+λλhC42)∥uh-u∥a2+2λhM2∥uh-u∥V∥σ-v∥W,∀v∈Wh.


## 3. Mixed Finite Element Multigrid Discretization Scheme Based on the Rayleigh Quotient Iteration

In this section, we develop the work in [8], noticing that in [8], when k>1,λkh1-λk,λkhl-1-λk,hl and λkhl-1-λk sholud be modified to their absolute values, respectively, and establish the following mixed finite element multi-grid discretization scheme based on the Rayleigh quotient iteration, and give a rigorous theoretical analysis. Suppose the partition satisfies the following conditions.Condition 1.
{
K
h
i
} is a family of regular meshes (see [19]) with the mesh diameter {hi} and hi=hi-1ti, ti∈(1,3) is arbitrarily chosen, i=1,2,…, and infiti>1.
Let{Vhi}0l and {Whi}0l be finite element spaces on {Khi}0l.Scheme 1.
Multigrid Discretization.Step 1.
Solve the eigenvalue problem (2.3)-(2.4) on VH×WH: find (λH,uH,σH)∈R×VH×WH, ∥uH∥a=1 such that
(3.1)a(uH,ψ)+b(ψ,σH)=λH(uH,ψ)D,∀ψ∈VH,b(uH,v)=0,∀v∈WH.Step 2.
u
h
0
⇐
u
H, λh0⇐λH, i⇐1.Step 3.
Solve an equation onVhi×Whi: find (u′,σ′)∈Vhi×Whi such that
(3.2)a(u′,ψ)+b(ψ,σ′)-λhi-1(u′,ψ)D=(uhi-1,ψ)D,∀ψ∈Vhi,b(u′,v)=0,∀v∈Whi.
Set uhi=u′/∥u′∥a, σhi=σ'/∥u′∥a.Step 4.
Compute the Rayleigh quotient(3.3)λhi=a(uhi,uhi)(uhi,uhi)D.Step 5.
Ifi=l, then output (λhl,uhl,σhl), stop; else, i⇐i+1, and return to Step 3.
Let(λH,uH,σH) be the kth eigenpair, and we use (λhl,uhl,σhl) as the kth approximation eigenpair of (2.3)-(2.4).
Next, we will discuss the validity of Scheme1.Lemma 3.1.
For any nonzerou,v∈V,
(3.4)∥u∥u∥a-v∥v∥a∥a≤2∥u-v∥a∥u∥a,∥u∥u∥a-v∥v∥a∥a≤2∥u-v∥a∥v∥a.Proof.
See [8].Denotedist(u,V)=infv∈V∥u-v∥a.Consider the eigenvalue problem (2.16) on Vh.Lemma 3.2.
Suppose thatμ and μh are the kth eigenvalue of T and Th, respectively, and (μ0,u0) is an approximate eigenpair, where μ0 is not an eigenvalue of Th, u0∈Vh,∥u0∥a=1, dist(u0,Mh(μ))≤1/2, maxk≤j≤k+q-1|(μj,h-μh)/(μ0-μj,h)|≤1/2, |μ0-μj,h|≥(ρ/2)(j≠k,k+1,…,k+q-1), and us∈Vh,uh∈Vh satisfy
(3.5)(μ0-Th)us=u0,uh=us∥us∥a.
Then
(3.6)dist(uh,M^h(μ))≤16ρ|μ0-μh|dist(u0,Mh(μ)),
where ρ=minμj≠μ|μj-μ| is the separation constant of the eigenvalue μ.Proof.
See [8]Since the convergence rate ofVhl-1 and Whl-1 approximating eigenfunctions is lower than that of Vhl and Whl approximating eigenfunctions, respectively, the approximation order of (λhl-1,uhl-1) is lower than that of (λhl,uhl). However, in general, the accuracy order of (λhl-1,uhl-1) will not exceed that of (λhl-1,uhl-1); therefore in the following Theorem 3.3 we assume that the accuracy order of (λhl-1,uhl-1) is lower than that of (λhl,uhl).Theorem 3.3.
Suppose that∥Th-T∥D→0(h→0), H is small properly, and Condition 1 holds. Let (λhl,uhl,σhl) be the approximate eigenpair obtained by Scheme 1, and let uhl-1 approximate u-∈M^(λ), λhl-1 approximate λ, and the accuracy order of (λhl-1,uhl-1) be lower than that of (λhl,uhl). Then there exists u∈M(λ) such that
(3.7)∥uhl-u∥a≤32ρC5C6(|λhl-1-λ|2+|λhl-1-λ|∥uhl-1-u-∥D)+(C2×q+3)∥(T-Thl)|M(λ)∥a,(3.8)|λhl-λ|≤(λhl+λλhlC42)∥uhl-u∥a2+2λhlM2∥uhl-u∥Vinfv∈Whl∥σ-v∥W,
where C5 and C6 are determined, respectively, by (3.10) and (3.14) in the following proof.Proof.
Letμ0=1/λhl-1, and u0=λhl-1Thluhl-1/∥λhl-1Thluhl-1∥a. Since u-∈M(λ), by calculation we deduce
(3.9)∥λhl-1Thluhl-1-u-∥a=∥λhl-1Thluhl-1-λTu-∥a=|λhl-1-λ|∥Thluhl-1∥a+λ∥Thl(uhl-1-u-)∥a+λ∥(Thl-T)u-∥a.
From Lemma 2.1, there exists a positive constant C5 depending only on ν,ν1,ν2,M1, and M2 such that
(3.10)∥Thlv∥a≤C5∥v∥D,∀v∈D.
Then
(3.11)∥λhl-1Thluhl-1-u-∥a≤C5(C4|λhl-1-λ|+λ∥uhl-1-u-∥D)+λ∥(Thl-T)|M(λ)∥a.
By Lemma 3.1, we derive
(3.12)dist(u0,M^(λ))≤∥u0-u-∥a≤2∥λhl-1Thluhl-1-u-∥a≤2C5(C4|λhl-1-λ|+λ∥uhl-1-u-∥D)+2λ∥(Thl-T)|M(λ)∥a.
Using the triangle inequality and (2.33), we deduce
(3.13)dist(u0,Mhl(λ))≤dist(u0,M^(λ))+C3∥(Thl-T)|M(λ)∥a.
According to the hypotheses of the theorem, we know that λhl-1→λ and λhl-1-λ are an infinitesimal of lower order comparing with λj,hl-λ. Hence, there exists a positive constant C6 independent of hl(l=1,2,…) such that for j=k,k+1,…,k+q-1 we have
(3.14)|μ0-μj,hl|=|λhl-1-λ+λ-λj,hlλj,hlλhl-1|≤C6|λhl-1-λ|.
Note that H is small enough and hl≪hl-1; from (3.13) and (3.12), we obtain
(3.15)dist(u0,Mhl(λ))≤12.
Noticing that λ=λk=λk+1=⋯=λk+q-1, we have
(3.16)|μj,hl-μhl|=|λhl-λj,hlλhlλj,hl|=|λhl-λ+λj-λj,hlλhlλj,hl|,
which together with (3.14), noting that λj,hl-λ is an infinitesimal of higher order comparing with λhl-1-λ, yields
(3.17)maxk≤j≤k+q-1|μj,hl-μhlμ0-μj,hl|≤12.
Since ρ is the separation constant, H is small enough, and hl≪hl-1, there holds
(3.18)|μ0-μj,hl|≥ρ2,j≠k,k+1,…,k+q-1.
For u′ in Step 3 of Scheme 1, from the definition of Th and Sh (taking i=l), we have
(3.19)a(λhl-1Thlu′,ψ)+b(ψ,λhl-1Shlu′)=λhl-1(u′,ψ)D,∀ψ∈Vhl,(3.20)b(λhl-1Thlu′,v)=0,∀v∈Whl,(3.21)a(Thluhl-1,ψ)+b(ψ,Shluhl-1)=(uhl-1,ψ)D,∀ψ∈Vhl,(3.22)b(Thluhl-1,v)=0,∀v∈Whl.
Hence, when i=l, Step 3 of Scheme 1 is equivalent to the following: find (u′,σ′)∈Vhl×Whl such that
(3.23)a(u′,ψ)+b(ψ,σ′)-λhl-1a(Thlu′,ψ)-λhl-1b(ψ,Shlu′)=a(Thluhl-1,ψ)+b(ψ,Shluhl-1),∀ψ∈Vhl,(3.24)b(u',v)=0,∀v∈Whl.
And set uhl=u′/∥u′∥a, σhl=σ′/∥u′∥a.
From (3.23), we obtain
(3.25)a(u′-λhl-1Thlu′-Thluhl-1,ψ)+b(ψ,σ′-λhl-1Shlu′-Shluhl-1)=0,∀ψ∈Vhl.
Combining (3.24), (3.20) and (3.22), we get
(3.26)b(u′-λhl-1Thlu′-Thluhl-1,v)=0,∀v∈Whl.
By (3.26) and taking ψ=u′-λhl-1Thlu′-Thluhl-1 in (3.25), we obtain
(3.27)a(u′-λhl-1Thlu′-Thluhl-1,u′-λhl-1Thlu′-Thluhl-1)=0.
Thus
(3.28)(1λhl-1-Thl)u′=1λhl-1Thluhl-1,uhl=u′∥u′∥a.
From (3.28) we know that the first term on the left-hand side of (3.25) is equal to 0, thus
(3.29)b(ψ,σ′-λhl-1Shlu′-Shluhl-1)=0,∀ψ∈Vhl,
then, using discrete inf-sup condition we get
(3.30)σ′=λhl-1Shlu′+Shluhl-1.
Thus Step 3 of Scheme 1 is equivalent to (3.28), (3.30), uhl=u′/∥u′∥a, and σhl=σ′/∥u′∥a.
Noting that(1/λhl-1)Thluhl-1=∥(1/λhl-1)Thluhl-1∥au0 differs from u0 by only a constant and selecting us=(λhl-1u′)/∥Thluhl-1∥a, we have
(3.31)(1λhl-1-Thl)us=u0,uhl=us∥us∥a.
By (3.15), (3.17), (3.18), and (3.31), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.13) and (3.14) into (3.6), we obtain
(3.32)dist(uhl,M^hl(λ))≤16ρC6(λhl-1-λ)(dist(u0,M^(λ))+C3∥(T-Thl)|M(λ)∥a).
Let the eigenfunctions {uj,hl}kk+q-1 be an orthonormal system of Mhl(λ) in the sense of a(·,·), then
(3.33)dist(uhl,Mhl(λ))=∥uhl-∑j=kk+q-1‍a(uhl,uj,hl)uj,hl∥a.
Let
(3.34)u*=∑j=kk+q-1‍a(uhl,uj,hl)uj,hl,
noting ∥uhl-u*∥a≤dist(uhl,M^hl(λ)), from (3.32) we deduce
(3.35)∥uhl-u*∥a≤16ρC6(λhl-1-λ)(dist(u0,M^(λ))+C3∥(T-Thl)|M(λ)∥a).
By Lemma 2.3, there exist {uj0}kk+q-1⊂M^(λ) such that uj,hl-uj0 satisfy (2.32). Let
(3.36)u=∑j=kk+q-1‍a(uhl,uj,hl)uj0,
then u∈M(λ). Using (2.32), we deduce
(3.37)∥u*-u∥a=∥∑j=kk+q-1‍a(uhl,uj,hl)(uj,hl-uj0)∥a≤C2q∥(Thl-T)|M(λ)∥a.
Combining (3.35) with the above inequality, we have
(3.38)∥uhl-u∥a≤16ρC6(λhl-1-λ)dist(u0,M^(λ))+(C2q+1)∥(T-Thl)|M(λ)∥a.
Substituting (3.12) into (3.38), we get (3.7).
From Step3 of Scheme 1, we know that b(uhl,σhl)=0, thus
(3.39)λhl=a(uhl,uhl)(uhl,uhl)D=a(uhl,uhl)+2b(uhl,σhl)(uhl,uhl)D.
Select λr=λhl, u*=uhl, and σ*=σhl. From Lemma 2.4, we get
(3.40)λhl-λ=a(uhl-u,uhl-u)+2b(uhl-u,σhl-σ)(uhl,uhl)D-λ(uhl-u,uhl-u)D(uhl,uhl)D.
Noting that ∀v∈Whl, b(uhl-u,v)=0, and (uhl,uhl)D=(1/λhl)a(uhl,uhl)=1/λhl, we have
(3.41)λhl-λ=λhla(uhl-u,uhl-u)+2λhlb(uhl-u,v-σ)-λλhl(uhl-u,uhl-u)D,∀v∈Whl.
Note ∥uhl-u∥D≤C4∥uhl-u∥a. From (3.41) we obtain (3.8).Forl=1, Scheme 1 is actually the two-grid discretization scheme established in [7]. By Theorem 3.3 we get the following conclusion directly.Theorem 3.4.
Suppose∥Th-T∥D→0(h→0). Let h0=H be small properly. Let (λh1,uh1,σh1) be an approximate eigenpair obtained by Scheme 1 (l=1). Then there exists u∈M(λ) such that
(3.42)∥uh1-u∥a≤32ρC5C6(|λh0-λ|2+|λh0-λ|C1∥(T-Th0)|M(λ)∥D)+(C2×q+3)∥(T-Th1)|M(λ)∥a,(3.43)|λh1-λ|≤(λh1+λλh1C42)∥uh1-u∥a2+2λh1M2∥uh1-u∥Vinfv∈Wh1∥σ-v∥W.Proof.
Consider Scheme1. Here l=1, uh0=uH. By Lemma 2.3, we know that uh0 approximates u-∈M^(λ), and the accuracy order of (λh0,uh0) is lower than (λh1,uh1). Hence, for l=1, the conditions of Theorem 3.3 hold. Select u- in the proof of Theorem 3.3 such that uh0-u- satisfies Lemma 2.3. Then from (2.30), we obtain
(3.44)∥uh0-u-∥D≤C1∥(T-Th0)|M(λ)∥D,
substituting it into (3.7), we obtain (3.42). From (3.8), we deduce (3.43).Theorem3.4 is actually Theorem  3.3 in [7], but we analyze in detail the constants appeared in the error estimates.From Theorem3.4 and Theorem 3.3, we know that λhl→λ (l→∞) and the convergence rate is high. Thus, we use θ×|λhl-λhl-1| as a posteriori error indicator of λhl-λ (details can be seen in Remark 4.5). Then we establish the following adaptive algorithm.Scheme 2 (Adaptive Algorithm).
Give an error toleranceε and choose the parameter 0<θ≤1, H, t1, and h1=Ht1.Step 1.
Solve (2.3)-(2.4) on VH×WH: find (λH,uH,σH)∈R×VH×WH, ∥uH∥a=1 such that
(3.45)a(uH,ψ)+b(ψ,σH)=λH(uH,ψ)D,∀ψ∈VH,b(uH,v)=0,∀v∈WH.Step 2.
u
h
0
⇐
u
H, λh0⇐λH, l⇐1.Step 3.
Solve an equation onVhl×Whl: find (u′,σ′)∈Vhl×Whl such that
(3.46)a(u′,ψ)+b(ψ,σ′)-λhl-1(u′,ψ)D=(uhl-1,ψ)D,∀ψ∈Vhl,b(u′,v)=0,∀v∈Whl.
Let uhl=u′/∥u′∥a, σhl=σ′/∥u′∥a.Step 4.
Compute the Rayleigh quotient(3.47)λhl=a(uhl,uhl)(uhl,uhl)D.Step 5.
Ifθ×|λhl-λhl-1|>ε, then select tl+1, hl+1=hltl+1, l⇐l+1 and return to Step 3; else output (λhl,uhl,σhl), stop.Remark 3.5.
In Scheme2, we use θ×|λhl-λhl-1| as a posteriori error indicator which is global. In order to cope with difficulties caused by local singularity of a complicated problem in calculation, so far, most algorithms designed a local a posteriori error indicator to establish adaptive algorithm with local mesh refinement (e.g., see [20–22]). However, because the eigenvalue problem of electric field is so complicated, that it is very difficult to obtain a local a posteriori error indicator of eigenfunction. Fortunately, the influence of local singularity can be avoided by using the weight method which is established by Costabel and Dauge to discrete the eigenvalue problem of electric field. And the performance of the weigh method is very good (see [3, 11]). Hence, without local mesh refinement, by using the weight method mentioned above our algorithm can also guarantee its high efficiency.


## 4. The Eigenvalue Problem of Electric Field

Consider the eigenvalue problem of electric field:(4.1)c2curlcurlu→=ω2u→,inΩ,divu→=0,inΩ,u→×γ→=0,on∈∂Ω,
where Ω is a polyhedron in R3, γ→ is the unit outward normal to ∂Ω.Physicallyu→ denotes the electric field, ω denotes the time frequency, and c is the speed of the light velocity. Usually, let λ=ω2/c2 named eigenvalue.Let(4.2)H(curl,Ω)={q→∈L2(Ω)3:curlq→∈L2(Ω)3},H0(curl,Ω)={q→∈H(curl,Ω):q→×γ→|∂Ω=0}.WhenΩ is a convex polyhedron, we define the following function space:
(4.3)χ={q→∈H0(curl,Ω):divq→∈L2(Ω)}.
Denote
(4.4)(q→,ψ→)0=∫Ωq→·ψ→dx,∥q→∥0=(q→,q→)01/2.(q→,ψ→)χ=(curlq→,curlψ→)0+(divq→,divψ→)0,∥q→∥χ=(q→,q→)χ1/2,
From [23, 24], we know that χ⊂H1(Ω)3; (q→,ψ→)χ is a coercive bilinear form on χ, and ∥q→∥χ is a norm.WhenΩ is a nonconvex polyhedron, the problem is relatively complicated. Let E denote a set of edges of reentrant dihedral angles on ∂Ω, and d=d(x) denote the distance to the set E: d(x)=dist(x,∪e∈Ee-). We introduce a weight function ωr which is a nonnegative smooth function of x. It can be represented by dr in reentrant edge and angular domain. We shall write ωr≃dr. Define the weighted functional spaces:
(4.5)Lr2(Ω)={v∈Lloc2(Ω):ωrv∈L2(Ω)},χr={q→∈H0(curl,Ω):divq→∈Lr2(Ω)}.
Denote
(4.6)(q→,ψ→)Lr2=∫Ωωr2q→·ψ→dx,∥q→∥Lr2=(q→,q→)Lr21/2.(q→,ψ→)χr=(curlq→,curlψ→)0+(divq→,divψ→)Lr2,∥q→∥χr=(q→,q→)χr1/2.
Let σΔD be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition:
(4.7){ϕ∈H01(Ω):Δϕ∈L2(Ω)}⊂∩s<σΔDHs(Ω),{ϕ∈H01(Ω):Δϕ∈L2(Ω)}⊂HσΔD(Ω).
From the regularity estimate, we know σΔD∈((3/2),2). Let rmin=2-σΔD.From [11, 25], we see that for all r∈(rmin,1), the seminorm ∥q→∥χr is a norm on χr, and χr∩H1(Ω)3 is dense in χr.In the following discussion, we will useχr,Lr2(Ω) for both nonconvex and convex domains. We select r∈(rmin,1) for non-convex domain and χr=χ,Lr2(Ω)=L2(Ω) for convex domain.By introducing the Lagrange multiplierσ, [6] changed (4.1) into the mixed variational formulation: find (λ,u→,σ)∈R+×χr×Lr2(Ω) such that
(4.8)(u→,ψ→)χr+(divψ→,σ)Lr2=λ(u→,ψ→)0,∀ψ→∈χr,(divu→,v)Lr2=0,∀v∈Lr2(Ω).LetKh be a regular simplex partition (tetrahedral partition) of Ω with the mesh diameter h. Define the finite element space Vh×Wh⊂χr×Lr2(Ω).Restricting (4.8) on the above-mentioned finite space, we obtain the discrete mixed variational form: find (λh,u→h,σh)∈R+×Vh×Wh such that
(4.9)(u→h,ψ→)χr+(divψ→,σh)Lr2=λh(u→h,ψ→)0,∀ψ→∈Vh,(divu→h,v)Lr2=0,∀v∈Wh.Set(4.10)V=χr,∥·∥V=∥·∥a=∥·∥χr,W=Lr2(Ω),∥·∥W=∥·∥Lr2,D=L2(Ω)3,∥·∥D=∥·∥0,a(q→,ψ→)=(q→,ψ→)χrb(ψ→,v)=(divψ→,v)Lr2.
Then (4.8) and (4.9) can be written as (2.3)-(2.4) and (2.5)-(2.6), respectively (it is needed to add  →  for the vector function, e.g., u,ψ should be written in the forms of u→,ψ→).We apply Schemes1 and 2 to the eigenvalue problem of electric field (4.8). Adding the symbol  →  for the vector function, we get a multi-grid discretization scheme and adaptive algorithm for mixed finite element of the eigenvalue problem of electric field which are still called Schemes 1 and 2.It is easy to know thata(·,·) and b(·,·) are continuous bilinear forms on V×V and V×W, respectively. V↪cD. (It is true obviously when Ω is convex; when Ω is non-convex, see [25].)Consider the source problem associated with (4.8): find (w→,p)∈χr×Lr2(Ω) such that
(4.11)(w→,ψ→)χr+(divψ→,p)Lr2=(f→,ψ→)0,∀ψ→∈χr,(divw→,v)Lr2=0,∀v∈Lr2(Ω).For (4.11) and its conforming finite element approximation, condition (C1) of Lemma 2.1 holds obviously; [3] has proved that condition (C2) holds; assume that the discrete inf-sup condition (C3) holds, then conditions of Lemma 2.1 hold. Thus we can define operators T,S,Th, and Sh. (4.8) and (4.9) can be written as (2.15) and (2.16)-(2.17), respectively.The following Lemma4.1 is cited from [3, 25].Lemma 4.1.
Equation (4.1) is equivalent to (4.8), and the solutions of (4.8) σ=S(λu→)=0 and u→∈χr, divu→=0.Note that Lemma4.1 shows that for the eigenvalue problem of electric field, the second term on the right-hand side in (3.8) is equal to 0.Theorem 4.2.
Suppose that the discrete inf-sup condition (C3) holds. Then∥T-Th∥D→0(h→0). Let (λh,u→h,σh) be the kth eigenpair of (4.9) with ∥u→h∥χr=1, and let λ be the kth eigenvalue of (4.8). Then λh→λ(h→0), and there exists an eigenfunction u→ corresponding to λ with ∥u→∥χr=1, such that
(4.12)|λh-λ|≤(λh+λλhC42)C22∥(Th-T)|M(λ)∥χr2,(4.13)∥u→-u→h∥χr≤C2∥(Th-T)|M(λ)∥χr,
let u→∈M^(λ), then there exists u→h∈Mh(λ) such that
(4.14)∥u→-u→h∥χr≤C3∥(Th-T)|M(λ)∥χr,
where C2, C3, and C4 are constants independent of h.Proof.
From the preceding discussion and hypotheses of the theorem, we know thatV↪cD, a(u,v) is symmetric, and the conditions of Lemma 2.1 hold. Besides, since Kh is a regular partition, when Ω is a convex polyhedron, χr⊂H1(Ω)3(χr=χ); when Ω is a non-convex polyhedron, χr∩H1(Ω)3 is dense in χr. Since C∞(Ω-)3 is dense in H1(Ω)3, thus, no matter Ω is convex or non-convex, C∞(Ω-)3 is dense in χr. For any given f→∈D, we have Tf→∈χr. Thus for any ε>0, according to the density, we know that there exists w→∈C∞(Ω-)3 such that
(4.15)∥Tf→-w→∥χr≤ε2.
Selecting h0>0 being small properly, when 0<h≤h0, we have
(4.16)∥w→-Ihw→∥χr≤Ch|w→|2≤ε2,
where Ih:C∞(Ω-)3→Vh is an interpolation operator. Thus
(4.17)infq→∈Vh∥Tf→-q→∥χr≤∥Tf→-Ihw→∥χr≤∥Tf→-w→∥χr+∥w→-Ihw→∥χr≤ε2+ε2=ε.
Namely, infq→∈Vh∥Tf→-q→∥χr→0(h→0). Hence (2.26) is true. Analogously, using the density of C∞(Ω-) in Lr2(Ω), we deduce that infv∈Wh∥Sf→-v∥Lr2→0(h→0), namely, (2.27), is true. Hence, from Theorem 2.2 and Lemma 2.3, we know that ∥T-Th∥D→0(h→0), λh→λ(h→0), (4.13), and (4.14) hold. From (2.38), (4.13), and Lemma 4.1, we get (4.12). The proof is completed.Denote(4.18)ελ(h)=supu→∈M^(λ)infψ→∈Vh∥u→-ψ→∥χr.
From Lemma 2.1, noting σ=0, we deduce
(4.19)∥(Th-T)|M(λ)∥χr=supu→∈M^(λ)∥(Th-T)u∥χr≤supu→∈M^(λ)Ceinfψ→∈Vh∥Tu→-ψ→∥χr≤λ-1Ceελ(h)≡C7ελ(h).Theorem 4.3.
Assume that the discrete inf-sup condition (C3) holds,h0=H is small properly, Condition 1 holds and supiti<3. Let (λhl,uhl) be an approximate eigenpair obtained by Scheme 1, then there exists u→∈M(λ) such that
(4.20)∥u→hl-u→∥χr≤2(C2×q+3)C7ελ(hl),(4.21)|λhl-λ|≤4(λhl+λλhlC42)(C2×q+3)2C72ελ(hl)2.Proof.
We use induction to complete the proof. Note that the conditions of Theorem2.2 hold. For l=1, Scheme 1 is actually two-grid discretization scheme. Substituting (4.19) into (4.12) and (4.13), we derive
(4.22)∥u→H-u→∥χr≤C2C7ελ(H),|λ-λH|≤(λH+λλHC42)C22C72ελ(H)2.
Combining (3.7) with l=1 and the above two inequalities, we know that there exists u→∈M(λ) such that
(4.23)∥uh1-u→∥χr≤32ρC5C6(|λh0-λ|2+|λh0-λ|∥uh0-u-∥D)+(C2×q+3)∥(T-Th1)|M(λ)∥χr≤32ρC5C6((λH+λλHC42)2C24C74ελ(H)4+(λH+λλHC42)C22C72ελ(H)2C2C4C7ελ(H)(λH+λλHC42)2)+(C2×q+3)C7ελ(h1)≤2(C2×q+3)C7ελ(h1).
Since σ=0, substituting (4.23) into (3.43), we deduce
(4.24)|λh1-λ|≤(λh1+λλh1C42)∥u→h1-u→∥χr2≤(λh1+λλh1C42)4(C2×q+3)2C72ελ(h1)2.
The above two inequalities show that Theorem 4.2 is true for l=1.
Suppose that the theorem is true forl-1, then by Theorem 3.3, we get
(4.25)∥u→hl-u→∥χr≤32ρC5C6((λhl-1+λλhl-1C42)242(C2×q+3)4C24C74ελ(hl-1)4+(λhl-1+λλhl-1C42)4(C2×q+3)2C72ελ(hl-1)22(C2×q+3)C7ελ(hl-1))+(C2×q+3)C7ελ(hl)≤2(C2×q+3)C7ελ(hl),
That is, (4.20) is valid.
From (4.20) and (3.8), we obtain (4.21). The proof is completed.Assume thatKh is a regular simplex partition (tetrahedral partition) of Ω with the mesh diameter h. Let Vh and Wh be the Pk+1-Pk finite element spaces as follows:
(4.26)Vh={q→∈C0(Ω¯)3:q→×γ→|∂Ω=0,q→|κ∈Pk+1(κ)3·∀κ∈Kh},Wh={v∈C0(Ω¯):v|κ∈Pk(κ),∀κ∈Kh,v|Eh=0}.
Here we set Eh=∪κ∈Kh,∂κ∩E≠ϕκ. v|Eh=0 means that v is equal to 0 on the tetrahedron where reentrant edge and angular point are adjacent. Considering finite element approximation of (4.11), for the 3-DP2-iso-P1 Taylor-Hood finite element, Ciarlet and Girault [26] have discussed that the discrete inf-sup condition (C3) holds when Ω is a convex domain; for the Pk+1-Pk element, Ciarlet and Hechme have proved that the discrete inf-sup condition (C3) holds when Ω is a polyhedron (see Section  2.2 in [4] and pp. 509 in [3]).From the above mentioned, we know that thePk+1-Pk element approximation of (4.11) satisfies the conditions of Theorem 4.2.LetσΔN be the smallest singular exponent in the Laplace problem with homogenous Neumman boundary condition, then σΔN∈(3/2,2). Denote τ=min(r-rmin,σΔN-1).Corollary 4.4.
Assume thath0=H is small properly, ti∈(1,3](i=1,2,.,l), and hi=hi-1ti(i=1,2,…,l). Let (λhl,uhl) be an approximate eigenpair of the Pk+1-Pk element obtained by Scheme 1. Then when Ω is a convex domain, there exists u→∈M(λ) such that
(4.27)∥u→hl-u→∥χr≤2(C2×q+3)C7C′hl,(4.28)|λhl-λ|≤4(λhl+λλhlC42)(C2×q+3)2C72C'2hl2.
When Ω is a non-convex domain, there exists u→∈M(λ) such that
(4.29)∥u→hl-u→∥χr≤2(C2×q+3)C7C′′hlμ,∀μ∈(0,τ),(4.30)|λhl-λ|≤4(λhl+λλhlC42)(C2×q+3)2C72C′′2hl2μ,∀μ∈(0,τ),
where C′ and C′′ are determined by (4.31) and (4.32) in the proof, respectively.Proof.
The hypotheses of Corollary4.4 imply that the conditions of Theorem 4.3 hold. When Ω is convex, for any u→∈M(λ) we have u→=T(λu→)∈H2(Ω) (see (44) in [1]). Thus there exists C′ independent of hl(l=1,2,…) such that
(4.31)ελ(hl)≤C′hl.
Substituting the above inequality into (4.20) and (4.21), we get (4.27) and (4.28), respectively.
WhenΩ is a non-convex domain, for any u→∈M(λ), by (36) in [3] we know that there exists C′′ independent of hl(l=1,2,…), such that
(4.32)ελ(hl)≤C′′hlμ.
Substituting the above inequality into (4.20) and (4.21), we derive (4.29) and (4.30), respectively.Remark 4.5.
From Corollary4.4, we see that the constants in the error estimates are not only independent of the mesh diameter but also independent of the iterative degrees. Thus, when l→∞, we have λhl→λ. Suppose that the precision order of (4.28) is optimal which cannot be improved any more, then
(4.33)λhl-λhl+1=λhl-λ+λ-λhl+1=O(hl2r),λhl-1-λhl=λhl-1-λ+λ-λhl=O(hl-12r),
where for convex domain r=1, while for non-convex domain r=μ∈(0,τ) but approximates τ arbitrarily. Therefore we have that λhl-λhl+1≤λhl-1-λhl. Then
(4.34)|λhl-λ|≤|λhl-λhl+1|+|λhl+1-λ|≤|λhl-1-λhl|+ϑ(hl+12r),
thus we can use η(λhl)=θ×|λhl-1-λhl| as a posteriori error indicator of |λhl-λ|, where θ∈(0,1].


## 5. Numerical Experiments

Consider the eigenvalue problem (4.1) of electric field, where Ω-=[0,π]×[0,π] is a square domain or Ω-=[-1,0]×[-1,0]∪[-1,1]×[0,1] is an L-shaped domain. For the square domain, the first five exact eigenvalues are λ1=λ2=1, λ3=2, and λ4=λ5=4; for the L-shaped domain, the first five eigenvalues are λ1≈1.47562182408, λ2≈3.53403136678, λ3=λ4=π2≈9.86960440109, and λ5≈11.3894793979, and the first eigenfunction has a strong singularity (see, e.g.,  [3]).We adopt a uniform isosceles right triangulation forΩ (the edge in each element is along three fixed directions, see Figure 1) to produce the meshes Khl with mesh diameter hl.Figure 1The definition ofP2-P1 mixed finite element spaces is given by
(5.1)Vhl={q→∈C0(Ω¯)2:q→×γ→|∂Ω=0,q→|κ∈P2(κ)2,∀κ∈Khl},Whl={v∈C0(Ω¯):v|κ∈P1(κ),∀κ∈Khl,v|Ehl=0}.
We make use of Matlab to compute the first five approximate eigenvalues by using Scheme 1 with P2-P1 element. The numerical results are listed in Tables 1, 2, and 3.Table 1
The results on the square by Scheme1 (P2-P1 element) for the eigenvalue problem of electric field (r=0).
k
H
l
h
l
λ
k
,
h
l
λ
k
h
l
η
(
λ
k
,
h
l
)
/
e
(
λ
k
,
h
l
)
η
(
λ
k
h
l
)
/
e
(
λ
k
h
l
)
1
2
/
8
1
2
/
32
1.0000001280750
1.00000012807483
248.41
248.41
1
2
/
8
2
2
/
64
1.0000000080355
1.00000000803501
14.94
14.94
1
2
/
8
3
2
/
128
—
1.00000000050340
—
14.96
3
2
/
8
1
2
/
32
2.0000017958515
2.00000179585163
245.62
245.62
3
2
/
8
2
2
/
64
2.0000001126165
2.00000011261761
14.95
14.95
3
2
/
8
3
2
/
128
—
2.00000000704840
—
14.98
4
2
/
8
1
2
/
32
4.0000081820714
4.00000818207150
242.08
242.08
4
2
/
8
2
2
/
64
4.0000005140035
4.00000051400385
14.92
14.92
4
2
/
8
3
2
/
128
—
4.00000003219500
—
14.97Table 2
The results on the L-shaped domain by Scheme1 (P2-P1 element) for the eigenvalue problem of electric field (r=0.5).
k
H
l
h
l
λ
k
,
h
l
λ
k
h
l
η
(
λ
k
,
h
l
)
/
e
(
λ
k
,
h
l
)
η
(
λ
k
h
l
)
/
e
(
λ
k
h
l
)
1
2
/
10
1
2
/
40
2.619901684020
2.619902307450
0.16
0.16
1
2
/
10
2
2
/
80
2.545994662814
2.545994677427
0.69
0.69
1
2
/
10
3
2
/
160
—
2.468072524066
—
0.78
2
2
/
10
1
2
/
40
3.540738971244
3.540738974231
5.75
5.75
2
2
/
10
2
2
/
80
3.536561194304
3.536561194305
1.65
1.65
2
2
/
10
3
2
/
160
—
3.534975905948
—
1.67
3
2
/
10
1
2
/
40
9.869612668412
9.869612668412
244.51
244.51
3
2
/
10
2
2
/
80
9.869604920374
9.869604920377
14.92
14.92
3
2
/
10
3
2
/
160
—
9.869604433619
—
14.92
5
2
/
10
1
2
/
40
11.392491049450
11.392491049225
7.11
7.11
5
2
/
10
2
2
/
80
11.390607786437
11.390607786437
1.67
1.67
5
2
/
10
3
2
/
160
—
11.389899921315
—
1.68Table 3
The results on the L-shaped domain by Scheme1 (P2-P1 element) for the eigenvalue problem of electric field (r=0.95).
k
H
l
h
l
λ
k
,
h
l
λ
k
h
l
η
(
λ
k
,
h
l
)
/
e
(
λ
k
,
h
l
)
η
(
λ
k
h
l
)
/
e
(
λ
k
h
l
)
1
2
/
10
1
2
/
40
1.550099678021
1.550100277590
1.88
1.88
1
2
/
10
2
2
/
80
1.512784318422
1.512784324775
1.00
1.00
1
2
/
10
3
2
/
160
—
1.492972425344
—
1.14
2
2
/
10
1
2
/
40
3.534598663496
3.534598663588
14.65
14.65
2
2
/
10
2
2
/
80
3.534154160497
3.534154160495
3.62
3.62
2
2
/
10
3
2
/
160
—
3.534055109225
—
4.17
3
2
/
10
1
2
/
40
9.869612641692
9.869612641692
243.23
243.23
3
2
/
10
2
2
/
80
9.869604919289
9.869604919288
14.90
14.90
3
2
/
10
3
2
/
160
—
9.869604433575
—
14.92
5
2
/
10
1
2
/
40
11.389671749018
11.389671749041
28.83
28.83
5
2
/
10
2
2
/
80
11.389519405478
11.389519405477
3.81
3.81
5
2
/
10
3
2
/
160
—
11.389487122250
—
4.18In Tables1–3, λ1,hl,λ2,hl,…,λ5,hl denote the first five eigenvalues obtained by using the mixed element on Khl directly; λ1hl,λ2hl,…,λ5hl denote the first five eigenvalues obtained by Scheme 1:
(5.2)e(λk,hl)=|λk,hl-λk|,e(λkhl)=|λkhl-λk|,η(λk,hl)=θ|λk,hl-λk,hl-1|,η(λkhl)=θ|λkhl-λkhl-1|,(θ=1).From Tables1–3, we can see that (1)  λkhl and λk,hl have the same accuracy. (2) It fails to find λk,hl by direct computation by using the mixed element on Khl with hl=2/128 in the case of the square domain and hl=2/160 in the case of the L-shaped domain (here Matlab shows that Sparse lu with 4 outputs (UMFPACK) failed), but Scheme 1 still works. (3)η(λkhl)=|λkhl-λkhl-1| is an efficient and reliable error indicator of λkhl.It can be seen from Tables1–3 that, in the calculation of error indicators, θ should be selected as follows: in the case of the square domain, θ=1/15; in the case of the L-shaped domain, θ is equal to 1,3/5,1/15, and 3/5, respectively when r=0.5, and θ is equal to 1,1/4,1/15 and 1/4 respectively, when r=0.95.Remark 5.1.
Taking Table1, for example, we illustrate how to select ti next. In Table 1, h0=2/8, h1=2/32, h2=2/64, and h3=2/128. According to hi=hi-1ti, by calculation, we get that t1≈1.80, t2≈1.22, and t3≈1.18.



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*Source: 190768-2012-11-08.xml*