# Highly Efficient Calculation Schemes of Finite-Element Filter Approach for the Eigenvalue Problem of Electric Field

Authors: Yu Zhang; Yidu Yang; Jie Liu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper discusses finite-element highly efficient calculation schemes for solving eigenvalue problem of electric field. Multigrid discretization is extended to the filter approach for eigenvalue problem of electric field. With this scheme one solves an eigenvalue problem on a coarse grid just at the first step, and then always solves a linear algebraic system on finer and finer grids. Theoretical analysis and numerical results show that the scheme has high efficiency. Besides, we use interpolation postprocessing technique to improve the accuracy of solutions, and numerical results show that the scheme is an efficient and significant method for eigenvalue problem of electric field.


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## Body

## 1. Introduction

In recent years, eigenvalue problems of electric field has attracted increasing attention in the fields of physics and mathematics, and its numerical methods (the filter approach, the parameterized approach, and the mixed approach) are also developed further (see [1–7]). Although the filter approach is an effective and important method for solving eigenvalue problems of electric field, its computation costs and accuracy of numerical solutions still need to be improved.In fact, it is really a challenging job to reduce the computation costs without decreasing the accuracy of finite-element solutions. As we know, two-grid discretization and multigrid discretization are reliable and important methods satisfying the above requirements. Two-grid discretization was first introduced by Xu for nonsymmetric and nonlinear elliptic problems, and so forth (see [8–10]). Later on, it was successfully applied to Stokes equations, semilinear eigenvalue problems and linear eigenvalue problems, and so forth (see [11–16]). Recently, Yang and Bi [16] established two-grid finite-element discretization and multigrid discretization Schemes based on shifted-inverse power method. Referecnces [6, 7] applied the two Schemes to the mixed approach for eigenvalue problem of electric field, and [17] applied them to conforming finite element for the Steklov eigenvalue problem. Based on the work mentioned above, this paper discusses two-grid discretization and multigrid discretization Schemes of the filter approach for eigenvalue problem of electric field and analyzes error estimates. They are extensions of Scheme  2 and Scheme  3 in [16], respectively.From 1989 to 1991, Lin and Yang firstly pointed out and proved that the function, obtained by using nodes of lower-order element as interpolation nodes to make a higher order interpolation of lower order finite-element solutions, can have global gradient superconvergence. The technique used to obtain global superconvergence was called finite-element interpolation postprocessing or finite-element interpolation correction (see reviews paper [18] and the references cited therein). For over 20 years, finite element interpolation postprocessing technique has been developed greatly and was applied to a variety of partial differential equations (see [19–25]). It is applied to this paper too. We give Theorem 4.1, and our numerical results show that interpolation postprocessing is an efficient and significant method for solving eigenvalue problems of electric field.The rest of this paper is organized as follows. In the next section, some preliminaries which are needed are provided. In Section3, two kinds of finite-element discretization schemes for eigenvalue problem of electric field are given and the error estimates are established. In Section 4, we introduce interpolation postprocessing technique. Finally, numerical experiments are presented.


## 2. Preliminaries

LetΩ⊂Rn  (n=2,3) be a bounded polyhedron domain with boundary ∂Ω. We denote by n the unit outward normal vector to ∂Ω, by u the electric field, and by ω the time frequency. Let c=3.0×108 m/s be the light velocity in vacuum, curl curl operator, and div divergence operator.Consider the following eigenvalue problem of electric field:(2.1)c2curlcurlu=ω2u,inΩ,divu=0,inΩ,u×n=0,on∈∂Ω.
Let λ=ω2/c2 named eigenvalue.Define function spaces as follows:(2.2)H(curl,Ω)={v∈L2(Ω)n:curlv∈L2(Ω)n},H0(curl,Ω)={v∈H(curl,Ω):v×n|∂Ω=0}.
When Ω is a convex polyhedron, we define the following function space:
(2.3)X={v∈H0(curl,Ω):divv∈L2(Ω)}.
Denote
(2.4)b(u,v)=(u,v)0=∫Ωu·vdx,∥u∥0=(u,u)01/2,a(u,v)=(u,v)X=(curlu,curlv)0+(divu,divv)0,∥u∥X=(u,u)X1/2.
Let σΔD∈(3/2,2) be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition:
(2.5){ϕ∈H1(Ω):Δϕ∈L2(Ω),ψ|∂Ω=0}⊂∩s<σΔDHs(Ω),{ϕ∈H1(Ω):Δϕ∈L2(Ω),ψ|∂Ω=0}⊄HσΔD(Ω).
Set γmin=2-σΔD and γ∈(γmin,1).WhenΩ is a nonconvex polyhedron, let E denote a set of edges of reentrant dihedral angles on ∂Ω, and let d=d(x) denote the distance to the set E:d(x)=dist(x,∪e∈Ee-). We introduce a weight function ωγ which is a nonnegative smooth function corresponding to x. It can be represented by dγ in reentrant edge and angular domain. We shall write ωγ≃dγ. Define the weighted functional spaces:
(2.6)Lγ2(Ω)={w∈Lloc2(Ω):ωγw∈L2(Ω)},Xγ={v∈L2(Ω)n:curlv∈L2(Ω)n,v×n|∂Ω=0,divv∈Lγ2(Ω)}.
Denote
(2.7)(w,v)Lγ2=∫Ωωγ2wvdx,∥w∥0,γ=(w,w)Lγ21/2,a(u,v)=(u,v)Xγ=(curlu,curlv)0+(divu,divv)Lγ2,∥u∥Xγ=(u,u)Xγ1/2.Note thatXγ=X when Ω is a convex polyhedron, namely, in the case of γ=0. Consider the variational formulation: Find (λ,u)∈R+×Xγ with ∥u∥Xγ=1, such that
(2.8)a(u,v)=λb(u,v),∀v∈Xγ.Letπh be a regular simplex partition, and let Xh be a space of piecewise polynomial of degree less than or equal to k defined on πh:
(2.9)Xh={v∈C0(Ω-)n:v×n|=∂Ω0,v|κ∈Pk(κ)n,∀κ∈πh}.
Then, Xh⊆Xγ.The discrete variational form of (2.8): Find (λh,uh)∈R+×Xh with ∥uh∥Xγ=1, such that
(2.10)a(uh,v)=λhb(uh,v),∀v∈Xh.
The eigenpairs of (2.1) must be that of (2.8). But the converse of this statement may not be true, namely, (2.8) has spurious pairs. Hence, (2.10) has spurious pairs.It is easy to prove thata(·,·) and b(·,·) are symmetric bilinear forms. Next we shall prove that a(·,·) is continuous and V-elliptic.From the definition ofa(·,·), we have
(2.11)|a(w,v)|=|∫Ωcurlw·curlvdx+∫Ωωγ2divwdivvdx|≤∥curlw∥0∥curlv∥0+∥divw∥0,γ∥divv∥0,γ≤(∥curlw∥02+∥divw∥0,γ2)(∥curlv∥02+∥divv∥0,γ2)≤∥w∥Xγ∥v∥Xγ.
Therefore, continuity of a(·,·) is valid. And
(2.12)a(w,w)=∥w∥Xγ2,
which indicates that a(·,·) is V-elliptic.Define operatorT:Xγ→Xγ satisfying
(2.13)a(Tf,v)=b(f,v),∀v∈Xγ.
Define operator Th:Xh→Xh satisfying
(2.14)a(Thf,v)=b(f,v),∀v∈Xh.
It is easy to prove that T:Xγ→Xγ, Th:Xh→Xh is self-adjoint completely continuous operator, respectively. Actually, for all f,g∈Xγ, we have
(2.15)a(f,Tg)=a(Tg,f)=b(g,f)=b(f,g)=a(Tf,g),
which shows that T:Xγ→Xγ is self-adjoint in the sense of inner product a(·,·). Similarly, we can prove that Th:Xh→Xh is self-adjoint in the sense of inner product a(·,·).From [2, 4], we get Xγ↪L2(Ω)3 (compactly imbedded). Hence, we derive that operator T:Xγ→Xγ is completely continuous. Obviously, Th:Xh→Xh is a finite-rank operator.By [3, 26], we know that (2.8) has the following equivalent operator form:
(2.16)Tu=μu.
Denote μk=1/λk, μk,h=1/λk,h.Then, the eigenvalues of (2.8) are sorted as
(2.17)0<λ1≤λ2≤⋯≤λk≤⋯↗+∞.
We can construct a complete orthogonal system of Xγ by using the eigenfunctions corresponding to {λk}:
(2.18)u1,u2,…,uk,….
Equation (2.10) has the following equivalent operator form:
(2.19)λhThuh=uh.
Then the eigenvalues of (2.10) are sorted as
(2.20)0<λ1,h≤λ2,h≤⋯≤λk,h≤⋯λNh,h,
and the corresponding eigenfunctions are
(2.21)u1,h,u2,h,…,uk,h,…,uNh,h,
where Nh=dimXh.In this paper,μk and μk,h, λk and λk,h are all called eigenvalues.Suppose that the algebraic multiplicity ofμk is equal to q. μk=μk+1=⋯=μk+q-1. Let M(μk) be the space spanned by all eigenfunctions corresponding to μk of T, and let Mh(μk) be the space spanned by all eigenfunctions corresponding to all eigenvalues of Th that converge to μk. Let M^(μk)={v:v∈M(μk),∥v∥Xγ=1}, M^h(μk)={v:v∈Mh(μk),∥v∥Xγ=1}. We also write M(λk)=M(μk), Mh(λk)=Mh(μk), M^(λk)=M^(μk), and M^h(λk)=M^h(μk).The Filter Approach
Let(λh,uh) be an eigenpair of (2.10), we know that some of these eigenvalues are “real,” but some are spurious (namely, not divergence free). We should filter out the spurious pairs to obtain “real” eigenpairs. Hence, ones designed a filter ratio:
(2.22)∥divuh∥0,γ∥curluh∥0.
The corresponding value of filter ratio is small for “real” pairs since the divergence part of the eigenvector is small, whereas it is large for spurious ones since the curl part small. Noting that when a multiple eigenvalue is dealt with, an additional step must be carried out (see [3, 5]).Next we introduce error estimates for the filter approach.Define∥(T-Th)|M(λk)∥Xγ=maxu∈M(λk)(∥(T-Th)u∥Xγ/∥u∥Xγ).Denote(2.23)δh(λk)=supu∈M^(λk)infv∈Xh∥u-v∥Xγ.
Let Ph:Xγ→Xh be orthogonal projection, namely,
(2.24)a(u-Phu,v)=0,∀v∈Xh.
Then, Th=PhT.Using the spectral theory (see [26]), [3] discussed error estimates for the filter approach and gave the following lemmas.Lemma 2.1.
∥
T
h
-
T
∥
X
γ
→
0  (h→0).Lemma 2.2.
Let(λk,h,uk,h) be the kth eigenpair of (2.10) with ∥uk,h∥Xγ=1. Let λk be the kth eigenvalue of (2.8). Then, there exists uk∈M^(λk) such that
(2.25)|λk,h-λk|≤C1δh2(λk),(2.26)∥uk,h-uk∥Xγ≤C2δh(λk).
For any uk∈M^(λk), there exists uh∈Mh(λk) such that
(2.27)∥uh-uk∥Xγ≤C3δh(λk),
where C1, C2, and C3 are constants independent of mesh diameter.In this paper, we will use the following lemma.Lemma 2.3.
Let(λ,u) be an eigenpair of (2.8), then for any w∈Xγ, ∥w∥0≠0, the Rayleigh quotient a(w,w)/∥w∥02 satisfies
(2.28)a(w,w)∥w∥02-λ=∥w-u∥Xγ2∥w∥02-λ∥w-u∥02∥w∥02.Proof.
The proof is completed by using the same proof steps as that of Lemma  9.1 in [26].


## 3. Two-Grid Discretization Scheme and Multigrid Discretization Scheme

Consider (2.19) on Xh (inner product a(·,·) and norm ∥·∥Xγ). We will discuss the high efficiency of two-grid discretization scheme and multigrid discretization scheme next.Lemma 3.1.
For all nonzerou,v∈Xγ,
(3.1)∥u∥u∥Xγ-v∥v∥Xγ∥Xγ≤2∥u-v∥Xγ∥u∥Xγ,∥u∥u∥Xγ-v∥v∥Xγ∥Xγ≤2∥u-v∥Xγ∥v∥Xγ.Proof.
See [16].Lemma 3.2.
Let(μ0,u0) be an approximation for (μk,uk), where μ0 is not an eigenvalue of Th, and u0∈Xh with ∥u0∥Xγ=1. Suppose that maxk≤j≤k+q-1|(μj,h-μk,h)/(μ0-μj,h)| ≤ 1/2, dist(u0,Mh(μk)) ≤ 1/2,  |μ0-μj,h| ≥ ρ/2,  (j≠k,k+1,…,k+q-1), and u∈Xh,  ukh∈Xh satisfy
(3.2)(μ0-Th)u=u0,ukh=u∥u∥Xγ.
Then
(3.3)dist(ukh,M^h(μk))≤16ρ|μ0-μk,h|dist(u0,Mh(μk)),
where ρ=minμj≠μk|μj-μk| is the separation constant of the eigenvalue μk.Proof.
See [16].


### 3.1. Two-Grid Discretization Scheme

Reference [16] established the two-grid discretization scheme based on shifted-inverse power method. Next, we will apply the scheme to eigenvalue problem of electric field.LetπH and πh be regular meshes (see [3]) with diameters H and h, respectively. Let δh(λk)=δH(λk)t2, t2∈[1+δ,3-δ], and δ be a properly small positive number.Scheme 1.
Two-grid Discretization.Step 1.
Solve (2.8) on a coarse grid πH: Find (λH,uH)∈R+×XH, such that ∥uH∥Xγ=1, and
(3.4)a(uH,v)=λHb(uH,v),∀v∈XH.
And obtain the “real” eigenpair (λk,H,uk,H) by filtering process.Step 2.
Solve a linear system on a fine gridπh: Find u∈Xh, such that
(3.5)a(u,v)-λk,Hb(u,v)=b(uk,H,v),∀v∈Xh.
And set ukh=u/∥u∥Xγ.Step 3.
Compute the Rayleigh quotient(3.6)λkh=a(ukh,ukh)b(ukh,ukh).
We use(λkh,ukh) as the approximate eigenpair of (2.1).Theorem 3.3.
Suppose thatH is properly small. Let (λkh,ukh) be the approximate eigenpair obtained by Scheme 1. Then there exists eigenpair (λk,uk) of (2.1), such that
(3.7)∥ukh-uk∥Xγ≤48ρC1C2C4C5δH3(λk)+3C2qδh(λk),(3.8)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
where C4, C5, and C6 are positive constants independent of mesh diameters, and these constants are decided by (3.11), (3.13), and (3.30) in the following proof.Proof.
We use Lemma3.2 to complete the proof. Select μ0=1/λk,H and u0 = λk,HThuk,H/∥λk,HThuk,H∥Xγ. Obviously,
(3.9)∥u0∥Xγ=1.
Noting that λk=λk+1=⋯λk+q-1, for j=k,k+1,…,k+q-1, we have
(3.10)|μj,h-μk,h|=|λk,h-λj,hλk,hλj,h|=|λk,h-λk+λj-λj,hλk,hλj,h|≤|λk,h-λk|+|λj-λj,h|λk,hλj,h,(3.11)|μ0-μj,h|=|λj,h-λk,Hλj,hλk,H|=|λj,h-λj+λk-λk,Hλj,hλk,H|≤C4|λk,H-λk|.
Combining the above two inequalities with (2.25) and noting that δh(λk) is a small quantity of higher order than δH(λk), we obtain
(3.12)maxk≤j≤k+q-1|μj,h-μk,hμ0-μj,h|≤12.
From Lemma 2.1, we know that ∥Th-T∥Xγ→0(h→0), then there exists a constant C5 independent of h, such that
(3.13)∥Thf∥Xγ≤C5∥f∥Xγ,∀f∈Xh.
Obviously, there exists uk∈M^(λk), such that
(3.14)∥uk,H-uk∥Xγ=dist(uk,H,M^(λk)).
Then, we derive
(3.15)∥λk,HThuk,H-uk∥Xγ=∥λk,HThuk,H-λkTuk∥Xγ≤∥(λk,H-λk)Thuk,H∥Xγ+∥λkTh(uk,H-uk)∥Xγ+∥λk(Th-T)uk∥Xγ≤C5|λk,H-λk|+C5λk∥uk,H-uk∥Xγ+∥(Ph-I)uk∥Xγ≤C5|λk,H-λk|+C5λk∥uk,H-uk∥Xγ+δh(λk).
Hence, by Lemma 3.1, (3.15) and (2.28), we have
(3.16)dist(u0,M^(λk))≤∥u0-uk∥Xγ≤2∥λk,HThuk,H-uk∥Xγ≤2C5|λk,H-λk|+2C5λk∥uk,H-uk∥Xγ+2δh(λk)≤3C5λk∥uk,H-uk∥Xγ+2δh(λk)=3C5λkdist(uk,H,M^(λk))+2δh(λk).
Combining the triangle inequality, (2.27) and (3.16), we deduce
(3.17)dist(u0,Mh(λk))≤dist(u0,M^(λk))+C3δh(λk)≤3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk).
Since H is small enough and δh(λk)=δH(λk)t2, from (2.26) and (3.17), we know
(3.18)dist(u0,Mh(λk))≤12.
For j≠k,  k+1,…,k+q-1, since H is small enough, ρ is the separation constant, we have
(3.19)|μ0-μj,h|≥ρ2.
From the Step 2 in Scheme 1 and (2.14), we get
(3.20)a(u,v)-λk,Ha(Thu,v)=a(Thuk,H,v),
namely,
(3.21)u-λk,HThu=Thuk,H.
Thus (1/λk,H)u-Thu=(1/λk,H)Thuk,H and ukh=u/∥u∥Xγ. Note that (1/λk,H)Thuk,H = ∥(1/λk,H)Thuk,H∥Xγu0 differs from u0 by only a constant; then Step 2 is equivalent to
(3.22)(1λk,H-Th)u=u0,ukh=u∥u∥Xγ.
From the arguments of (3.9), (3.12), (3.18), (3.19), and (3.22), we see that the conditions of Lemma 3.2 hold. Hence, substituting (3.11) and (3.17) into (3.3), we obtain
(3.23)dist(ukh,M^h(λk))≤16ρC4|λk,H-λk|(3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk)).
Let eigenvectors {uj,h}kk+q-1 be an orthogonal basis of Mh(λk) (in the sense of inner product a(·,·)), then
(3.24)dist(ukh,Mh(λk))=∥ukh-∑j=kk+q-1‍a(ukh,uj,h)uj,h∥Xγ.
Set
(3.25)u*=∑j=kk+q-1‍a(ukh,uj,h)uj,h.
From (3.23), we directly get
(3.26)∥ukh-u*∥Xγ=dist(ukh,Mh(λk))≤16ρC4|λk,H-λk|(3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk)).
From Lemma 2.2, we know that there exist {uj0}kk+q-1⊂M(λk) making uj,h-uj0 satisfy (2.26). Let uk=∑j=kk+q-1a(ukh,uj,h)uj0, then
(3.27)∥u*-uk∥Xγ=∥∑j=kk+q-1‍a(ukh,uj,h)(uj,h-uj0)∥Xγ≤(∑j=kk+q-1‍∥uj,h-uj0∥Xγ2)1/2≤C2∑j=kk+q-1‍δh(λj)≤C2qδh(λk).
Combining (3.26) and (3.27), we obtain
(3.28)∥ukh-uk∥Xγ≤∥ukh-u*∥Xγ+∥u*-uk∥Xγ≤48ρC4C5λk|λk,H-λk|dist(uk,H,M^(λk))+3C2qδh(λk).
Besides, by (2.26), we easily know
(3.29)dist(uk,H,M^(λk))≤C2δH(λk),
which together with (3.28) and (2.25) leads to (3.7).
From the continuous embedding ofXγ into L2(Ω)n, we get that there exists a constant C6 independent of meshes, such that
(3.30)∥u∥02≤C6∥u∥Xγ2.
Equation (3.7) indicates that ukh converges to uk in the sense of norm ∥·∥Xγ, then ukh converges to uk in the sense to norm ∥·∥0; thus, 1/∥ukh∥02→1/∥uk∥02=λk. Therefore, when h is small enough, we have
(3.31)|λkh-λk|≤∥ukh-uk∥Xγ2∥ukh∥02+λk∥ukh-uk∥02∥ukh∥02≤2λk∥ukh-uk∥Xγ2+2λk2∥ukh-uk∥02≤2λk(1+C6λk)∥ukh-uk∥Xγ2.
The proof of Theorem 3.3 is completed.LetσΔN be the smallest singular exponent in the Laplace problem with homogenous Neumann boundary condition, then σΔN∈(3/2,2). Denote τ=min(γ-γmin,σΔN-1).Corollary 3.4.
Suppose thatH is properly small. Let (λkh,ukh) be an approximate eigenpair obtained by Scheme 1. Then there exists an eigenpair (λk,uk) of (2.1), such that when Ω is a convex domain,
(3.32)∥ukh-uk∥Xγ≤6qC2C′h,(3.33)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
when Ω is a nonconvex domain,
(3.34)∥ukh-uk∥Xγ≤6qC2C′′hμ,∀μ∈(0,τ),(3.35)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
where C′ and C′′ are stated in the proof as follows.Proof.
From [1, 4], we know that when Ω is a convex domain, there exists a constant C′ independent of h, such that
(3.36)δh(λk)≤C′h.
Substituting the above inequality into (3.7), and noting that δh(λk) is an infinitesimal of lower order comparing with δH(λk)3, we know that (3.32) is valid.
And whenΩ is a nonconvex domain, there exists a constant C′′ independent of h, such that
(3.37)δh(λk)≤C′′hμ,
where μ∈(0,τ).
Substituting the above inequality into (3.7), we know that (3.34) is valid.


### 3.2. Multigrid Discretization Scheme

Next, we will discuss finite-element multigrid discretization scheme based on Rayleigh quotient iteration method. Assume that partition satisfies the following condition.Condition (A).
{
π
h
i
}
1
l is a family of regular meshes (see [3]) with diameters hi, δhi+1(λk)=δhi(λk)ti+1, ti+1∈[1+δ,3-δ], i=1,2,…, and δ is a properly small positive number.Let{Xhi}1l be the finite-element spaces defined on {πhi}1l. Further, let πH=πh1, XH=Xh1.Scheme 2.
Multigrid Discretization.Step 1.
Solve (2.8) on a coarse grid πH: Find (λH,uH)∈R+×XH, with ∥uH∥Xγ=1, such that
(3.38)a(uH,v)=λHb(uH,v),∀v∈XH.
And obtain the “real” eigenpair (λk,H,uk,H) by filtering process.Step 2.
u
k
h
1
=
u
k
,
H,  λkh1=λk,H,  i⇐2.Step 3.
Solve a linear system on a fine gridπhi: Find u∈Xhi, such that
(3.39)a(u,v)-λkhi-1b(u,v)=b(ukhi-1,v),∀v∈Xhi.
Set ukhi=u/∥u∥Xγ.Step 4.
Compute the Rayleigh quotient(3.40)λkhi=a(ukhi,ukhi)b(ukhi,ukhi),Step 5.
Ifi=l, then output (λkhl,ukhl), stop. Else, i⇐i+1, and return to Step 3.We use(λkhl,ukhl) obtained by Scheme 2 as the approximate eigenpair of (2.1). Next, we will discuss the efficiency of Scheme 2.Theorem 3.5.
Suppose thatH is properly small and Condition (A) holds. Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair (λk,uk) of (2.1), such that
(3.41)∥ukhl-uk∥Xγ≤48ρC4C5λk|λkhl-1-λk|dist(ukhl-1,M^(λk))+3C2qδhl(λk),(3.42)|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ,l≥2.Proof.
We use induction to complete the proof of (3.41).
Forl=2, Scheme 2 is actually Scheme 1. Hence, (3.41) is easily obtained from (3.28).
Suppose that (3.41) holds for l=3,4,…,l-1. Next, we shall prove that (3.41) holds for l.
Selectμ0=1/λkhl-1, μk,hl=1/λk,hl, and u0=λkhl-1Thlukhl-1/∥λkhl-1Thlukhl-1∥Xγ. Using the proof method of Theorem 3.3, we deduce
(3.43)|μ0-μk,hl|≤C4|λkhl-1-λk|,maxk≤j≤k+q-1|μj,hl-μk,hlμ0-μj,hl|≤12.
Using the triangle inequality and (2.27), we get
(3.44)dist(u0,Mhl(λk))≤dist(u0,M^(λk))+C3δhl(λk),
and together with the induction assumption, yields
(3.45)dist(u0,Mhl(λk))≤12.
From Step 3 of Scheme 2, we know that ukhl satisfies
(3.46)(1λkhl-1-Th)u=u0,ukhl=u∥u∥Xγ.
From the above arguments, we know that the conditions of Lemma 3.2 hold.
Defineu* and uk as those in Theorem 3.3 (using ukhl instead of ukh, uj,hl instead of uj,h), then
(3.47)u*=∑j=kk+q-1‍a(ukhl,uj,hl)uj,hl,uk=∑j=kk+q-1‍a(ukhl,uj,hl)uj0,
where uj,hl-uj0 satisfies (2.26). We can derive by Lemma 3.2 and the proof of (3.11) that
(3.48)∥ukhl-u*∥Xγ=dist(ukhl,Mhl(λk))≤16ρ|μ0-μk,hl|dist(u0,Mhl(λk))≤16ρC4|λkhl-1-λk|dist(u0,Mhl(λk)).
Substituting (3.44) into the above inequality, we deduce
(3.49)∥ukhl-u*∥Xγ≤16ρC4|λkhl-1-λk|(dist(u0,M^(λk))+C3δhl(λk)).
Like the proof method of (3.27), we get
(3.50)∥u*-uk∥Xγ≤C2qδhl(λk).
From the above two inequalities, we obtain
(3.51)∥ukhl-uk∥Xγ≤16ρC4|λkhl-1-λk|dist(u0,M^(λk))+2C2qδhl(λk).
There exists a constant C5 independent of hl such that
(3.52)∥Thlf∥Xγ≤C5∥f∥Xγ,∀f∈Xhl.
Like the proof method of (3.16), we can derive
(3.53)dist(u0,M^(λk))≤3C5λkdist(ukhl-1,M^(λk))+2δhl(λk).
Combining (3.51) and (3.53), we know that (3.41) is valid. Like the proof method of (3.8), we get (3.42), namely, Theorem 3.5 is valid.Corollary 3.6.
Suppose that Condition (A) holds andh1 (namely, H) is properly small. Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair (λk,uk) of (2.1), such that the following error estimates hold: when Ω is a convex domain,
(3.54)∥ukhl-uk∥Xγ≤6qC2C′hl,|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ2;
when Ω is a nonconvex domain,
(3.55)∥ukhl-uk∥Xγ≤6qC2C′′hlμ,∀μ∈(0,τ),|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ2,
where the C′ and C′′ are the ones in Corollary 3.4.


## 3.1. Two-Grid Discretization Scheme

Reference [16] established the two-grid discretization scheme based on shifted-inverse power method. Next, we will apply the scheme to eigenvalue problem of electric field.LetπH and πh be regular meshes (see [3]) with diameters H and h, respectively. Let δh(λk)=δH(λk)t2, t2∈[1+δ,3-δ], and δ be a properly small positive number.Scheme 1.
Two-grid Discretization.Step 1.
Solve (2.8) on a coarse grid πH: Find (λH,uH)∈R+×XH, such that ∥uH∥Xγ=1, and
(3.4)a(uH,v)=λHb(uH,v),∀v∈XH.
And obtain the “real” eigenpair (λk,H,uk,H) by filtering process.Step 2.
Solve a linear system on a fine gridπh: Find u∈Xh, such that
(3.5)a(u,v)-λk,Hb(u,v)=b(uk,H,v),∀v∈Xh.
And set ukh=u/∥u∥Xγ.Step 3.
Compute the Rayleigh quotient(3.6)λkh=a(ukh,ukh)b(ukh,ukh).
We use(λkh,ukh) as the approximate eigenpair of (2.1).Theorem 3.3.
Suppose thatH is properly small. Let (λkh,ukh) be the approximate eigenpair obtained by Scheme 1. Then there exists eigenpair (λk,uk) of (2.1), such that
(3.7)∥ukh-uk∥Xγ≤48ρC1C2C4C5δH3(λk)+3C2qδh(λk),(3.8)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
where C4, C5, and C6 are positive constants independent of mesh diameters, and these constants are decided by (3.11), (3.13), and (3.30) in the following proof.Proof.
We use Lemma3.2 to complete the proof. Select μ0=1/λk,H and u0 = λk,HThuk,H/∥λk,HThuk,H∥Xγ. Obviously,
(3.9)∥u0∥Xγ=1.
Noting that λk=λk+1=⋯λk+q-1, for j=k,k+1,…,k+q-1, we have
(3.10)|μj,h-μk,h|=|λk,h-λj,hλk,hλj,h|=|λk,h-λk+λj-λj,hλk,hλj,h|≤|λk,h-λk|+|λj-λj,h|λk,hλj,h,(3.11)|μ0-μj,h|=|λj,h-λk,Hλj,hλk,H|=|λj,h-λj+λk-λk,Hλj,hλk,H|≤C4|λk,H-λk|.
Combining the above two inequalities with (2.25) and noting that δh(λk) is a small quantity of higher order than δH(λk), we obtain
(3.12)maxk≤j≤k+q-1|μj,h-μk,hμ0-μj,h|≤12.
From Lemma 2.1, we know that ∥Th-T∥Xγ→0(h→0), then there exists a constant C5 independent of h, such that
(3.13)∥Thf∥Xγ≤C5∥f∥Xγ,∀f∈Xh.
Obviously, there exists uk∈M^(λk), such that
(3.14)∥uk,H-uk∥Xγ=dist(uk,H,M^(λk)).
Then, we derive
(3.15)∥λk,HThuk,H-uk∥Xγ=∥λk,HThuk,H-λkTuk∥Xγ≤∥(λk,H-λk)Thuk,H∥Xγ+∥λkTh(uk,H-uk)∥Xγ+∥λk(Th-T)uk∥Xγ≤C5|λk,H-λk|+C5λk∥uk,H-uk∥Xγ+∥(Ph-I)uk∥Xγ≤C5|λk,H-λk|+C5λk∥uk,H-uk∥Xγ+δh(λk).
Hence, by Lemma 3.1, (3.15) and (2.28), we have
(3.16)dist(u0,M^(λk))≤∥u0-uk∥Xγ≤2∥λk,HThuk,H-uk∥Xγ≤2C5|λk,H-λk|+2C5λk∥uk,H-uk∥Xγ+2δh(λk)≤3C5λk∥uk,H-uk∥Xγ+2δh(λk)=3C5λkdist(uk,H,M^(λk))+2δh(λk).
Combining the triangle inequality, (2.27) and (3.16), we deduce
(3.17)dist(u0,Mh(λk))≤dist(u0,M^(λk))+C3δh(λk)≤3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk).
Since H is small enough and δh(λk)=δH(λk)t2, from (2.26) and (3.17), we know
(3.18)dist(u0,Mh(λk))≤12.
For j≠k,  k+1,…,k+q-1, since H is small enough, ρ is the separation constant, we have
(3.19)|μ0-μj,h|≥ρ2.
From the Step 2 in Scheme 1 and (2.14), we get
(3.20)a(u,v)-λk,Ha(Thu,v)=a(Thuk,H,v),
namely,
(3.21)u-λk,HThu=Thuk,H.
Thus (1/λk,H)u-Thu=(1/λk,H)Thuk,H and ukh=u/∥u∥Xγ. Note that (1/λk,H)Thuk,H = ∥(1/λk,H)Thuk,H∥Xγu0 differs from u0 by only a constant; then Step 2 is equivalent to
(3.22)(1λk,H-Th)u=u0,ukh=u∥u∥Xγ.
From the arguments of (3.9), (3.12), (3.18), (3.19), and (3.22), we see that the conditions of Lemma 3.2 hold. Hence, substituting (3.11) and (3.17) into (3.3), we obtain
(3.23)dist(ukh,M^h(λk))≤16ρC4|λk,H-λk|(3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk)).
Let eigenvectors {uj,h}kk+q-1 be an orthogonal basis of Mh(λk) (in the sense of inner product a(·,·)), then
(3.24)dist(ukh,Mh(λk))=∥ukh-∑j=kk+q-1‍a(ukh,uj,h)uj,h∥Xγ.
Set
(3.25)u*=∑j=kk+q-1‍a(ukh,uj,h)uj,h.
From (3.23), we directly get
(3.26)∥ukh-u*∥Xγ=dist(ukh,Mh(λk))≤16ρC4|λk,H-λk|(3C5λkdist(uk,H,M^(λk))+(C3+2)δh(λk)).
From Lemma 2.2, we know that there exist {uj0}kk+q-1⊂M(λk) making uj,h-uj0 satisfy (2.26). Let uk=∑j=kk+q-1a(ukh,uj,h)uj0, then
(3.27)∥u*-uk∥Xγ=∥∑j=kk+q-1‍a(ukh,uj,h)(uj,h-uj0)∥Xγ≤(∑j=kk+q-1‍∥uj,h-uj0∥Xγ2)1/2≤C2∑j=kk+q-1‍δh(λj)≤C2qδh(λk).
Combining (3.26) and (3.27), we obtain
(3.28)∥ukh-uk∥Xγ≤∥ukh-u*∥Xγ+∥u*-uk∥Xγ≤48ρC4C5λk|λk,H-λk|dist(uk,H,M^(λk))+3C2qδh(λk).
Besides, by (2.26), we easily know
(3.29)dist(uk,H,M^(λk))≤C2δH(λk),
which together with (3.28) and (2.25) leads to (3.7).
From the continuous embedding ofXγ into L2(Ω)n, we get that there exists a constant C6 independent of meshes, such that
(3.30)∥u∥02≤C6∥u∥Xγ2.
Equation (3.7) indicates that ukh converges to uk in the sense of norm ∥·∥Xγ, then ukh converges to uk in the sense to norm ∥·∥0; thus, 1/∥ukh∥02→1/∥uk∥02=λk. Therefore, when h is small enough, we have
(3.31)|λkh-λk|≤∥ukh-uk∥Xγ2∥ukh∥02+λk∥ukh-uk∥02∥ukh∥02≤2λk∥ukh-uk∥Xγ2+2λk2∥ukh-uk∥02≤2λk(1+C6λk)∥ukh-uk∥Xγ2.
The proof of Theorem 3.3 is completed.LetσΔN be the smallest singular exponent in the Laplace problem with homogenous Neumann boundary condition, then σΔN∈(3/2,2). Denote τ=min(γ-γmin,σΔN-1).Corollary 3.4.
Suppose thatH is properly small. Let (λkh,ukh) be an approximate eigenpair obtained by Scheme 1. Then there exists an eigenpair (λk,uk) of (2.1), such that when Ω is a convex domain,
(3.32)∥ukh-uk∥Xγ≤6qC2C′h,(3.33)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
when Ω is a nonconvex domain,
(3.34)∥ukh-uk∥Xγ≤6qC2C′′hμ,∀μ∈(0,τ),(3.35)|λkh-λk|≤2λk(1+C6λk)∥ukh-uk∥Xγ2,
where C′ and C′′ are stated in the proof as follows.Proof.
From [1, 4], we know that when Ω is a convex domain, there exists a constant C′ independent of h, such that
(3.36)δh(λk)≤C′h.
Substituting the above inequality into (3.7), and noting that δh(λk) is an infinitesimal of lower order comparing with δH(λk)3, we know that (3.32) is valid.
And whenΩ is a nonconvex domain, there exists a constant C′′ independent of h, such that
(3.37)δh(λk)≤C′′hμ,
where μ∈(0,τ).
Substituting the above inequality into (3.7), we know that (3.34) is valid.


## 3.2. Multigrid Discretization Scheme

Next, we will discuss finite-element multigrid discretization scheme based on Rayleigh quotient iteration method. Assume that partition satisfies the following condition.Condition (A).
{
π
h
i
}
1
l is a family of regular meshes (see [3]) with diameters hi, δhi+1(λk)=δhi(λk)ti+1, ti+1∈[1+δ,3-δ], i=1,2,…, and δ is a properly small positive number.Let{Xhi}1l be the finite-element spaces defined on {πhi}1l. Further, let πH=πh1, XH=Xh1.Scheme 2.
Multigrid Discretization.Step 1.
Solve (2.8) on a coarse grid πH: Find (λH,uH)∈R+×XH, with ∥uH∥Xγ=1, such that
(3.38)a(uH,v)=λHb(uH,v),∀v∈XH.
And obtain the “real” eigenpair (λk,H,uk,H) by filtering process.Step 2.
u
k
h
1
=
u
k
,
H,  λkh1=λk,H,  i⇐2.Step 3.
Solve a linear system on a fine gridπhi: Find u∈Xhi, such that
(3.39)a(u,v)-λkhi-1b(u,v)=b(ukhi-1,v),∀v∈Xhi.
Set ukhi=u/∥u∥Xγ.Step 4.
Compute the Rayleigh quotient(3.40)λkhi=a(ukhi,ukhi)b(ukhi,ukhi),Step 5.
Ifi=l, then output (λkhl,ukhl), stop. Else, i⇐i+1, and return to Step 3.We use(λkhl,ukhl) obtained by Scheme 2 as the approximate eigenpair of (2.1). Next, we will discuss the efficiency of Scheme 2.Theorem 3.5.
Suppose thatH is properly small and Condition (A) holds. Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair (λk,uk) of (2.1), such that
(3.41)∥ukhl-uk∥Xγ≤48ρC4C5λk|λkhl-1-λk|dist(ukhl-1,M^(λk))+3C2qδhl(λk),(3.42)|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ,l≥2.Proof.
We use induction to complete the proof of (3.41).
Forl=2, Scheme 2 is actually Scheme 1. Hence, (3.41) is easily obtained from (3.28).
Suppose that (3.41) holds for l=3,4,…,l-1. Next, we shall prove that (3.41) holds for l.
Selectμ0=1/λkhl-1, μk,hl=1/λk,hl, and u0=λkhl-1Thlukhl-1/∥λkhl-1Thlukhl-1∥Xγ. Using the proof method of Theorem 3.3, we deduce
(3.43)|μ0-μk,hl|≤C4|λkhl-1-λk|,maxk≤j≤k+q-1|μj,hl-μk,hlμ0-μj,hl|≤12.
Using the triangle inequality and (2.27), we get
(3.44)dist(u0,Mhl(λk))≤dist(u0,M^(λk))+C3δhl(λk),
and together with the induction assumption, yields
(3.45)dist(u0,Mhl(λk))≤12.
From Step 3 of Scheme 2, we know that ukhl satisfies
(3.46)(1λkhl-1-Th)u=u0,ukhl=u∥u∥Xγ.
From the above arguments, we know that the conditions of Lemma 3.2 hold.
Defineu* and uk as those in Theorem 3.3 (using ukhl instead of ukh, uj,hl instead of uj,h), then
(3.47)u*=∑j=kk+q-1‍a(ukhl,uj,hl)uj,hl,uk=∑j=kk+q-1‍a(ukhl,uj,hl)uj0,
where uj,hl-uj0 satisfies (2.26). We can derive by Lemma 3.2 and the proof of (3.11) that
(3.48)∥ukhl-u*∥Xγ=dist(ukhl,Mhl(λk))≤16ρ|μ0-μk,hl|dist(u0,Mhl(λk))≤16ρC4|λkhl-1-λk|dist(u0,Mhl(λk)).
Substituting (3.44) into the above inequality, we deduce
(3.49)∥ukhl-u*∥Xγ≤16ρC4|λkhl-1-λk|(dist(u0,M^(λk))+C3δhl(λk)).
Like the proof method of (3.27), we get
(3.50)∥u*-uk∥Xγ≤C2qδhl(λk).
From the above two inequalities, we obtain
(3.51)∥ukhl-uk∥Xγ≤16ρC4|λkhl-1-λk|dist(u0,M^(λk))+2C2qδhl(λk).
There exists a constant C5 independent of hl such that
(3.52)∥Thlf∥Xγ≤C5∥f∥Xγ,∀f∈Xhl.
Like the proof method of (3.16), we can derive
(3.53)dist(u0,M^(λk))≤3C5λkdist(ukhl-1,M^(λk))+2δhl(λk).
Combining (3.51) and (3.53), we know that (3.41) is valid. Like the proof method of (3.8), we get (3.42), namely, Theorem 3.5 is valid.Corollary 3.6.
Suppose that Condition (A) holds andh1 (namely, H) is properly small. Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 2. Then there exists an eigenpair (λk,uk) of (2.1), such that the following error estimates hold: when Ω is a convex domain,
(3.54)∥ukhl-uk∥Xγ≤6qC2C′hl,|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ2;
when Ω is a nonconvex domain,
(3.55)∥ukhl-uk∥Xγ≤6qC2C′′hlμ,∀μ∈(0,τ),|λkhl-λk|≤2λk(1+C6λk)∥ukhl-uk∥Xγ2,
where the C′ and C′′ are the ones in Corollary 3.4.


## 4. Interpolation Postprocessing Technique

In this section, we apply interpolation postprocessing technique to the filter approach for eigenvalue problem of electric field.Letπ2h be a regular simplex mesh of Ω. When n=2, the mesh πh is obtained by dividing each element of the mesh π2h into four congruent triangular elements; when n=3, the mesh πh is obtained by connecting the midpoints on each edge of the tetrahedral element, which divides each element of tetrahedralization π2h into eight tetrahedral elements.LetIh:C0(Ω¯)n→Xh with k=1 be a piecewise linear node interpolation operator on πh. Let I2h(2):C0(Ω¯)n→X2h with k=2 be a piecewise quadratic node interpolation operator on π2h by using the corners of the mesh πh as interpolation nodes.Scheme 3.
Interpolation Postprocessing Technique.Step 1.
Use linear finite-element filter approach to solve the problem (2.1) on the mesh πh, and obtain the “real” eigenpair (λk,h,uk,h).Step 2.
Onπ2h, use the value of the function uk,h on the corners of the mesh πh as interpolation conditions to construct a piecewise quadratic interpolation I2h(2)uk,h.Step 3.
Compute the Rayleigh quotient:(4.1)λk,hr=a(I2h(2)uk,h,I2h(2)uk,h)b(I2h(2)uk,h,I2h(2)uk,h).
Here,(λk,hr,I2h(2)uk,h) is the eigenpair corrected.We develop the work in [18] to get the following theorem.Theorem 4.1.
Let(λk,hr,I2h(2)uk,h) be an approximate eigenpair obtained by Scheme 3. Assume that M(λk)⊂H2+α(Ω) and there exists an uk∈M(λk) such that ∥Ihuk-uk,h∥Xγ≤Ch1+α, for some α>0. Then
(4.2)∥I2h(2)uk,h-uk∥Xγ≤Ch1+α,(4.3)|λk,hr-λk|≤Ch2+2α.Proof.
From boundedness of interpolationI2h(2), we have
(4.4)∥I2h(2)uk-I2h(2)uk,h∥Xγ=∥I2h(2)Ihuk-I2h(2)uk,h∥Xγ≤C∥Ihuk-uk,h∥Xγ≤Ch1+α,
by triangle inequality and interpolation error estimate, we get
(4.5)∥I2h(2)uk,h-uk∥Xγ≤∥I2h(2)uk,h-I2h(2)uk∥Xγ+∥I2h(2)uk-uk∥Xγ≤Ch1+α,
namely, (4.2) is valid. Combining (2.28) and (4.2), we know that (4.3) is valid.Remark 4.2.
Generally, to 2nd-order elliptic eigenvalue problems, condition∥Ihuk-uk,h∥H1(Ω)≤Ch1+α is valid (see [18–25]). But to eigenvalue problems of electric field, it is very difficult to prove that ∥Ihuk-uk,h∥Xγ≤Ch1+α. In Section 5, we will verify this theorem by the numerical experiments.


## 5. Numerical Experiments

In this section, we consider numerical solutions of problem (2.1) on the L-shaped domain [-1,0]×[-1,0]∪[-1,1]×[0,1] and on the square domain [0,π]×[0,π]. The smallest five exact eigenvalues are λ1≈1.47562182408,λ2≈3.53403136678,λ3=λ4≈9.86960440109,λ5≈11.3894793979, and λ1 = λ2 = 1, λ3 = 2, λ4=λ5=4, respectively.We adopt a uniform isosceles right triangulation forΩ (the edge in each element is along three fixed directions, see Figures 1 and 2) to produce the meshes πhl with mesh diameter hl.Figure 1
The L-shaped domainΩ.Figure 2
The square domainΩ.Here the weight isω=dγ(d=x12+x22). In the numerical experiments, when Ω is the L-shaped domain, let γ=0.5 or 0.95; when Ω is the square domain, we choose γ=0. And we use the numerical integral formula with accuracy of order 2 in our experiments.From the following tables, we know that these three schemes are reliable for solving Maxwell eigenvalue problems. In addition, the accuracy of solutions is improved highly by these schemes.Example 5.1.
Solve problem (2.1) on the L-shaped domain [-1,0]×[-1,0]∪[-1,1]×[0,1] by using Scheme 1 with quadratic finite element. The eigenvalues obtained by Scheme 1 can be seen in [27].Example 5.2.
Solve problem (2.1) on the square domain [0,π]×[0,π] and the L-shaped domain [-1,0]×[-1,0]∪[-1,1]×[0,1] by using Scheme 2 with quadratic finite element.We compute the first five approximate eigenvalues by using Scheme2. The numerical results are listed in Tables 1–3; here λkh,(k=1,2,…,5) denote the first five “real” eigenvalues obtained by Scheme 2.Table 1
The results on square by Scheme2 for eigenvalue problem of electric field (γ=0): Set h1=H=2/8,h2=h1/4,hi=hi-1/2,i=3,4,….
l
λ
1
h
l
λ
3
h
l
λ
4
h
l
2
1.000000128059
2.000001795837
4.000008182019
3
1.000000008035
2.000000112617
4.000000514003
4
1.000000000503
2.000000007048
4.000000032195
5
1.000000000033
2.000000000442
4.000000002014Table 2
The results on L-shape domain by Scheme2 for eigenvalue problem of electric field (γ=0.5): Set h1=H=2/8,h2=h1/4,hi=hi-1/2,i=3,4,….
l
λ
1
h
l
λ
2
h
l
λ
3
h
l
λ
5
h
l
2
1.821670160961
3.541016446362
9.869624525814
11.392623917025
3
1.747568405543
3.536558996225
9.869605667282
11.390593949705
4
1.690869606785
3.534929019047
9.869604480461
11.389874997804
5
1.646185577455
3.534344665373
9.869604406059
11.389617861812Table 3
The results on L-shape domain by Scheme2 for eigenvalue problem of electric field (γ=0.95): Set h1=H=2/8,h2=h1/4,hi=hi-1/2,i=3,4,….
l
λ
1
h
l
λ
2
h
l
λ
3
h
l
λ
5
h
l
2
1.442172956105
3.534742382807
9.869624451909
11.390286991090
3
1.424394098212
3.534137592831
9.869605664957
11.389497235196
4
1.437944305790
3.534047685310
9.869604480377
11.389493626284
5
1.473860172568
3.534033892611
9.869604406055
11.389480149393From Tables1–3, we see that Scheme 2 is highly efficient for solving eigenvalue problem of electric field.Example 5.3.
Solve problem (2.1) on the square domain [0,π]×[0,π] and the L-shaped domain [-1,0]×[-1,0]∪[-1,1]×[0,1] by using Scheme 3.In Tables4–6, λk,h(k=1,2,…,5) denote the first five “real” eigenvalues obtained by linear element filter approach directly, λk,hr(k=1,2,…,5) denote the first five “real” eigenvalues obtained by Scheme 3, λk,2h(2)(k=1,2,…,5) denote the eigenvalues obtained by quadratic element filter approach directly.Table 4
The results on square by Scheme3 for eigenvalue problem of electric field (γ=0).
k
h
λ
k
,
h
λ
k
,
h
r
λ
k
,
2
h
(
2
)
1
2
/
32
1.000803263068
1.000002233535
1.000002031741
1
2
/
64
1.000200802028
1.000000140453
1.000000128059
1
2
/
128
1.000050199677
1.000000008794
1.000000008035
1
2
/
256
1.000012549868
1.000000000550
1.000000000503
3
2
/
32
2.004819363681
2.000031493140
2.000028466776
3
2
/
64
2.001204798563
2.000001982275
2.000001795837
3
2
/
128
2.000301197210
2.000000124127
2.000000112617
3
2
/
256
2.000075299154
2.000000007762
2.000000007048
4
2
/
32
4.012861235873
4.000136435109
4.000129172679
4
2
/
64
4.003213391903
4.000008625833
4.000008182018
4
2
/
128
4.000803229708
4.000000540798
4.000000514003
4
2
/
256
4.000200800066
4.000000033828
4.000000032195Table 5
The results on L-shape domain by Scheme3 for eigenvalue problem of electric field (γ=0.5).
k
h
λ
k
,
h
λ
k
,
h
r
λ
k
,
2
h
(
2
)
1
2
/
32
2.011577759579
1.943699371956
1.922638950089
1
2
/
64
1.898872106646
1.846871485478
1.821288431396
1
2
/
128
1.820217989064
1.778546113807
1.747565968425
1
2
/
256
1.761155941126
1.726985873864
1.690868193292
2
2
/
32
3.572934947967
3.554733448789
3.552868211051
2
2
/
64
3.550750124217
3.542510419003
3.541016085614
2
2
/
128
3.541430954385
3.537608542327
3.536558996172
2
2
/
256
3.537364244380
3.535586886107
3.534929019046
3
2
/
32
9.901333191694
9.869943232779
9.869921180772
3
2
/
64
9.877532690720
9.869626032863
9.869624525812
3
2
/
128
9.871586259667
9.869605763991
9.869605667282
3
2
/
256
9.870099853293
9.869604486559
9.869604480461
5
2
/
32
11.460149814384
11.399622345282
11.398934132563
5
2
/
64
11.410106013972
11.393258438593
11.392624308452
5
2
/
128
11.396038772982
11.391051676895
11.390593949719
5
2
/
256
11.391771543060
11.390164853843
11.389874997804Table 6
The results on L-shape domain by Scheme3 for eigenvalue problem of electric field (γ=0.95).
k
h
λ
k
,
h
λ
k
,
h
r
λ
k
,
2
h
(
2
)
1
2
/
32
1.508480704616
1.465444529094
1.450478468682
1
2
/
64
1.466277799473
1.448244634750
1.442139772480
1
2
/
128
1.448544881396
1.440581165707
1.424272599464
1
2
/
256
1.439836736426
1.434823427838
1.490040481056
2
2
/
32
3.540250087930
3.528411456749
3.543755446693
2
2
/
64
3.514437651295
3.458119260291
3.534742392880
2
2
/
128
3.538194948635
3.533617471424
3.534137592922
2
2
/
256
3.535161265275
3.534215865622
3.534047685310
3
2
/
32
9.901333191694
9.869943232779
9.869918238162
3
2
/
64
9.877532690720
9.869626032863
9.869624447221
3
2
/
128
9.871586259667
9.869605763991
9.869605663393
3
2
/
256
9.870099853293
9.869604486559
9.869604480222
5
2
/
32
11.460149814384
11.399622345282
11.390599176482
5
2
/
64
11.410106013972
11.393258438593
11.390317414966
5
2
/
128
11.396038772982
11.391051676895
11.389497235201
5
2
/
256
11.391771543060
11.390164853843
11.389493626284To the square domain, eigenfunctions are smooth enough. And from Table4, we see that λk,hr obtained by interpolation postprocessing technique achieve the accuracy order of quadratic finite element. To the L-shape domain, eigenfunctions are not smooth enough, generally. For example, the first eigenfunction has a strong singularity to L-shape domain (see [28]). Tables 5 and 6 show that the accuracy of λ3,hr is improved obviously, and the improvement of λ1,hr is not obvious.Remark 5.4.
Wang established two-grid discretization scheme of finite-element parameterized approach for eigenvalue problem of electric field (see [29]). And she also proved error estimates of the Scheme. It will still be meaningful to extend the multigrid discretization scheme and the interpolation postprocessing technique discussed in our paper to parameterized approach.



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*Source: 529498-2012-12-26.xml*