# A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems

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

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## Abstract

This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine gridKh is reduced to the solution of an eigenvalue problem on a much coarser grid KH and the solution of a linear algebraic system on the fine grid Kh. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking H=O(h4), and when using the Pk+1-Pk element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking H=O(h3). Finally, numerical experiments are presented to support the theoretical analysis.


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## Body

## 1. Introduction

To improve the efficiency of finite element method, Xu introduced a two-scale discretization scheme and applied it to nonsymmetric and nonlinear elliptic problems (see [1–3]). Later on, this scheme attracted the attention of academic circles and has been successfully applied to Stokes equations (see [4–7]), semilinear eigenvalue problems (see [8]), and linear differential operator eigenvalue problems, and so forth.Up to now, two kinds of finite element two-scale discretization schemes have been developed for linear differential operator eigenvalue problems. The first kind is established by Xu and Zhou [9] in 2001, whose idea correlates with the iterative Galerkin method which was established by Lin and Xie [10] and Sloan [11], but it bases on the finite element spaces on two different scale grids. The scheme of Xu and Zhou has been applied to the conforming finite element method for electric structure eigenvalue problem (see [12–14]), the conforming/nonconforming finite element method for non-self-adjoint eigenvalue problem (see [15, 16]), the conforming/nonconforming finite element method for Steklov eigenvalue problem (see [17, 18]), the mixed finite element method for Stokes eigenvalue problem, and biharmonic equations eigenvalue problem (see [19, 20]). Another group of two-scale discretization scheme is proposed by Yang and Bi [21]. They established the conforming/nonconforming finite element two-scale discretization scheme based on the shifted-inverse power method.With two-scale discretization schemes, the solution of an eigenvalue problem on a fine gridKh is reduced to the solution of an eigenvalue problem on a much coarser grid KH and the solution of a linear algebraic system on the fine grid Kh, and the resulting solution still maintains an asymptotically optimal accuracy. Thus, the computational efficiency is improved.Influenced by the work mentioned above, in this paper we establish a new finite element two-scale discretization scheme for mixed variational formulation of eigenvalue problem and apply it to Stokes eigenvalue problem and eigenvalue problem of electric field. The research of this paper has the following features.(1)
Our two-scale discretization scheme is a combinative production of the mixed finite element method and the shifted-inverse power method (see [22, Algorithm 27.2]). Comparing with the scheme in [19, 20], the scheme in this paper is more efficient: the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking H=O(h4) when solving Stokes eigenvalue problem and H=O(h3) when solving eigenvalue problem of electric field; however, with the scheme in [19, 20] the resulted solution maintains an asymptotically optimal accuracy by taking H=O(h2).(2)
The literatures of high-efficient numerical method for eigenvalue problem of electric field are not too many by now, thus they seem to be very valuable. Our two-scale discretization scheme is a new and highly efficient method for eigenvalue problem of electric field.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 eigenvalue problem mixed variational formulation (2.3)-(2.4) in general form, the finite element two-scale discretization scheme based on the shifted-inverse power method is established, and the validity of this scheme is proved theoretically. In Sections 4 and 5, the scheme established in Section 3 is applied to Stokes eigenvalue problem and eigenvalue problem of electric field, respectively. Finally, numerical experiments are presented in Section 6.


## 2. Preliminaries

LetV, W, and D be three real Hilbert spaces with inner products and norms (·,·)V, ∥·∥V, (·,·)W, ∥·∥W, and (·,·)D, ∥·∥D, respectively. We 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.In scientific and engineering computations, many eigenvalue problems for differential equation have the following 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,(2.8)b(w,v)=0,∀v∈W.Givenf∈D, find (wh,ph)∈Vh×Wh satisfying
(2.9)a(wh,ψ)+b(ψ,ph)=(f,ψ)D,∀ψ∈Vh,(2.10)b(wh,v)=0,∀v∈Wh.As for the mixed finite element method for boundary value problems, Brezzi and Babuska, and others have established a systematic theory. Denote(2.11)V0={u∈V:b(u,v)=0,∀v∈W},Vh0={u∈Vh:b(u,v)=0,∀v∈Wh}.Theorem 2.1 (Brezzi-Babuska Theorem).
Suppose that(1)
a(·,·) and b(·,·) are continuous bilinear forms, that is,
(2.12)|a(q,ψ)|≤M1‖q‖V‖ψ‖V,∀q,ψ∈V,|b(ψ,v)|≤M2‖ψ‖V‖v‖W,∀ψ∈V,v∈W;(2)
there existsν1>0, such that
(2.13)a(q,q)≥ν1‖q‖V2,∀0≠q∈V0;(3)
inf-sup condition: there existsν2>0, such that
(2.14)sup⁡ψ∈V,ψ≠0b(ψ,v)‖ψ‖V≥ν2‖v‖W,∀v∈W;
then there exists a unique solution to the problem (2.7)-(2.8), and
(2.15)‖w‖V+‖p‖W≤C‖f‖D,where constantC just depends on ν1,ν2 and M1,M2. Furthermore, suppose that(4)
there exists a constantμ~1>0, such that
(2.16)a(q,q)≥μ~1‖q‖V2,∀0≠q∈Vh0;(5)
discrete inf-sup condition: there exists a constantμ~2>0, such that
(2.17)sup⁡ψ∈Vh,ψ≠0b(ψ,v)‖ψ‖V≥μ~2‖v‖W,∀v∈Wh.Then there exists a unique solution (wh,ph) to the problem (2.9)-(2.10); moreover, the following error estimate is valid:
(2.18)‖w-wh‖V+‖p-ph‖W≤C{inf⁡q∈Vh‖w-q‖V+inf⁡v∈Wh‖p-v‖W},
where C just depends on μ~1,μ~2 and M1,M2.Sincea(·,·) is a symmetric, continuous, and V-elliptic bilinear form on V×V, b(·,·) is a continuous bilinear form, then conditions (1), (2), and (4) of Brezzi-Babuska Theorem hold. Suppose inf-sup condition and discrete inf-sup condition hold. Then by Brezzi-Babuska Theorem, we know that (2.7)-(2.8) and (2.9)-(2.10) are uniquely solvable for each f∈D. Thus we can define the corresponding linear bounded operators:T
:
D
→
V
,
S
:
D
→
W
: for all f∈D(2.19)a(Tf,ψ)+b(ψ,Sf)=(f,ψ)D,∀ψ∈V,(2.20)b(Tf,v)=0,∀v∈W,Th:D→Vh⊂V,Sh:D→Wh⊂W: for all f∈D,
(2.21)a(Thf,ψ)+b(ψ,Shf)=(f,ψ)D,∀ψ∈Vh,(2.22)b(Thf,v)=0,∀v∈Wh.
It is obvious that (2.3)-(2.4) have an equivalent operator form:
(2.23)λTu=u,(2.24)σ=S(λu).
Equations (2.5)-(2.6) have an equivalent operator form:
(2.25)λhThuh=uh,(2.26)σh=Sh(λhuh).
It is easy to prove that T:D→D,Th:D→D are self-adjoint operators. In fact, for all f,g∈D, taking ψ=Tg,v=Sg in (2.7)-(2.8) we obtain
(2.27)a(Tf,Tg)+b(Tg,Sf)=(f,Tg)D,b(Tf,Sg)=0.
Exchanging f and g, we get
(2.28)a(Tg,Tf)+b(Tf,Sg)=(g,Tf)D,(2.29)b(Tg,Sf)=0,
then
(2.30)(f,Tg)D=a(Tf,Tg)+b(Tg,Sf)+b(Tf,Sg)=a(Tg,Tf)+b(Tf,Sg)+b(Tg,Sf)=(g,Tf)D+0=(Tf,g)D.
It shows that T:D→D is self-adjoint in the sense of inner product (·,·)D. Analogously, it can be proved that Th:D→D is self-adjoint in the sense of inner product (·,·)D.Assume thatV↪D (compact imbedded), then it is easy to prove that the operator 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
(2.31)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
(2.32)0<λ1≤λ2≤⋯≤λk≤⋯↗+∞,
and the corresponding eigenfunctions are
(2.33)u1,σ1,u2,σ2,…,uk,σk,…,
where (ui,uj)D=δij.The eigenvalues of (2.5)-(2.6) can be sorted as
(2.34)0<λ1,h≤λ2,h≤⋯≤λk,h≤⋯≤λNh,h,
and the corresponding eigenfunctions are
(2.35)u1,h,σ1,h,u2,h,σ2,h,…,uk,h,σk,h,…,uNh,h,σNh,h,
where (ui,h,uj,h)D=δij.It is obvious thata(·,·) is an inner product on V,∥·∥a=a(·,·) and ∥·∥V are equivalent norms. Let uh=ui,h in (2.5); then
(2.36)a(ui,h,uj,h)+b(uj,h,σh)=λi,h(ui,h,uj,h)D.
From (2.6), we get b(uj,h,σh)=0. Then
(2.37)a(ui,h,uj,h)=λi,h(ui,h,uj,h)D=λi,hδij;
therefore, {ui,h/∥ui,h∥a} is a completely normal eigenvector system on Vh in the sense of 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. Let μ 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 the kth eigenvalue, too. Denote M(λ)=M(μ), Mh(λ)=Mh(μ), and M^(λ)=M^(μ). Define
(2.38)‖(T-Th)|M(λ)‖D=max⁡u∈M(λ),u≠0‖(T-Th)u‖D‖u‖D,‖(T-Th)|M(λ)‖a=max⁡u∈M(λ),u≠0‖(T-Th)u‖a‖u‖a.The convergence and error estimate about mixed element method of eigenvalue problem have been studied by [23–25]. From these literatures we easily know that the following results are valid.Lemma 2.2.
Suppose that the conditions of Brezzi-Babuska Theorem hold, and∥T-Th∥D→0(h→0). Let (λh,uh,σh) be the kth eigenpair of (2.5)-(2.6), ∥uh∥a=1, and λ  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
(2.39)|λh-λ|+‖uh-u‖D≤C‖(T-Th)|M(λ)‖D,(2.40)‖σ-σh‖W≤‖Sh(λu)-S(λu)‖W+C‖(T-Th)|M(λ)‖D,(2.41)‖u-uh‖a≤C‖(Th-T)|M(λ)‖a.
Let u∈M^(λ); then there exists uh∈Mh(λ) such that
(2.42)‖u-uh‖a≤C‖(Th-T)|M(λ)‖a.Proof.
From the spectral approximation theory (see [23]) we have (2.39).
Letu satisfy (2.39), and σ=S(λu). Next we will prove that this eigenpair satisfies (2.40)-(2.41). From Brezzi-Babuska Theorem and (2.39), we get
(2.43)‖S(λu-λhuh)‖W≤C‖λu-λhuh‖D≤C‖(T-Th)|M(λ)‖D,(2.44)‖T(λu-λhuh)‖V≤C‖λu-λhuh‖D≤C‖(T-Th)|M(λ)‖D.
Using the triangle inequality and (2.43), we deduce
(2.45)‖σ-σh‖W-‖Sh(λu)-S(λu)‖W=|‖S(λu)-Sh(λhuh)‖W-‖Sh(λu)-S(λu)‖W|≤‖Sh(λu-λhuh)‖W≤C‖(T-Th)|M(λ)‖D,
that is, (2.40) is valid.
Using the triangle inequality and (2.44), we get
(2.46)‖u-uh‖a-‖Th(λu)-T(λu)‖a=|‖T(λu)-Th(λhuh)‖a-‖Th(λu)-T(λu)‖a|≤‖Th(λu-λhuh)‖a≤C‖(T-Th)|M(λ)‖D≤C‖(T-Th)|M(λ)‖a,
which together with ∥Th(λu)-T(λu)∥a≤C∥(T-Th)|M(λ)∥a yields (2.41).
Let the eigenfunctions{uj,h} be an orthonormal system of Mh(λ) in the sense of inner product a(·,·). Then, from (2.41) and Lemma 3.1 we know that there exists a basis {uj0} of M(λ) satisfying ∥uj0∥a=1 and the following result is valid:
(2.47)‖uj0-uj,h‖a≤C‖(Th-T)|M(λ)‖a.
For any u∈M^(λ), we write u=∑j=kk+q-1‍αjuj0. By calculation, we get
(2.48)1=‖u‖a2=∑j=kk+q-1‍αj2+∑i≠j,i,j=kk+q-1‍αiαja(ui0,uj0).
From (2.47), when i≠j we have a(ui0,uj0)=a(ui0,uj0)-a(ui,h,uj,h)→0(h→0), and thus we get ∑j=kk+q-1‍αj2→1(h→0).
Denoteuh=∑j=kk+q-1‍αjuj,h, then uh∈Mh(λ). From (2.47), we deduce
(2.49)‖u-uh‖a≤C∑j=kk+q-1‍‖uj,h-uj0‖a≤C‖(Th-T)|M(λ)‖a,
that is, (2.42) is valid. The proof is completed.For(u*,σ*)∈V×W,u*≠0, define the Rayleigh quotient
(2.50)λr=a(u*,u*)+2b(u*,σ*)(u*,u*)D.The following lemma is an extension of [23, Lemma 9.1].Lemma 2.3.
Let(λ,u,σ) be an eigenpair of (2.3)-(2.4); then for all (u*,σ*)∈V×W, (u*,σ*)≠(0,0) with its Rayleigh quotient satisfying
(2.51)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.
From (2.3)-(2.4), we deduce
(2.52)a(u*-u,u*-u)+2b(u*-u,σ*-σ)-λ(u*-u,u*-u)D=a(u*,u*)-2a(u*,u)+a(u,u)+2b(u*,σ*)-2b(u*,σ)-2b(u,σ*)+2b(u,σ)-λ(u*,u*)D+2λ(u*,u)D-λ(u,u)D.=a(u*,u*)+2b(u*,σ*)-λ(u*,u*)D-2(a(u*,u)+b(u*,σ)-λ(u*,u)D)-2b(u,σ*)+(a(u,u)+b(u,σ)-λ(u,u)D)+b(u,σ).=a(u*,u*)+2b(u*,σ*)-λ(u*,u*)D-0-0+0+0.
By dividing by (u*,u*)D on both sides of the above identity, we obtain (2.51).Taking(u*,σ*)=(uh,σh) in (2.51) and using (2.4) and (2.6), we deriveLemma 2.4.
Let(λ,u,σ) and (λh,uh,σh) be the kth eigenpair of (2.3)-(2.4) and (2.5)-(2.6), respectively; then
(2.53)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,(2.54)|λh-λ|≤C(‖uh-u‖a2+‖uh-u‖a‖σ-v‖W),∀v∈Wh.


## 3. The Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems

This paper establishes the following finite element two-scale discretization scheme based on the shifted-inverse power method.Scheme 1.
One has the following.
Step1. Solve the eigenvalue problem (2.3)-(2.4) on a coarse grid KH: 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.
Step2. Solve a equation on a fine grid Kh: find (u′,σ′)∈Vh×Wh such that
(3.2)a(u′,ψ)+b(ψ,σ′)-λH(u′,ψ)D=(uH,ψ)D,∀ψ∈Vh,b(u′,v)=0,∀v∈Wh.
Set uh=u′/∥u′∥a,σh=σ′/∥u′∥a.
Step3. Compute the Rayleigh quotient
(3.3)λh=a(uh,uh)(uh,uh)D.Next we will discuss the validity of Scheme1.Lemma 3.1.
For any nonzero elementsu,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 [21].Denotedist(u,V)=infv∈V∥u-v∥a.Consider the eigenvalue problem (2.25) on the space 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 [21].Theorem 3.3.
Suppose that the conditions of Brezzi-Babuska Theorem hold and∥Th-T∥D→0(h→0). Let (λh,uh,σh) be the approximate eigenpair obtained by the two-scale discretization scheme and H small properly. Then there exists u∈M(λ) such that
(3.7)‖uh-u‖a≤C(|λH-λ|2+|λH-λ|‖(T-TH)|M(λ)‖D+‖(T-Th)|M(λ)‖a),(3.8)|λh-λ|≤C(‖uh-u‖a2+∥uh-u∥ainf⁡v∈Wh‖σ-v‖W).Proof.
We use Lemma3.2 in the proof.
Selectμ0=1/λH and u0=λHThuH/∥λHThuH∥a. Let u0∈M(λ) such that uH-u0 satisfies (2.39) and (2.41). By calculation we deduce
(3.9)‖λHThuH-u0‖a=‖λHThuH-λTu0‖a≤C(|λH-λ|+‖uH-u0‖D+‖(T-Th)u0‖a)≤C(|λH-λ|+‖(T-TH)|M(λ)‖D+‖(T-Th)|M(λ)‖a);
thus, using Lemma 3.1, we get
(3.10)dist⁡(u0,M^(λ))≤‖u0-u0‖u0‖a‖a≤C‖λHThuH-u0‖a≤C(|λH-λ|+‖(T-TH)|M(λ)‖D+‖(T-Th)|M(λ)‖a).
Using the triangle inequality and (2.42), we derive
(3.11)dist⁡(u0,Mh(λ))≤dist⁡(u0,M^(λ))+C‖(T-Th)|M(λ)‖a.
From Lemma 2.2 we know λH→λ,λj,h→λ; then
(3.12)|μ0-μj,h|=|λH-λ+λ-λj,hλj,hλH|.
When H is small enough, noting that h≪H, from (3.11) and (3.10) we get
(3.13)dist⁡(u0,Mh(λ))≤12.
Having in mind that λ=λk+1=⋯=λk+q-1 we have
(3.14)|μj,h-μh|=|λh-λj,hλhλj,h|=|λh-λ+λj-λj,hλhλj,h|,
which together with (3.12), noting that λj,h-λ is an infinitesimal of higher order comparing with λH-λ, yields
(3.15)max⁡k≤j≤k+q-1|μj,h-μhμ0-μj,h|≤12.
Since ρ is the separation constant, H is small enough, and h≪H, there holds
(3.16)|μ0-μj,h|≥ρ2,j≠k,k+1,…,k+q-1.
For u′ in Step 2 of Scheme 1, from the definition of Th and Sh we have
(3.17)a(ThλHu′,ψ)+b(ψ,ShλHu′)=λH(u′,ψ)D,∀ψ∈Vh,(3.18)b(Thu′,v)=0,∀v∈Wh.(3.19)a(ThuH,ψ)+b(ψ,ShuH)=(uH,ψ)D,∀ψ∈Vh,(3.20)b(ThuH,v)=0,∀v∈Wh.
Hence, Step 2 of Scheme 1 is equivalent to (u′,σ′)∈Vh×Wh,
(3.21)a(u′,ψ)+b(ψ,σ′)-λHa(Thu′,ψ)-λHb(ψ,Shu′)=a(ThuH,ψ)+b(ψ,ShuH),∀ψ∈Vh,(3.22)b(u′,v)=0,∀v∈Wh,uh=u′/∥u′∥a,σh=σ′/∥u′∥a.
From (3.21) we obtain
(3.23)a(u′-λHThu′-ThuH,ψ)+b(ψ,σ′-λHShu′-ShuH)=0,∀ψ∈Vh.
Combining (3.22), (3.18), and (3.20), we get
(3.24)b(u′-λHThu′-ThuH,v)=0,∀v∈Wh.
By (3.24), taking ψ=u′-λHThu′-ThuH in (3.23), we obtain
(3.25)a(u′-λHThu′-ThuH,u′-λHThu′-ThuH)=0.
Thus
(3.26)(1λH-Th)u′=1λHThuH,uh=u′‖u′‖a.
From (3.26) we know that the first term on the left-hand side of (3.23) is equal to 0; thus
(3.27)b(ψ,σ′-λHShu′-ShuH)=0,∀ψ∈Vh;
then, using discrete inf-sup condition, we obtain
(3.28)σ′=λHShu′+ShuH.
Thus Step  2 of Scheme 1 is equivalent to (3.26), (3.28), and uh=u′/∥u′∥a,σh=σ′/∥u′∥a. Noting that λH-1ThuH=∥λH-1ThuH∥au0 differs from u0 by only a constant and denoting us=u′/∥λH-1ThuH∥a, we have
(3.29)(1λH-Th)us=u0,uh=us‖us‖a.
By (3.13), (3.15), (3.16), and (3.29), we see that the conditions of Lemma 3.2 hold. Thus, substituting (3.11) and (3.12) into (3.6), we obtain
(3.30)dist⁡(uh,M^h(λ))≤C|λH-λ|(dist⁡(u0,M^(λ))+‖Tu-Thu‖a).
Let the eigenfunctions {uj,h}kk+q-1 be an orthonormal system of Mh(λ) (in the sense of a(·,·)). Then
(3.31)dist⁡(uh,Mh(λ))=‖uh-∑j=kk+q-1‍a(uh,uj,h)uj,h‖a.
Let
(3.32)u*=∑j=kk+q-1‍a(uh,uj,h)uj,h,
and noting that ∥uh-u*∥a≤dist(uh,M^h(λ)), from (3.30) we deduce
(3.33)‖uh-u*‖a≤C|λH-λ|(dist⁡(u0,M^(λ))+‖Tu-Thu‖a).
By Lemma 2.2, there exists {uj0}kk+q-1⊂M(λ) such that uj,h-uj0 satisfies (2.41). Let
(3.34)u=∑j=kk+q-1‍a(uh,uj,h)uj0;
then u∈M(λ). Using (2.41) we deduce
(3.35)‖u*-u‖a=‖∑j=kk+q-1‍a(uh,uj,h)(uj,h-uj0)‖a≤C(∑j=kk+q-1‍‖uj,h-uj0‖a2)1/2≤C‖(Th-T)|M(λ)‖a.
Combining (3.33) with the previous inequality, we have
(3.36)‖uh-u‖a≤C(|λH-λ|dist⁡(u0,M^(λ))+‖(Th-T)|M(λ)‖a).
Substituting (3.10) into (3.36), we get (3.7).
We know thatb(uh,σh)=0 from Step 2 of two-scale scheme; then
(3.37)λh=a(uh,uh)(uh,uh)D=a(uh,uh)+2b(uh,σh)(uh,uh)D.
Select λr=λh,u*=uh,σ*=σh. From Lemma 2.3, we get
(3.38)λh-λ=a(uh-u,uh-u)+2b(uh-u,σh-σ)(uh,uh)D-λ(uh-u,uh-u)D(uh,uh)D.
Noting that, for all v∈Wh,b(uh-u,v)=0, we have
(3.39)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.
Since V↪D (continuously imbedded), ∥uh-u∥D≤C∥uh-u∥a. Then from (3.39) we obtain (3.8).


## 4.   Two-Scale Discretization Scheme for Stokes Eigenvalue Problem

Consider the Stokes eigenvalue problem:(4.1)-Δu→+∇σ=λu→,inΩ,(4.2)div⁡u→=0,inΩ,(4.3)u→=0,on⁡∂Ω,
where Ω is a polygonal domain in R2,u→=(u1,u2) denotes the fluid velocity, and σ denotes the pressure.In this paper, we use the symbol→ to stand for vector function. For the function σ in Hm(Ω), let
(4.4)‖σ‖m=(∑|α|≤m‍∫Ω|∂ασ|2)1/2,α={α1,α2},|α|=α1+α2.
For vector function u→=(u1,u2), define
(4.5)‖u→‖m=(‖u1‖m2+‖u2‖m2)1/2,|u→|m=(|u1|m2+|u2|m2)1/2.Using Green's formula, we derive the mixed variational form associated with (4.1)–(4.3).Find(λ,u→,σ)∈R×H01(Ω)2×L02(Ω) with |u→|1=1 such that
(4.6)∫Ω∑i=12‍▿ui⋅▿ψi-∫Ωdiv⁡ψ→σ=λ∫Ωu→⋅ψ→,∀ψ→∈H01(Ω)2,(4.7)-∫Ωdiv⁡u→v=0,∀v∈L02(Ω).LetVh⊂H01(Ω)2,Wh⊂L02(Ω) be two mixed finite element spaces. The mixed finite element form is as follows.Seek(λh,u→h,σh)∈R×Vh×Wh with |u→h|1=1 such that
(4.8)∫Ω∑i=12‍▿uhi⋅▿ψi-∫Ωdiv⁡ψ→hσh=λ∫Ωu→h⋅ψ→,∀ψ→∈Vh,(4.9)-∫Ωdiv⁡u→hv=0,∀v∈Wh.Denote(4.10)V=H01(Ω)×H01(Ω),W=L02(Ω)={v∈L2(Ω):∫Ωv=0},D=L2(Ω)×L2(Ω),a(u→,ψ→)=∫Ω∑i=12‍▿ui⋅▿ψi,b(u→,v)=-∫Ωdiv⁡u→v,(u→,ψ→)D=∫Ωu→⋅ψ→.
Let ∥u→∥a=a(u→,u→). It is clear that ∥u→∥a=|u→|1 is a norm. Then (4.6)-(4.7) and (4.8)-(4.9) can be written in the forms of (2.3)-(2.4) and (2.5)-(2.6), respectively (we need to add → for the vector function, e.g., u,ψ should be written in the forms of u→,ψ→).We apply Scheme1 to the Stokes eigenvalue problem (4.6)-(4.7). Adding the symbol → for the vector function we get two-scale discretization scheme of mixed finite element for solving the Stokes eigenvalue problem (4.6)-(4.7), which is still called Scheme 1.Consider the associated source and approximate source problems.Find(w→,p)∈V×W such that
(4.11)a(w→,ψ→)+b(ψ→,p)=(f→,ψ→)D,∀ψ→∈V,(4.12)b(w→,v)=0,∀v∈W.Seek(w→h,ph)∈Vh×Wh such that
(4.13)a(w→h,ψ→)+b(ψ→,ph)=(f→,ψ→)D,∀ψ→∈Vh,(4.14)b(w→h,v)=0,∀v∈Wh.From [26] we know that (4.11)-(4.12) satisfy conditions (1)–(3) in Brezzi-Babuska Theorem; therefore, there exists a unique solution (w→,p)∈V×W to the problem (4.11)-(4.12) and the following estimate is valid:
(4.15)‖w→‖1+‖p‖0≤Cp‖f→‖0.
Condition (4) in Brezzi-Babuska Theorem holds since Vh⊂V and Wh⊂W. Suppose that condition (5) in Brezzi-Babuska Theorem (discrete inf-sup condition) is valid, then there exists a unique solution (w→h,ph)∈Vh×Wh to the problem (4.13)-(4.14), and the following error estimate is valid (see [27–29]):
(4.16)‖w→-w→h‖1+‖p-ph‖0≤C(inf⁡ψ→∈Vh‖w→-ψ→‖1+inf⁡v∈Wh‖p-v‖0).We assume that the following a prior estimate holds: for anyf→∈D,w→∈H1+r(Ω)×H1+r(Ω),p∈Hr(Ω), and
(4.17)‖w→‖1+r+‖p‖r≤C‖f→‖0,
where 0<r≤1 is a number determined by the maximal inner angle ω of Ω. When ω<π, r=1 (see [30]).Suppose that the following estimate holds: for anyw→∈H1+r(Ω)×H1+r(Ω), and for any p∈Hr(Ω),
(4.18)inf⁡ψ→∈Vh‖w→-ψ→‖V+inf⁡v∈Wh‖p-v‖W≤Chr(‖w→‖1+r+‖p‖r).From Section2, we know that (4.6)-(4.7) and (4.8)-(4.9) have the following equivalent operator forms, respectively,
(4.19)λTu→=u→,σ=S(λu→),(4.20)λhThu→h=u→h,σh=Sh(λhu→h).
Moreover, T and Th are all self-adjoint compact operators.Theorem 4.1.
Assume that discrete inf-sup condition, (4.17) and (4.18) hold; let H be properly small; and (λh,u→h,σh)  an approximate eigenpair obtained by Scheme 1. Then there exists u→∈M(λ) such that
(4.21)‖u→h-u→‖a≤C(H4r+hr),(4.22)|λh-λ|≤C(H8r+h2r).Proof.
From Brezzi-Babuska Theorem, (4.17), and (4.18), we deduce
(4.23)‖Tf→-Thf→‖V+‖Sf→-Shf→‖W≤Chr‖f→‖D,∀f→∈D.
By virtue of Nitsche technique (see [29]) and (4.17), we derive
(4.24)‖Tf→-Thf→‖D≤Chr(inf⁡q∈Vh‖Tf→-q‖V+inf⁡v∈Wh‖Sf→-v‖W)≤Ch2r‖f→‖D,∀f∈D.
Using (4.24), we have
(4.25)‖T-Th‖D=sup⁡f→∈D,∥f→∥D=1‖Tf→-Thf→‖D≤Ch2r⟶0(h⟶0).
Hence, the conditions in Lemma 2.2 and Theorem 3.3 hold.
From (2.23), we know that
(4.26)|λh-λ|+‖u→h-u→‖D≤C‖(T-Th)|M(λ)‖D≤Ch2r.
From (4.23), (4.18), and (4.17), we have
(4.27)‖(T-Th)|M(λ)‖a=sup⁡u→∈M(λ),u→≠0‖Tu→-Thu→‖a‖u→‖a≤Chr,(4.28)inf⁡v∈Wh‖σ-v‖W≤Chr‖σ‖r≤Chr‖u→‖D.
Substituting (4.26) and (4.27) into (3.7), we obtain (4.21). Substituting (4.21) and (4.28) into (3.8), we obtain (4.22).


### 4.1.  Mini Mixed Finite Element

Consider two-scale discretization scheme of Mini mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).Mini element was established by Arnold et al. in 1984 (see [31]). Let Kh be a regular triangulation of Ω under the meaning of paper [32], and
(4.29)Sh={v∈C(Ω-):v|κ∈P1,κ∈Kh},S0h=Sh⋂‍H01(Ω).
For any κ∈Kh, let N1, N2, and N3 denote barycentric coordinates. Denote Bh={v:v|κ∈span{N1N2N3},κ∈Kh}, and set
(4.30)Vh=(S0h⨁Bh)2,Wh=Sh⋂‍L02(Ω).From [31], we know that Mini element satisfies discrete inf-sup condition. From the interpolation theory in Sobolev space, we conclude that (4.18) is valid. Hence, for r satisfying (4.17), Scheme 1 for Mini mixed finite element is effective. Theorem 4.1 is valid.


### 4.2.P1-P1 Mixed Finite Element

Consider two-scale discretization scheme ofP1-P1 mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).LetK2h be a regular triangulation of Ω, and Kh is the product of refining K2h in the middle point. The P1-P1 mixed finite element space is defined by
(4.31)Vh=[Sh⋂‍H01(Ω)]2,Wh=S2h⋂‍L02(Ω),
where S2h and Sh are piecewise continuous linear polynomial spaces defined on K2h and Kh, respectively.From [33, Proposition 3.3], we know that P1-P1 element satisfies discrete inf-sup condition. By the interpolation theory in Sobolev space, we conclude that (4.18) holds. Therefore, for r satisfying (4.17), Scheme 1 for P1-P1 mixed finite element is effective. Theorem 4.1 is valid.


## 4.1.  Mini Mixed Finite Element

Consider two-scale discretization scheme of Mini mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).Mini element was established by Arnold et al. in 1984 (see [31]). Let Kh be a regular triangulation of Ω under the meaning of paper [32], and
(4.29)Sh={v∈C(Ω-):v|κ∈P1,κ∈Kh},S0h=Sh⋂‍H01(Ω).
For any κ∈Kh, let N1, N2, and N3 denote barycentric coordinates. Denote Bh={v:v|κ∈span{N1N2N3},κ∈Kh}, and set
(4.30)Vh=(S0h⨁Bh)2,Wh=Sh⋂‍L02(Ω).From [31], we know that Mini element satisfies discrete inf-sup condition. From the interpolation theory in Sobolev space, we conclude that (4.18) is valid. Hence, for r satisfying (4.17), Scheme 1 for Mini mixed finite element is effective. Theorem 4.1 is valid.


## 4.2.P1-P1 Mixed Finite Element

Consider two-scale discretization scheme ofP1-P1 mixed finite element for the Stokes eigenvalue problem (4.1)–(4.3) (Scheme 1).LetK2h be a regular triangulation of Ω, and Kh is the product of refining K2h in the middle point. The P1-P1 mixed finite element space is defined by
(4.31)Vh=[Sh⋂‍H01(Ω)]2,Wh=S2h⋂‍L02(Ω),
where S2h and Sh are piecewise continuous linear polynomial spaces defined on K2h and Kh, respectively.From [33, Proposition 3.3], we know that P1-P1 element satisfies discrete inf-sup condition. By the interpolation theory in Sobolev space, we conclude that (4.18) holds. Therefore, for r satisfying (4.17), Scheme 1 for P1-P1 mixed finite element is effective. Theorem 4.1 is valid.


## 5. Two-Scale Discretization Scheme for Eigenvalue Problem of Electric Field

Consider the eigenvalue problem of electric field:(5.1)c2curl⁡curl⁡u→=ω2u→,inΩ,(5.2)div⁡u→=0,inΩ,(5.3)u→×γ→=0,on⁡∈∂Ω,
where Ω is a polyhedron in R3 and γ→ is the outward normal unit vector on ∂Ω.In physics,u→ in the above eigenvalue problem of electric field denotes electric field, ω denotes the time frequency, and c is the speed of the light. Usually we set λ=ω2/c2 which is called eigenvalue.The spacesH(curl, Ω), H0(curl, Ω) are defined in the usual way:
(5.4)H(curl⁡,Ω)={q→∈L2(Ω)3:curl⁡q→∈L2(Ω)3},H0(curl⁡,Ω)={q→∈H(curl⁡,Ω):q→×γ→|∂Ω=0}.WhenΩ is convex polyhedron, we define the following function space:
(5.5)χ={q→∈H0(curl⁡,Ω):div⁡q→∈L2(Ω)}.
Denote
(5.6)(q→,ψ→)0=∫Ω‍q→⋅ψ→dx,‖q→‖0=(q→,q→)01/2.(q→,ψ→)χ=(curl⁡q→,curl⁡ψ→)0+(div⁡q→,div⁡ψ→)0,‖q→‖χ=(q→,q→)χ1/2.
From [34, 35], we see that χ⊂H1(Ω)3; (q→,ψ→)χ is a coercive bilinear form in χ, and ∥q→∥χ is a norm.On the other hand, whenΩ is nonconvex the maximal interior angle belongs to (π,2π). In this situation the problem is relatively complicated. Let E denote a set of reentrant edge with dihedral angles belonging to (π,2π), and let d 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 with respect to x. It can be represented by dr in reentrant edge and angular domain. We write ωr≃dr. Define the weight function space:
(5.7)Lr2(Ω)={v∈Lloc⁡2(Ω):ωrv∈L2(Ω)},χr={q→∈H0(curl⁡,Ω):div⁡q→∈Lr2(Ω)}.
Denote
(5.8)(q→,ψ→)Lr2=∫Ω‍ωr2q→⋅ψ→dx,‖q→‖Lr2=(q→,q→)Lr21/2,(q→,ψ→)χr=(curl⁡q→,curl⁡ψ→)0+(div⁡q→,div⁡ψ→)Lr2,‖q→‖χr=(q→,q→)χr1/2.
Let σΔN be the following smallest singular exponent in the Laplace problem with homogenous Dirichlet boundary condition:
(5.9){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 [36, 37], we know that, for all r∈(rmin,1), the seminorm |q→|χr is a norm in χr, and χr∩H1(Ω)3 is dense in χr.In the following discussion, we will useχr,Lr2(Ω) both for non-convex and convex domain. We take r∈(rmin,1) for non-convex domain; otherwise, we take χr=χ,Lr2(Ω)=L2(Ω).By introducing Lagrange multiplierσ, [36, 38, 39] changed (5.1)–(5.3) into the mixed variational formulation: find (λ,u→,σ)∈R+×χr×Lr2(Ω) such that
(5.10)(u→,ψ→)χr+(div⁡ψ→,σ)Lr2=λ(u→,ψ→)0,∀ψ→∈χr,(5.11)(div⁡u→,v)Lr2=0,∀v∈Lr2(Ω).LetKh be a regular simplex partition (tetrahedral partition) of Ω with the mesh diameter h. Define the Pk+1-Pk finite element space as follows:
(5.12)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 zero on the tetrahedron where reentrant edge and angular point are adjacent.Restricting (5.10)-(5.11) to the previous finite element space, we get discrete mixed variational form: find (λh,u→h,σh)∈R+×χh×Mh such that
(5.13)(u→h,ψ→)χr+(div⁡ψ→,σh)Lr2=λh(u→h,ψ→)0,∀ψ→∈χh,(5.14)(div⁡u→h,v)Lr2=0,∀v∈Mh.Set(5.15)V=χr,‖⋅‖V=‖⋅‖χr,W=Lr2(Ω),‖⋅‖W=‖⋅‖Lr2,D=L2(Ω)3,‖⋅‖D=‖⋅‖0,a(q→,ψ→)=(q→,ψ→)χr,b(ψ→,v)=(div⁡ψ→,v)Lr2.
Then (5.10)-(5.11) and (5.13)-(5.14) can be written in the forms of (2.3)-(2.4) and (2.5)-(2.6), respectively (we need to add → for the vector function, e.g., u,ψ should be written in the forms of u→,ψ→).We apply Scheme1 to the eigenvalue problem of electric field (5.10)-(5.11). Adding the symbol → for the vector function we get two-scale discretization scheme of mixed finite element for solving the eigenvalue problem of electric field (5.10)-(5.11) which is still called Scheme 1.It is easy to know thata(·,·) and b(·,·) are continuous bilinear forms on V×V and V×W, respectively. V is compact embedded in D (when Ω is convex, it is valid obviously; when Ω is non-convex, see [36]).Consider the source problem corresponding to (5.10)-(5.11).Find(w→,p)∈χr×Lr2(Ω) such that
(5.16)(w→,ψ→)χr+(div⁡ψ→,p)Lr2=(f→,ψ→)0,∀ψ→∈χr,(5.17)(div⁡w→,v)Lr2=0,∀v∈Lr2(Ω).For the problem (5.16)-(5.17) and its Pk+1-Pk element approximation, people have already proved the conditions in Brezzi-Babuska Theorem hold (see [38, 40, 41]).Therefore, we can define operatorsT,S,Th,Sh; moreover, (5.10)-(5.11) and (5.13)-(5.14) can be written in the forms of (2.23)-(2.24) and (2.25)-(2.26), respectively.Lemma5.1 is cited from the literature [36, 38].Lemma 5.1.
(5.1)–(5.3) is equivalent to (5.10)-(5.11), and the solutions of (5.10)-(5.11), (u→,σ), satisfy σ=S(λu→)=0 and u→∈χr with divu→=0.Denote(5.18)ελ(h)=sup⁡u→∈M^(λ)inf⁡ψ→∈Vh‖Tu→-ψ→‖a.For thePk+1-Pk element approximation of (5.16)-(5.17), [38] proved that the condition of [24, Theorem 1] (i.e., [38, Theorem 4.3]) is valid, hence, there holds the following.Lemma 5.2.
For thePk+1-Pk element approximation of (5.16)-(5.17), there exists r(h)>0,  r(h)→0(h→0) such that
(5.19)sup⁡f→∈D,f→≠0‖(T-Th)f→‖V‖f→‖D≤r(h).This lemma is very important. It tells us that∥T-Th∥D→0(h→0). Based on this lemma, [38] also proved the following conclusion.Lemma 5.3.
For thePk+1-Pk element approximation of (5.10)-(5.11), the following estimate is valid:
(5.20)|λ-λh|≤Cελ(h)2,(5.21)‖u→h-u→‖a≤Cελ(h).Theorem 5.4.
Let(λh,u→h) be the approximate Pk+1-Pk element eigenpair obtained by Scheme 1; then there exists u→∈M(λ) such that
(5.22)‖u→h-u→‖a≤C(ελ(H)3+ελ(h)),(5.23)|λh-λ|≤C(ελ(H)6+ελ(h)2).Proof.
We use Theorem3.3 to complete the proof. From (5.19) we see that the condition in Theorem 3.3 holds, and by Lemma 5.1 we know σ=0.
From (2.18), we deduce
(5.24)‖(T-Th)|M(λ)‖a=sup⁡u→∈M^(λ)‖Tu→-Thu→‖a≤Csup⁡u→∈M^(λ)inf⁡ψ→∈Vh‖Tu→-ψ→‖a≤Cελ(h).
From (5.20), we derive
(5.25)|λH-λ|≤Cελ(H)2.
Substituting (5.24) and (5.25) into (3.7), we get (5.22).
Sinceσ=0, (3.8) can be simplified to
(5.26)|λh-λ|≤C‖u→h-u→‖a2.
Substituting (5.22) into (5.26), we obtain (5.23).LetσΔN be the smallest singular exponent in the Laplace problem with homogenous Neumann boundary condition, then σΔN∈(3/2,2). Denote τ=min(r-rmin,σΔN-1).Corollary 5.5.
Under the condition of Theorem5.4, if Ω is a convex polyhedron there holds
(5.27)‖u→h-u→‖a≤C(H3+h),(5.28)|λh-λ|≤C(H6+h2);
if Ω is a non-convex polyhedron, then
(5.29)‖u→h-u→‖a≤C(H3μ+hμ),∀μ∈(0,τ),(5.30)|λh-λ|≤C(H6μ+h2μ),∀μ∈(0,τ).Proof.
WhenΩ is a convex polyhedron, for any u→∈M(λ) we have u→=T(λu→)∈H2(Ω) (see [35], or [42, equation (44)]); therefore
(5.31)ελ(H)≤CH,ελ(h)≤Ch.
Substituting these two formulae into (5.22) and (5.23), we obtain (5.27) and (5.28), respectively. When Ω is non-convex, for all u→∈M(λ), from [38, equation (36)] we know that
(5.32)ελ(H)≤CHμ,ελ(h)≤Chμ.
Substituting the above two formulae into (5.22) and (5.23), we obtain (5.29) and (5.30), respectively.


## 6. Numerical Experiments

In the following two examples, letλ1,H,λ2,H,…,λ4,H be the first four eigenvalues computed by using mixed finite method directly on coarse mesh KH and λ1,h,λ2,h,…,λ4,h the first four eigenvalues computed by using mixed finite method directly on fine mesh Kh. Let λ1h,λ2h,…,λ4h denote the first four eigenvalues computed by Scheme 1 on the meshes KH and Kh.Example 6.1.
Consider the Stokes eigenvalue problem (4.1)–(4.3), where Ω⊂R2 is a unit square domain. The smallest eigenvalue λ1 is approximately equal to 52.3446911 for this problem.We adopt a uniform isosceles right triangulation for the domainΩ (the edge in each element is along three fixed directions), and we give an initial mesh in Figure 1 and refine the initial mesh in a uniform way (each triangle is divided into four congruent triangles) repeatedly to get meshes KH and Kh.Figure 1We solve this problem by Scheme1  with Mini element. The numerical results are shown in Table 1.Table 1
The results on the square by Scheme1 (Mini element) for Stokes eigenvalue problem.
k
H
h
λ
k
,
H
λ
k
,
h
λ
k
h
1
2
8
2
64
57.50602342019
52.42255785241
52.42147553267
1
2
10
2
100
55.61324143591
52.37622690147
52.37604993774
1
2
12
2
144
54.59828590012
52.35983048690
52.35978975974
2
2
8
2
64
103.5158846668
92.30714521284
92.29553836981
2
2
10
2
100
99.36963738012
92.19619098643
92.19429134425
2
2
12
2
144
97.12904494102
92.15848299868
92.15804753694
3
2
8
2
64
110.7048253409
92.43050152401
92.39531304531
3
2
10
2
100
103.8058661064
92.24032470162
92.23510195554
3
2
12
2
144
100.1402638604
92.17887213187
92.17771885852
4
2
8
2
64
159.7476076282
128.8864896622
128.7085892117
4
2
10
2
100
148.7986378758
128.4767873556
128.4133574197
4
2
12
2
144
142.5976379771
128.3089668708
128.3076834410From Table1, we conclude that Scheme 1 can efficiently solve Stokes eigenvalue problem.Example 6.2.
Consider the eigenvalue problem of electric field (5.1)–(5.3), where Ω is a square domain [0,π]×[0,π] or an L-shaped domain [-1,0]×[-1,0]∪[-1,1]×[0,1]. For the square domain, the first four exact eigenvalues are λ1=1,λ2=1,λ3=2,λ4=4 for this problem. For the L-shaped domain, the first four approximate eigenvalues are λ1≈1.475622,λ2≈3.534031,λ3≈9.869604,λ4≈9.869604.We adopt a uniform isosceles right triangulation forΩ (the edge in each element is along three fixed directions, see Figure 2 for the L-shaped domain, and see Figure 1 for the square domain [0,π]×[0,π]) to produce the meshes KH and Kh.Figure 2We useP2-P1 mixed finite element to solve this problem. The definition of P2-P1 mixed finite element space is given by
(6.1)Vh={q→∈C0(Ω¯)2:q→×γ→|∂Ω=0,q→|κ∈P2(κ)2,∀κ∈Kh},Wh={v∈C0(Ω¯):v|κ∈P1(κ),∀κ∈Kh,v|Eh=0}.We compute the first four approximate eigenvalues by using Scheme1 with P2-P1 element on the mesh KH and Kh. The numerical results are listed in Tables 2, 3, and 4.Table 2
The results on the square by Scheme1 (P2-P1 element) for eigenvalue problem of electric field (r=0).
k
H
h
λ
k
,
H
λ
k
,
h
λ
k
h
1
2
4
2
72
1.000489112
1.000000005
1.000000005
1
2
6
2
72
1.000099609
1.000000005
1.000000005
1
2
8
2
72
1.000031944
1.000000005
1.000000005
2
2
4
2
72
1.000490210
1.000000005
1.000000005
2
2
6
2
72
1.000099685
1.000000005
1.000000005
2
2
8
2
72
1.000031956
1.000000005
1.000000005
3
2
4
2
72
2.006497695
2.000000070
2.000000070
3
2
6
2
72
2.001365268
2.000000070
2.000000070
3
2
8
2
72
2.000442889
2.000000070
2.000000070
4
2
4
2
72
4.028293524
4.000000321
4.000000333
4
2
6
2
72
4.006078511
4.000000321
4.000000321
4
2
8
2
72
4.001988912
4.000000321
4.000000321Table 3
The results on the L-shaped domain by Scheme1 (P2-P1 element) for eigenvalue problem of electric field (r=0.5).
k
H
h
λ
k
,
H
λ
k
,
h
λ
k
h
1
2
4
2
72
3.004954998
2.557217952
2.557241250
1
2
6
2
72
2.896205625
2.557217952
2.557225625
1
2
8
2
72
2.842087274
2.557217952
2.557221723
2
2
4
2
72
3.686002173
3.536967001
3.536967768
2
2
6
2
72
3.623341929
3.536967001
3.536967079
2
2
8
2
72
3.595097794
3.536967001
3.536967017
3
2
4
2
72
9.938579092
9.869605192
9.869605219
3
2
6
2
72
9.884458574
9.869605192
9.869605192
3
2
8
2
72
9.874470046
9.869605192
9.869605192
4
2
4
2
72
9.938761731
9.869605192
9.869605221
4
2
6
2
72
9.884463697
9.869605192
9.869605192
4
2
8
2
72
9.874472304
9.869605192
9.869605192Table 4
The results on the L-shaped domain by Scheme1 (P2-P1 element) for eigenvalue problem of electric field (r=0.95).
k
H
h
λ
k
,
H
λ
k
,
h
λ
k
h
1
2
4
2
72
1.913411786
1.517146260
1.517159275
1
2
6
2
72
1.779172346
1.517146260
1.517148955
1
2
8
2
72
1.723917701
1.517146260
1.517147418
2
2
4
2
72
3.589241343
3.534187594
3.534187611
2
2
6
2
72
3.558323120
3.534187594
3.534187595
2
2
8
2
72
3.547766251
3.534187594
3.534187595
3
2
4
2
72
9.937421407
9.869605190
9.869605213
3
2
6
2
72
9.884270121
9.869605190
9.869605191
3
2
8
2
72
9.874420606
9.869605190
9.869605191
4
2
4
2
72
9.937916802
9.869605191
9.869605216
4
2
6
2
72
9.884282799
9.869605191
9.869605190
4
2
8
2
72
9.874424604
9.869605191
9.869605190From Tables2–4, we conclude that Scheme 1 can efficiently solve the eigenvalue problem of electric field.In this paper,C denotes a positive constant independent of h, which may stand for different values at its different occurrences.



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*Source: 812914-2012-07-15.xml*