# A Multilevel Correction Scheme for the Steklov Eigenvalue Problem

Authors: Qichao Zhao; Yidu Yang; Hai Bi
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Combining the correction technique proposed by Lin and Xie and
the shifted inverse iteration, a multilevel correction scheme for the Steklov eigenvalue
problem is proposed in this paper. The theoretical analysis and numerical experiments
indicate that the scheme proposed in this paper is efficient for both simple and multiple eigenvalues of the Steklov eigenvalue problem.


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## Body

## 1. Introduction

Steklov eigenvalue problems have important applications in physics and engineering, for instance, in the study of surface waves (see [1]), in the analysis of stability of mechanical oscillators immersed in a viscous fluid (see [2]), and in the study of the vibration modes of a structure in contact with an incompressible fluid (see [3]). Thus, finite element methods for Steklov eigenvalue problems have attracted the attention of mathematics and physics community. Reference [4] first studied the convergence and error estimation of finite element approximations and [5–8] made in-depth research; after that, [9–15] discussed the highly efficient finite element methods.Recently, an efficient multilevel method based on the correction step was proposed by Lin and Xie [16, 17] and further successfully applied to Helmholtz transmission eigenvalue problems [18], convection-diffusion eigenvalue problems [19], and the Steklov problem [15]. The multilevel correction method proposed by Lin and Xie can be regarded as a combination of two-grid method and the extended/generalized finite element method which was developed in 1990s (see [20, 21]).The shifted inverse iteration method is a basic approach for solving matrix eigenvalue problems (see Algorithm 27.3 in [22]). Now, two-grid methods based on the shifted inverse iteration have been established (see [23, 24]) and applied to Steklov eigenvalue problems (see [11]).In this paper, we combine the correction technique of Lin and Xie and the shifted inverse iteration to establish a new efficient multilevel scheme for the Steklov eigenvalue problem which is suitable not only for simple eigenvalues but also for multiple eigenvalues. Our scheme can be described as follows: (1) solve the Steklov eigenvalue problem in the coarsest finite element space; (2) implement the shifted inverse iteration once in an augmented space by using the previously obtained eigenvalue as the shift and eigenfunction as the iteration initial value; (3) solve the Steklov eigenvalue problem again in a new space which is constructed by combining the coarsest finite element space with the obtained eigenfunction approximation in (2), and then return to (2) for next loop. Furthermore, we prove the scheme can reach the optimal order that the same as solving the corresponding boundary value problem. Our scheme is easy to realize under the package of iFEM [25] with Matlab, and the numerical results are satisfactory.In this paper,C (with or without subscripts) denotes a positive constant independent of mesh diameters and correction times.


## 2. Preliminaries

LetHs(Ω) and Hs(∂Ω) denote the usual Sobolev spaces with real-order s with norms ·s and ·s,∂Ω on Ω and ∂Ω, respectively. Here H0(∂Ω)=L2(∂Ω).Consider the following Steklov eigenvalue problem:(1)-Δu+u=0,inΩ,∂u∂ν=λu,on∂Ω,where Ω⊂R2 is a bounded polygonal domain with the maximum interior angle ω and ∂u/∂ν is the outward normal derivative on ∂Ω.The variational form of (1) is given as follows: find λ∈R and 0≠u∈H1(Ω) such that a(u,u)=1 and (2)au,v=λbu,v,∀v∈H1Ω,where (3)au,v=∫Ω∇u·∇v+uvdx,bu,v=∫∂Ωuvds.It is easy to know that a(·,·) is a symmetric, continuous, and H1(Ω)-elliptic bilinear form on H1(Ω)×H1(Ω) satisfying (4)aw,v≤C1w1v1,∀w,v∈H1Ω,aw,w≥c1w12,∀w∈H1Ω.So we can use a(·,·) and ·a=a(·,·)1/2 as the inner product and norm on H1(Ω), respectively. We can also use b(·,·) and ·b=b(·,·)1/2 as the inner product and norm on L2(∂Ω), respectively.From [4, 26] we know that problem (2) has an eigenvalue sequence {λj}1∞ satisfying (5)0<λ1≤λ2≤⋯≤λj≤⋯,limj→∞⁡λj=∞and the corresponding eigenfunctions (6)u1,u2,…,uj,…,where a(ui,uj)=δij. The eigenvalues λj are repeated according to their multiplicity in the sequence {λj}1∞.LetTh be a regular triangulation of Ω. Denote the diameter of an element K∈Th by hK and the mesh diameter h=maxK∈Th⁡hK. Let the finite element space Vh⊂H1(Ω) be a piecewise polynomial space on Th.Then the finite element approximation of (2) is given as follows: find λh∈R and 0≠uh∈Vh such that uha=1 and (7)auh,vh=λhbuh,vh,∀vh∈Vh.It is well known that (7) has eigenvalues (see, e.g., [4, 26]) (8)0<λ1,h≤λ2,h≤⋯≤λj,h≤⋯≤λNh,hNh=dim⁡Vh,and the associated eigenfunctions (9)u1,h,u2,h,…,uj,h,…,uNh,h,where a(ui,h,uj.h)=δij, 1≤i,j≤Nh.Consider the following boundary value problems (10) and (11) associated with (2) and (7), respectively.Findψ∈H1(Ω) such that ψa=1 and (10)aψ,v=bf,v,∀v∈H1Ω.Findψh∈Vh such that ψha=1 and (11)aψh,vh=bf,vh,∀vh∈Vh.The following regularities of the Steklov eigenvalue problem are valid.Lemma 1.
Iff∈L2(∂Ω), there exists a unique solution ψ∈H1+r/2(Ω) to (10) satisfying (12)ψ1+r/2≤C2f0,∂Ω.If f∈H1/2(∂Ω), there exists a unique solution ψ∈H1+r(Ω) to (10) satisfying (13)ψ1+r≤C2f1/2,∂Ω,where C2 is a positive constant.Proof.
See (4.10) in [27], Proposition 4.4 in [3], and Lemma 2.1 in [11].By (13), we know that uj∈H1+r(Ω), r=1, if Ω is convex; otherwise r<π/ω and r can be closed to π/ω arbitrarily.Then, from (10) and (11), we can define two linear bounded operators T:L2(∂Ω)→H1(Ω) and Th:L2(∂Ω)→Vh such that (14)aTf,v=bf,v,∀v∈H1Ω,(15)aThf,vh=bf,vh,∀vh∈Vh.It is obvious that Tf and Thf are the solution of (10) and (11), respectively. We know that T:H1(Ω)→H1(Ω) and Th:H1(Ω)→Vh are completely continuous operators (see, e.g., [11]). From [4, 26], we know that (2) and (7) have the following equivalent operator forms: (16)Tu=μu,Thuh=μhuh,where μ=1/λ and μh=1/λh. From (15), we can deduce that Th:L2(∂Ω)→Vh is bounded; that is, (17)Thfa≤C3fb,∀f∈L2∂Ω,Thfa≤C3fa,∀f∈H1Ω,where C3 is a positive constant independent of h and f.Denote the eigenfunction space corresponding toλi by (18)Mλi=w∈H1Ω∣wisaneigenfunctionof2correspondingtoλiandwa=1.Suppose that the multiplicity of λi is q; that is, λi=λi+1=⋯=λi+q-1. We use (λj,h,uj,h) to denote the eigenpair approximation for (λj,uj), j=i,…,i+q-1. Let (19)Mhλi=span⁡ui,h,ui+1,h,…,ui+q-1,h,and let M(μi)=M(λi) and Mh(μi)=Mh(λi).Denote(20)ηah=supf∈L2∂Ω,f0,∂Ω=1⁡infvh∈Vh⁡Tf-vha,δhλi=supw∈Mλi⁡infvh∈Vh⁡w-vha.From Lemma 1 and the interpolation error estimate, we get (21)ηah⟶0,ash⟶0.For two linear spacesA and B, we define (22)dist⁡u,A=infv∈A⁡u-va,∀u∈B,Θ^A,B=supw∈A,wa=1⁡infv∈B⁡w-va,Φ^A,B=supw∈A,wb=1⁡infv∈B⁡w-vb.Define the gaps between M(λi) and Mh(λi) in ·a as (23)ΘMλi,Mhλi=max⁡Θ^Mλi,Mhλi,Θ^Mhλi,Mλiand in ·b as (24)ΦMλi,Mhλi=max⁡Φ^Mλi,Mhλi,Φ^Mhλi,Mλi.Lemma 2.
The following estimates are valid:(25)ΘMλi,Mhλi≤Cλiδhλi,(26)ΦMλi,Mhλi≤Cλiηahδhλi,(27)λj,h-λi≤Cλiδh2λi,for j=i,…,i+q-1. Here and hereafter C(λi) is a positive constant depending on λi but independent of h.Proof.
See [4] or P. 699 in [26].We need the following result (cf. [11, 23]) for the shifted inverse iteration method.Lemma 3.
Let(μ0,u0) be an approximation for (μi,ui), where μ0 is not an eigenvalue of Th, and u0∈Vh with u0a=1. Suppose that dist(u0,Mh(μi))≤1/2, μ0-μj,h≥ρ/2 for j≠i,…,i+q-1, and u∈Vh, ujh∈Vh satisfy (28)μ0-Thu=u0,ujh=uua.Then (29)dist⁡ujh,Mhλi≤C4λ0-λi+Cλiδh2λidist⁡u0,Mhλi,where ρ=minμj≠μi⁡μj-μi is a separation constant of μi and C4 is a positive constant independent of h.Proof.
From (3.2) in [23], (2.25) in [11], and (27), we have (30)dist⁡ujh,Mhλi≤16ρ1λ0-1λj,hdist⁡u0,Mhλi≤C4λ0-λi+Cλiδh2λidist⁡u0,Mhλi.The proof is completed.


## 3. One Correction Step Based on the Shifted Inverse Iteration

We first generate a coarse meshTH with the mesh size H. The coarse linear finite element space VH is defined on TH. Then we define a sequence of triangulations Thk of Ω determined as follows. Set h1=H and let Thk be obtained from Thk-1 via regular refinement. Then we construct the linear finite element spaces such that (31)VH=Vh1⊂Vh2⊂⋯⊂Vhn,δhk+1λi=δhk1+γλi,where γ∈(0,2].Assume that we have the eigenpair approximations(λ~j,hk,u~j,hk)∈R×VH,hk (j=i,…,i+q-1), where the eigenvalues {λ~j,hk}j=ii+q-1 are the approximations of the eigenvalue λi of (2) and VH,hk is an extended space (see Algorithm 4). Now we give the following one correction step to improve the accuracy of the current eigenpair approximations {λ~j,hk,u~j,hk}j=ii+q-1.Algorithm 4 (one correction step).
Step
1. For j=i,…,i+q-1, do the following.
Findu-j,hk+1∈Vhk+1 such that (32)au-j,hk+1,v-λ~i,hkbu-j,hk+1,v=bu~j,hk,v,∀v∈Vhk+1,and set u^j,hk+1=u-j,hk+1/u-j,hk+1a.
Step
2. Define a new finite element space VH,hk+1=VH⊕span{u^i,hk+1,…,u^i+q-1,hk+1} and solve the following Steklov eigenvalue problem.
Find(λ~j,hk+1,u~j,hk+1)∈R×VH,hk+1 such that u~j,hk+1a=1 and (33)au~j,hk+1,v=λ~j,hk+1bu~j,hk+1,v,∀v∈VH,hk+1.Output the eigenvalues {λ~j,hk+1}j=ii+q-1 and the corresponding orthonormal eigenfunctions {u~j,hk+1}j=ii+q-1 with respect to a(·,·).
We adopt the notations in [15–17] to simplify and summarize Algorithm 4 as (34)λ~j,hk+1,u~j,hk+1j=ii+q-1=CorrectionVH,λ~j,hk,u~j,hkj=ii+q-1,Vhk+1.Theorem 5.
Suppose thatH is properly small. Assume the eigenpairs {λ~j,hk,u~j,hk}j=ii+q-1 in Algorithm 4 have the following error estimates: (35)ΘMλi,MH,hkλi≤C0δhkλi,ΦMλi,MH,hkλi≤C0ηaHδhkλi,λi-λ~j,hk≤C0δhk2λi.Then, after one correction step, the resulting eigenpair approximations {λ~j,hk+1,u~j,hk+1}j=ii+q-1 have the following error estimates: (36)ΘMλi,MH,hk+1λi≤C0δhk+1λi,(37)ΦMλi,MH,hk+1λi≤C0ηaHδhk+1λi,(38)λi-λ~j,hk+1≤C0δhk+12λi,where (39)C0=max⁡2q3/2C5C2λi,4q3C52C3λi,Cλiis a positive constant dependent on λi but independent of k and h, and C5=1 when λi is a simple eigenvalue, and when λi is a multiple eigenvalue C5 is a positive constant independent of k and h (see Lemma 6 at the back).Proof.
From (35), we know there exists a basis {uj}j=ii+q-1 of M(λi) such that (40)uj-u~j,hka≤C0δhkλi,(41)uj-u~j,hkb≤C0ηaHδhkλi,(42)λi-λ~i,hk≤C0δhk2λi.By (15), we know that (32) is equivalent to the following: (43)au-j,hk+1,v-λ~i,hkaThk+1u-j,hk+1,v=aThk+1u~j,hk,v,∀v∈Vhk+1,and u^j,hk+1=u-j,hk+1/u-j,hk+1a; that is, (44)1λ~i,hk-Thk+1u-j,hk+1=λ~i,hk-1Thk+1u~j,hk,u^j,hk+1=u-j,hk+1u-j,hk+1a.Select uj,hk+10=λ~i,hkThk+1u~j,hk/λ~i,hkThk+1u~j,hka; then uj,hk+10a=1 and uj,hk+10∈Vhk+1. Note that λ~i,hk-1Thk+1u~j,hk=λ~i,hk-1Thk+1u~j,hkauj,hk+10 differs from uj,hk+10 by only a constant; then (32) is equivalent to (45)1λ~i,hk-Thk+1u-j,hk+1=uj,hk+10,u^j,hk+1=u-j,hk+1u-j,hk+1a.Let λj,hk+1 be the jth eigenvalue of (7) with h=hk+1; then thanks to Lemma 3 we have (46)dist⁡u^j,hk+1,Mhk+1λi≤16ρ1λ~i,hk-1λj,hk+1·dist⁡uj,hk+10,Mhk+1λi≤C4λ~i,hk-λi+Cλiδhk+12λi·dist⁡uj,hk+10,Mhk+1λi≤C0C4δhk2λi+C4Cλiδhk+12λi·dist⁡uj,hk+10,Mhk+1λi.Then, using the triangle inequality and Lemma 2, we get (47)dist⁡uj,hk+10,Mhk+1λi≤dist⁡uj,hk+10,Mλi+Cλiδhk+1λi.Combining (46) and (47), we obtain (48)dist⁡u^j,hk+1,Mhk+1λi≤C0C4δhk2λi+C4Cλiδhk+12λi·dist⁡uj,hk+10,Mλi+C4Cλi·C0δhk2λi+Cλiδhk+12λiδhk+1λi.
Let the eigenvectors{uj,hk+1}j=ii+q-1 be an orthonormal basis of Mhk+1(λi) with respect to a(·,·). Denote (49)u^j∗=∑m=ii+q-1au^j,hk+1,um,hk+1um,hk+1;then (50)u^j,hk+1-u^j∗a=dist⁡u^j,hk+1,Mhk+1λi.By (25), there exists {uj′}j=ii+q-1⊂M(λi) satisfying (51)uj,hk+1-uj′a≤Cλiδhk+1λi.Let (52)u^j=∑m=ii+q-1au^j,hk+1,um,hk+1um′.Noting that {u~j,hk}j=ii+q-1 is linearly independent, it can be deduced from (45), (49), and (52) that {u^j}j=ii+q-1 is linearly independent.
Using (49), (51), and (52), we deduce that (53)u^j-u^j∗a=∑m=ii+q-1au^j,hk+1,um,hk+1um′-um,hk+1a≤∑m=ii+q-1um′-um,hk+1a21/2≤∑m=ii+q-1C2λiδhk+12λi1/2≤q1/2Cλiδhk+1λi.Thus, using the triangle inequality, we have (54)u^j,hk+1-u^ja≤dist⁡u^j,hk+1,Mhk+1λi+q1/2Cλiδhk+1λi,which together with (48) yields (55)u^j,hk+1-u^ja≤C4C0δhk2λi+C4Cλiδhk+12λi·dist⁡uj,hk+10,Mλi+Cλi·C4C0δhk2λi+C4Cλiδhk+12λi+q1/2·δhk+1λi.
Noting thatuja=1 and using Lemma 3.1 in [23], we have (56)dist⁡uj,hk+10,Mλi≤uj,hk+10-uja≤2λ~i,hkThk+1u~j,hk-uja.Since uj=λjTuj and u~j,hka=1, from (17), (41), and (42), we have (57)λ~i,hkThk+1u~j,hk-uja=λ~i,hkThk+1u~j,hk-λjThk+1u~j,hk+λjThk+1u~j,hk-λjThk+1uj+λjThk+1uj-λjTuja≤λ~i,hk-λjThk+1u~j,hka+λjThk+1u~j,hk-uja+λjThk+1-Tuja≤C0C3δhk2λi+C3λiu~j,hk-ujb+δhk+1λi≤C0C3δhk2λi+C0C3λiηaHδhkλi+δhk+1λi.From (56) and (57), we get (58)dist⁡uj,hk+10,Mλi≤2C0C3δhk2λi+2C0C3λiηaHδhkλi+2δhk+1λi.Combining (55) and (58), reminding that δhk+1=δhk1+γ, γ∈(0,2], we can obtain (59)u^j,hk+1-u^ja≤C0C4δhk2λi+C4Cλiδhk+12λi·2C0C3δhk2λi+2C0C3λiηaHδhkλi+2δhk+1λi+C0C4Cλiδhk2λi+C4C2λiδhk+12λi+q1/2Cλiδhk+1λi=2C02C3C4δhk1+γλi·δhk3-γλi+2C02C3C4λiηaHδhk1+γλiδhk2-γλi+2C0C4δhk2λiδhk+1λi+2C0C3C4Cλi·δhk2λiδhk+12λi+2C0C3C4CλiλiηaH·δhkλiδhk+12λi+2C4Cλiδhk+13λi+C0C4Cλiδhk2λi+C4C2λiδhk+12λi+q1/2Cλiδhk+1λi=2C02C3C4δhk3-γλi+2C02C3C4λiηaHδhk2-γλi+2C0C4δhk2λi+2C0C3C4Cλiδhk2λiδhk+1λi+2C0C3C4CλiλiηaHδhkλiδhk+1λi+2C4Cλiδhk+12λi+C0C4Cλiδhk2λi+C4C2λiδhk+12λi+q1/2Cλiδhk+1λi≤2C02C3C4δHλi+2C02C3C4λiηaH+2C0C4δH2λi+2C0C3C4CλiδH3λi+2C0C3C4CλiλiηaHδH2λi+2C4CλiδH2λi+C0C4CλiδH2λi+C4C2λiδH2λi+q1/2Cλiδhk+1λi.When H is properly small, we have (60)2C02C3C4δHλi+2C02C3C4λiηaH+2C0C4δH2λi+2C0C3C4CλiδH3λi+2C0C3C4CλiλiηaHδH2λi+2C4CλiδH2λi+C0C4CλiδH2λi+C4C2λiδH2λi≤q1/2Cλi;then we obtain the following error estimate: (61)u^j,hk+1-u^ja≤2q1/2Cλiδhk+1λi.
Now we come to estimate the error for the eigenpairs{λ~j,hk+1,u~j,hk+1}j=ii+q-1. Based on the error estimate theory of eigenvalue problem by finite element methods (see, e.g., [4, 26]), from Lemmas 2 and 6 at the back, (61), and the definition of the space VH,hk+1, we have the following estimates: (62)ΘMλi,MH,hk+1λi≤Cλi·supw∈Mλi⁡infvH,hk+1∈VH,hk+1⁡w-vH,hk+1a≤Cλi·supβj⁡∑j=ii+q-1βju^j-∑m=ii+q-1βmu^m,hk+1a≤Cλi·supβj⁡∑j=ii+q-1βju^j-u^j,hk+1a≤qC5Cλi·max⁡u^i,hk+1-u^ia,…,u^i+q-1,hk+1-u^i+q-1a≤2q3/2C5C2λiδhk+1λi,ΦMλi,MH,hk+1λi≤Cλi·supw∈Mλi⁡infvH,hk+1∈VH,hk+1⁡w-vH,hk+1b≤Cλi·η~aHsupw∈Mλi⁡infvH,hk+1∈VH,hk+1⁡w-vH,hk+1a≤2q3/2C5C2λiη~aHδhk+1λi,where (63)η~aH=supf∈L2∂Ω,f0,∂Ω=1⁡infv∈VH,hk+1⁡Tf-va≤supf∈L2∂Ω,f0,∂Ω=1⁡infv∈VH⁡Tf-va=ηaH.From (62) and (63), we obtain (36) and (37).
From (27) and (36), we have (64)λ~j,hk+1-λi≤Cλisupw∈Mλi⁡infvH,hk+1∈VH,hk+1⁡w-vH,hk+1a2≤4q3C52C3λiδhk+12λi.Hence, estimate (38) holds.Lemma 6.
For the multiple eigenvalueλi, assume that λ~j,hk-λi=O(δhk2(λi)) and δhk+1(λi)=δhk(λi)1+ε, where j=i,…,i+q-1, ε∈(ξ,2], and ξ∈(0,2) is a positive constant. Let {u^j}j=ii+q-1 be defined as (52); then {u^j}j=ii+q-1 is a basis of M(λi). For any w∈M(λi), (65)w=∑j=ii+q-1βju^j,there holds (66)βj≤C5,forj=i,…,i+q-1,where C5 is a constant independent of k and h.Proof.
From the proof of Theorem5 we know that {u^j}j=ii+q-1 is linearly independent, so it can be taken as a basis of M(λi). For any w∈M(λi), w=∑j=ii+q-1βju^j, we have (67)1=wa2=∑m=ii+q-1βmu^ma2=∑m=ii+q-1βm2u^ma2+∑m,j=i,m≠ji+q-1βmβjau^m,u^j.Noticing that the norm of self-adjoint operator is equal to its spectral radius, then (68)μ0-Thk+1-1a=minj⁡μ0-μj,hk+1-1≡μ0-μj0,hk+1-1,where μ0=1/λ0=1/λ~i,hk.
Foruj in (40)-(41), by calculation, we deduce (69)μ0-Thk+1-1λ~i,hkThk+1u~j,hk-μ0-μj0,hk+1-1uja=μ0-Thk+1-1λ~i,hkThk+1u~j,hk-uj+μ0-Thk+1-1uj-μ0-μj0,hk+1-1uja=μ0-Thk+1-1λ~i,hkThk+1u~j,hk-uj+μ0-Thk+1-1·uj-∑m=ii+q-1auj,um,hk+1um,hk+1+μ0-Thk+1-1∑m=ii+q-1auj,um,hk+1um,hk+1-μ0-μj0,hk+1-1uj-∑m=ii+q-1auj,um,hk+1um,hk+1-μ0-μj0,hk+1-1∑m=ii+q-1auj,um,hk+1um,hk+1a≤μ0-Thk+1-1λ~i,hkThk+1u~j,hk-uja+μ0-Thk+1-1aδhk+1λi+μ0-μj0,hk+1-1δhk+1λi+∑m=ii+q-1auj,um,hk+1μ0-μm,hk+1-1um,hk+1-∑m=ii+q-1auj,um,hk+1μ0-μj0,hk+1-1um,hk+1a≤μ0-Thk+1-1λ~i,hkThk+1u~j,hk-uja+μ0-Thk+1-1aδhk+1λi+μ0-μj0,hk+1-1δhk+1λi+μ0-μj0,hk+1-1∑m=ii+q-1auj,um,hk+1·μ0-μm,hk+1-1μ0-μj0,hk+1-1-1um,hk+1a,where (70)μ0-μm,hk+1-1μ0-μj0,hk+1-1-1=μ0-μj0,hk+1μ0-μm,hk+1-1=μm,hk+1-μj0,hk+1μ0-μm,hk+1.Since λ~j,hk-λi=O(δhk2(λi)), there exists C5′ such that (71)λ~j,hk-λi≥C5′δhk2λi,forj=i,…,i+q-1.Noting that μ0=1/λ0=1/λ~i,hk, then, for m=i,…,i+q-1, we have (72)μm,hk+1-μj0,hk+1μ0-μm,hk+1=λ0λj0,hk+1-λm,hk+1λj0,hk+1λm,hk+1-λ0≤2λ0Cλiδhk+12λiλj0,hk+1C5′-Cλiδhk2ελiδhk2λi=2λ0Cλiδhk2+2ελiλj0,hk+1C5′-Cλiδhk2ελiδhk2λi=2λ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi.
From (45), we have (73)u¯j,hk+1=μ0-Thk+1-1λ~i,hkThk+1u~j,hkλ~i,hkThk+1u~j,hka,u^j,hk+1=μ0-Thk+1-1λ~i,hkThk+1u~j,hkμ0-Thk+1-1λ~i,hkThk+1u~j,hka.Since VH,hk⊂Vhk⊂Vhk+1, we get (74)λ~j,hk≥λj,hk+1,forj=i,…,i+q-1.Note that (75)uj=μ0-μj0,hk+1-1ujμ0-μj0,hk+1-1uja=-μ0-μj0,hk+1-1ujμ0-μj0,hk+1-1uja;then, from Lemma 3.1 in [23], (57), and (68)–(73), we get (76)u^j,hk+1+uja=μ0-Thk+1-1λ~i,hkThk+1u~j,hkμ0-Thk+1-1λ~i,hkThk+1u~j,hka-μ0-μj0,hk+1-1ujμ0-μj0,hk+1-1ujaa≤2·1μ0-μj0,hk+1-1ujaμ0-Thk+1-1·λ~i,hkThk+1u~j,hk-μ0-μj0,hk+1-1uja≤2μ0-μj0,hk+1μ0-Thk+1-1λ~i,hkThk+1u~j,hk-uja+μ0-Thk+1-1aδhk+1λi+μ0-μj0,hk+1-1·δhk+1λi+μ0-μj0,hk+1-1·2qλ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi≤2λ~i,hkThk+1u~j,hk-uja+4δhk+1λi+4qλ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi≤2C0C3δhk2λi+2C0C3λiηaHδhkλi+6δhk+1λi+4q·λ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi.For i≤m,j≤i+q-1, from (76), we have (77)au^m,hk+1,u^j,hk+1=au^m,hk+1+um,u^j,hk+1-aum,u^j,hk+1=au^m,hk+1+um,u^j,hk+1-aum,u^j,hk+1+uj+aum,uj≤u^m,hk+1+uma+u^j,hk+1+uja+aum,uj≤4C0C3δhk2λi+4C0C3λiηaHδhkλi+12δhk+1λi+8qλ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi+aum,uj.By a(u~m,hk,u~j,hk)=δmj, from (40), we get (78)aum,uj=aum-u~m,hk,uj+au~m,hk,uj-u~j,hk+au~m,hk,u~j,hk≤2C0δhkλi+δmj;thus (79)au^m,hk+1,u^j,hk+1≤4C0C3δhk2λi+4C0C3λiηaHδhkλi+12δhk+1λi+8qλ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi+2C0δhkλi+δmj;then, from (61) and (79), we have (80)au^m,u^j=au^m-u^m,hk+1,u^j+au^m,hk+1,u^j-u^j,hk+1+au^m,hk+1,u^j,hk+1≤4q1/2Cλiδhk+1λi+4C0C3δhk2λi+4C0C3λiηaHδhkλi+12δhk+1λi+8q·λ0Cλiδhk2ελiλj0,hk+1C5′-Cλiδhk2ελi+2C0δhkλi+δmj.Thus there exists C5′′ such that (81)au^m,u^j≤C5′′δhkλimin⁡1,2ε,m≠j.Then, from (67), we have (82)∑m=ii+q-1βm2=1-∑m,j=i,m≠ji+q-1βmβjau^m,u^j≤1+q∑m=ii+q-1βm2C5′′δhkλimin⁡1,2ε,thus, we get(83)∑m=ii+q-1βm2≤11-qC5′′δhkλimin⁡1,2ε;that is, {βm}m=ii+q-1 are uniformly bounded from above with respect to hk.


## 4. Multilevel Correction Scheme for the Steklov Eigenvalue Problem

Based on Algorithm4, we introduce the following multilevel correction scheme.Algorithm 7 (multilevel correction scheme).
Step
1. Construct a series of finite element spaces VH=Vh1,Vh2,…,Vhn such that (31) holds.
Step
2. Compute q eigenpair approximations {λj,h1,uj,h1}j=ii+q-1 for the eigenpairs {λj, uj}j=ii+q-1 by solving the following Steklov eigenvalue problem.
Find(λj,h1,uj,h1)∈R×Vh1 such that a(uj,h1,uj,h1)=1 and (84)auj,h1,v=λj,h1buj,h1,v,∀v∈Vh1.
Step
3. λ~j,h1⇐λj,h1 and u~j,h1⇐uj,h1, for j=i,…,i+q-1.
Fork=1,…,n-1, do (85)λ~j,hk+1,u~j,hk+1j=ii+q-1=CorrectionVH,λ~j,hk,u~j,hkj=ii+q-1,Vhk+1.
Finally, we obtainq eigenpair approximations {λ~j,hn,u~j,hn}j=ii+q-1∈R×VH,hn.Theorem 8.
After implementing Algorithm7, if H is small enough, the resultant eigenpair approximations {λ~j,hn,u~j,hn}j=ii+q-1 have the following error estimates: (86)ΘMλi,MH,hnλi≤C0δhnλi,(87)ΦMλi,MH,hnλi≤C0ηaHδhnλi,(88)λi-λ~j,hn≤C0δhn2λi.Proof.
We use induction to prove Theorem8.
First, forn=1, since C(λi)≤C0, we can obtain (86)–(88) from (84) and Lemma 2.
Suppose that (86)–(88) hold for n-1. Then from Theorem 5 we know that (86)–(88) are valid for n.Corollary 9.
Under the conditions of Theorem8, further assume that {Vhi}i=1n is a series of linear or bilinear finite element spaces, and M(λ)⊂H1+σ(Ω), r≤σ≤1. Then (86)–(88) are valid and (89)ηaH≤CHr/2,δhnλi≤Chnσ.Remark 10.
Whenq=1, we obtain error estimates (86) and (88) for simple eigenvalues. Comparing (86) and (88) with the estimates (4.6), (4.7) in [15], we know that our algorithm achieves the optimal convergence order and is as efficient as Algorithm 4.1 in [15].Remark 11.
With the iteration times increasing, the approximate eigenvalues are closer to the exact ones which lead the equation (32) more closer to singular, although it is not difficult to solve numerically (see [22]). However, we fix the shift in Algorithms 4 and 7 and get a new scheme. In Section 5, we will report a numerical experiment when the shift is fixed as λ~j,hk=λ~j,h1.Remark 12.
Reference [15, 17] analyzed that the computational costs of their algorithms are almost the same as those of solving the corresponding boundary value problem. Comparing our Algorithm 7 with Algorithm 7.2 in [15], it is not difficult to see that the computational work of these two algorithms are almost the same.


## 5. Numerical Examples

In this section, we consider the Steklov eigenvalue problem (1) on the unit square Ω=(-0.5,0.5)2 and the L-shape domain Ω=(0,2)2∖[1,2]2, respectively. In our numerical experiments, we use algorithms in [15], our Algorithm 7, and Algorithm 7 with fixed shift λ~j,hk=λ~j,h1 to solve (1), and the resulting approximations are denoted by λj,hk+1X, λj,hk+1S, and λj,hk+1F, respectively. Our programs are performed under the package of Chen (cf. [25]).Example 1.
We solve (1) on the unit square Ω=(-0.5,0.5)2 by three algorithms with the linear conforming element. Since the exact eigenvalues are unknown, we use λ1≈0.24007908542 and λ2=λ3≈1.49230313453 obtained by the spectral element method (see [28]) as the reference eigenvalues. We depict the error curves of three algorithms in Figure 1.Figure 1
Ω
=
(
-
0.5,0.5
)
2, error curves for the 1st (a) and 2nd and 3rd (b) eigenvalues.
(a)
(b)Example 2.
We solve (1) on the L-shape domain Ω=(0,2)2∖[1,2]2 by three algorithms with the linear conforming element. Since the exact eigenvalues are unknown, we use λ1≈0.34141604255 and λ5≈1.70092884408 obtained by the spectral element method (see [28]) as the reference values. The error curves of three algorithms are depicted in Figure 2.Figure 2
Ω
=
(
0,2
)
2
∖
[
1,2
]
2, error curves for the 1st (a) and 5th (b) eigenvalues.
(a)
(b)From Figures1 and 2 we can see that our algorithm achieves the optimal convergence order and is as efficient as that in [15].


## 6. Concluding Remark

In this paper, we combined the correction technique proposed by Lin and Xie and the shifted inverse iteration to establish a new efficient multilevel correction scheme for the Steklov eigenvalue problem which is suitable not only for simple eigenvalues but also for multiple eigenvalues. The algorithm and analysis in this paper are suitable for the Steklov eigenvalue problem in three dimensional domains and also suitable for general second-order self-adjoint elliptic eigenvalue problems.



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*Source: 791298-2015-08-20.xml*