# The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

Authors: Fubiao Lin; Junying Cao; Zhixin Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.


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## Body

## 1. Introduction

As an important equation in quantum mechanics, Schrödinger eigenvalue problems have important physical background modern electronic structure computations [1, 2]. Thus, finite element methods for solving this problem become an important topic which has attracted the attention of mathematical and physical fields: a priori error estimate is discussed in [3], some posteriori error estimates and adaptive algorithms have been studied in [4–7], and, in addition, it also includes two-scale method [8–12] and the extrapolation methods [13–16].It is worth noting that some researchers [5, 6, 17] constructed a series of efficient algorithms to solve PDE eigenvalue problems with angular singularity. For elliptic boundary value problem, Xu and Zhou [18] combined two-grid finite element discretization scheme with the local defect correction to propose a general and powerful parallel-computing technique. This technique has been used and developed by many scholars, for instance, it can be used to solve Stokes equation (see [19, 20]), Especially, Xu and Zhou [21], Dai and Zhou [22], and Bi et al. [23–25] developed this method and established local and parallel three-scale finite element discretizations for symmetric elliptic singular eigenvalue problems.As a matter of fact, due to the influence of Coulomb potentials, the convergence order of three-dimensional numerical methods and the computational efficiency of numerical methods will further deteriorate [26]. Therefore, one of the most direct and effective methods is to transform the three-dimensional problem into one-dimensional problem. Inspired by [27–29] and others references, it is necessary to further study the high-precision numerical method for singular problems. Therefore, in this paper, we turn to discuss finite element multiscale discretization based on local defect correction. We further apply local defect-correction technique proposed by Xu and Zhou to Schrödinger eigenvalue problems, and our work has the following features. (1) We first extend local and parallel three-scale finite element discretizations for symmetric eigenvalue problems established by Dai and Zhou [22] to solve Schrödinger eigenvalue problem. (2) Based on [23], we establish a new multiscale finite element discretization method by local refinement, and this scheme repeatedly makes defect correction on finer and finer local meshes to make up for accuracy loss caused by abrupt changes of local mesh size in three-scale scheme. (3) For the two-scale algorithms in [8, 10], we prove the local error estimates of eigenfunctions. (4) Our scheme is simple and easy to carry out, and theoretical analysis and numerical experiment verify its efficiency to solve the singular Schrödinger eigenvalue problem.The rest of this paper is organized in the following way. in Section2, we will briefly introduce Schrödinger eigenvalue problem and the associated dimension reduction scheme. In Section 3, we will establish the multiscale finite element method. The error estimates of eigenvalues and eigenfunctions will be studied in Section 4. Several numerical experiments are presented in Section 5 to demonstrate the accuracy and efficiency of our algorithm. Some concluding remarks are given in Section 6.


## 2. Dimension Reduction Scheme

Consider the Schrödinger eigenvalue problem:(1)−12Δψ+Vψ=λψ,inℝ3,(2)limx⟶∞ψ=0,where V is the effective potential and x=x1,x2,x3∈ℝ3.Applying a truncation from a unbounded domainℝ3 to a bounded spherical domain B3:=x∈ℝ3:x<R, we find(3)−12Δψ+Vψ=λψ,inB3,ψ=0,on∂B3.Using the spherical coordinate transformation [27], problems (1) and (2) are equivalent to(4)−12∂tt+12∂tuk+kk+12uk+R24t+12V0ruk=λkR24t+12uk,(5)uk1=0.This problem has singularities towardst=−1.Next, we introduce the weighted Sobolev spaces onΩ:=−1,1:(6)Lω2Ω≔v:∫Ωωv2dt<∞,Hω,k1Ω≔v:∂tmv∈Lω2Ωifk=0,∂tmv∈Lωm2Ωifk≥1,m=0,1,v1=0,with ωt:=t+12. For simplicity of notations in the reminder, we omit the subscript k in uk and λk and denote Hω,k1Ω by Hω1Ω for short.The variational form of (4) and (5) is to find λ∈ℝ and nonzero u∈Hω1Ω, satisfying(7)au,v=λbu,v,∀v∈Hω1Ω,where(8)au,v=∫Ω12t+12u′v′+kk+12uv+R24t+12V0r+μuvdt,bu,v=∫ΩR24t+12uvdt,with μ>0. According to Theorem 1 in [27], a⋅,⋅ and b⋅,⋅ define the inner product in Hω1Ω and Lω2Ω, respectively. Let ⋅a and ⋅b be the norm induced by the inner products a⋅,⋅ and b⋅,⋅. Let ⋅a and ⋅b.ForD⊂Ω0⊂Ω, we use D⊂⊂Ω0 to mean that dist∂D\∂Ω,∂Ω0\∂Ω>0.Assume thatπhΩ=τ is a mesh of Ω with mesh-size function hx whose value is the diameter hτ of the element τ containing x, and hΩ=maxx∈Ωhx is the mesh diameter of πhΩ. We write hΩ as h for simplicity. Let VhΩ⊂CΩ¯, defined on πhΩ, be a space of piecewise polynomials, and Vh0Ω=VhΩ∩Hω1Ω. Given G⊂Ω, we define πhG and VhG to be the restriction of πhΩ and VhΩ to G, respectively, and(9)Vh0G=v∈Vh0Ω:v|∂G\∂Ω=0,V0hG=v∈Vh0Ω:suppv\∂Ω⊂⊂G.For anyG⊂Ω mentioned in this paper, we assume that it aligns with πhΩ when necessary.In this part,C denotes a positive constant independent of h, which may not be the same constant in different places. For simplicity, we use the symbol x≲y to mean that x≤Cy.We adopt the following assumptions similar as in [18] for meshes and finite element space.(A0) There existsν≥1 such that hΩν≲hx,∀x∈Ω.(A1) There existsr≥1 such that, for w∈Hω1Ω∩H1+tΩ,(10)infv∈Vh0Ωh−1w−vb+w−va≲htw1+t,0≤t≤r.(A2)Inverse Estimate. For any v∈VhΩ1, va,Ω1≲h−1vb,Ω1.(A3)Superapproximation. For G⊂Ω1, let ω˜∈C∞Ω¯ with supp ω˜\∂Ω⊂⊂G. Then, for any w∈VhG, there exists v∈Vh0G such that h−1ω˜w−va,G≲wa,G.The finite element approximation of (7) is given; find λh∈C and uh∈Vh0Ω,uha=1, satisfying(11)auh,v=λhbuh,v,∀v∈Vh0Ω.Define the solution operatorT:Lω2Ω⟶Hω1Ω and Th:Lω2Ω⟶Vh0Ω as follows:(12)aTg,v=bg,v,∀v∈Hω1Ω,(13)aThg,v=bg,v,∀v∈Vh0Ω.Problems (7) and (11) have the equivalent operator forms (14) and (15), respectively:(14)Tu=λ−1u,(15)Thuh=λh−1uh.The following regularity assumption is needed in theoretical analysis. For anyf∈Lω2Ω, Tf∈Hω1Ω∩H1+γΩ satisfies(16)Tf1+γ≤CΩfb.According to [30] and Section 5.5 in [31], the above assumption is reasonable.For someG⊂Ω, we need the following local regularity assumption.R(G): for any f∈Lω2G, there exists a ϕ∈Hω1G∩H1+γG satisfying (17)aϕ,v=bf,v,∀v∈Hω1G,ϕ1+γ,G≤CGfb,G,where CΩ and CG are two priori constants.Define the Ritz projectionPh:Hω1Ω⟶Vh0Ω by(18)au−Phu,v=0,∀v∈Vh0Ω.Then,Th=PhT (see [32]).LetMλ be the space spanned by all generalized eigenfunctions corresponding to λ of T, Mhλ be the space spanned by all generalized eigenfunctions corresponding to all eigenvalues of Th that converge to λ.We also need the lemma as follows (see [8, 10]).Lemma 1.
Letλ,u be an eigenpair of (7). Then, for all w∈H01Ω, w≠0,(19)aw,wbw,w−λ=aw−u,w−ubw,w−λbw−u,w−ubw,w.The a priori error estimates of the finite element approximations (11) can be found in [3, 32].Lemma 2.
Assume thatMλ⊂Hr+sΩ (0<s<1). Then,(20)λh−λ≲h2r+2s−2,and let uh∈Mhλ with uhb=1; then, there is u∈Mλ such that(21)uh−ua≲hr+s−1,uh−ub≲hr+s−1+γ.The authors in [18, 33] studied the local behavior of finite element. The following results are given in [18].Lemma 3.
Suppose thatf∈Lω2Ω and G⊂⊂Ω0⊂Ω. If w∈VhΩ0 satisfies(22)aw,v=bf,v,∀v∈V0hΩ0,then(23)wa,G≲wb,Ω0+fb,Ω0.Proof.
Letp≥2ν−1 be an integer, and let(24)D⊂⊂Ωp⊂⊂Ωp−1⊂⊂⋯⊂⊂Ω1⊂⊂Ω0.
ChooseD1⊂Ω satisfying D⊂⊂D1⊂⊂Ωp and ω˜∈C∞Ω¯ such that suppω˜⊂⊂Ωp and ω˜≡1 on D1¯. Then, from (A3), there exists v∈V0hΩp such that(25)ω˜2w−va,Ωp≲hΩ0wa,Ωp,so we have(26)aw,ω˜2w−v≲hΩ0wa,Ωp2,(27)bf,v≲fb,Ω0vb,Ωp≲fb,Ω0hΩ0wa,Ωp+ω˜wa,Ω.
Sincev∈V0hΩp⊂V0hΩ0, the definition w implies(28)aw,ω˜2w=aw,ω˜2w−v+bf,v.
A simple calculation shows that(29)aω˜w,ω˜w≲aw,ω˜2w+wb,Ω02,∀w∈Hω1Ω.
It follows from (26)–(29) that(30)ω˜wa,Ω2≲aw,ω˜2w+wb,Ω02=aw,ω˜2w−v+wb,Ω02+bf,v≲hΩ0wa,Ωp2+wb,Ω02+fb,Ω0hΩ0w1,Ωp+ω˜wa,Ω,and thus,(31)wa,D≲hΩ01/2wa,Ωp+wb,Ω0+fb,Ω0.
Similarly, we can obtain(32)wa,Ωj≲hΩ01/2wa,Ωj−1+wb,Ω0+fb,Ω0,j=1,2,…,p.
By using (31) and (32), we get from (A0) and (A2) and inverse estimate that(33)wa,D≲hΩ0p+1/2wa,Ω0+wb,Ω0+fb,Ω0≲hΩ0p+1/2h−1wb,Ω0+wb,Ω0+fb,Ω0≲wb,Ω0+fb,Ω0.
This completes the proof.Lemma 4.
Suppose thatG⊂⊂Ω0⊂Ω. Then, the following estimates are valid:(34)hγu−Phua,Ω+u−Phub,Ω≲hγinfv∈Vh0Ωu−va,Ω,(35)u−Phua,G≲infv∈Vh0Ωu−va,Ω0+hγu−Phua,Ω.Proof.
By (15), we obtain(36)Thλhuh−λua≲λhuh−λub.
By the definitions ofT, Th, and Ph, we deduce that(37)λTu=u,λThu=Phu.
LetPhΩ0 be the finite element projection onto V0hΩ0; then,(38)aPhu−PhΩ0u,v=0,∀v∈V0hΩ0.
According to Lemma3, we have(39)Phu−PhΩ0ua,G≲Phu−PhΩ0ub,Ω0.
Then, by using (14) and (39), we conclude that(40)u−Phua,D≲u−PhΩ0u+PhΩ0u−Phua,D≲infv∈Vh0Ωu−va,Ω0+u−Phub,Ω.
Thus, we derive (35) from (39).


## 3. Multiscale Discretizations Based on Local Defect Correction

Consider the eigenvalue problem (7) which has an isolated singular point t=−1 (e.g., see Figure 1).Figure 1
Finite element meshes.LetD⊂⊂Ω be a given subdomain containing the singular point z, and we introduce domains:(41)Ω⊃Ω1⊃Ω2⊃⋯⊃Ωl⊃⊃D.LetπHΩ be a shape-regular grid, which is made up of simplices, with size H∈0,1, πwΩ be a refined mesoscopic shape-regular grid (from πHΩ), and πhΩi be a locally refined grid (from πhi−1Ωi−1) that satisfies h−1=H, h0=w,hi≪hi−1i=0,1,…,l (Figure 1 shows πHΩ, πwΩ, and πh1Ω1. Let VH0Ω,Vw0Ω, and Vhi0Ωi1l be finite element spaces of degree less than or equal to r defined on πHΩ, πwΩ, and πhiΩi1l, respectively.Based on algorithmB0 in [22], we establish the following three-scale discretization scheme.Scheme 1.
(three-scale discretizations based on local defect correction).Step 1: solve (7) on a globally coarse grid πHΩ; find λH∈C,uH∈VH0Ω such that uH0=1 and(42)auH,v=λHbuH,v,∀v∈VH0Ω.Step 2: solve two linear boundary value problems on a globally mesoscopic gridπwΩ; find uw∈Vw0Ω such that(43)auw,v=λHbuH,v,∀v∈Vw0Ω,and then, compute the Rayleigh quotientλw=auw,uw/buw,uw.Step 3: solve two linear boundary value problems on a locally fine gridπh1Ω1; find eh1∈Vh10Ω1 such that(44)aeh1,v=λwbuw,v−auw,v,∀v∈Vh10Ω1.Step 4: set(45)uw,h1=uw+eh1,onΩ¯1,uw,inΩ∖Ω¯1,and compute the Rayleigh quotient:(46)λw,h1=auw,h1,uw,h1buw,h1,uw,h1.
We useλw,h1,uw,h1 obtained by Scheme 1 as the approximate eigenpair of (7).
It is obvious thatλw,uw in Scheme 1 can be viewed as approximate eigenpairs obtained by the two-grid discretization scheme in [8, 10] from πHΩ and πwΩ.
UsingScheme 1, abrupt changes of mesh size will appear near ∂Ω1. Influenced by the technique on the transition layer proposed by [23], we repeatedly use the local defect-correction technique to establish the following multiscale discretization scheme.Scheme 2.
(multiscale discretizations based on local defect correction).Step 1: the same as that of Step 1 ofScheme 1.Step 2: the same as that of Step 2 ofScheme 1.Step 3:uw,h0⇐uw and λw,h0⇐λw.Step 4: fori=1,2,…,l, execute Step 5 and Step 6.Step 5: solve linear boundary value problems on locally fine gridπhiΩi; find ehi∈Vhi0Ωi such that(47)aehi,v=λw,hi−1buw,hi−1,v−auw,hi−1,v,∀v∈Vhi0Ωi.Step 6: set(48)uw,hi=uw,hi−1+ehi,onΩ¯i,uw,hi−1,inΩ∖Ω¯i,and compute(49)λw,hi=auw,hi,uw,hibuw,hi,uw,hi.
We useλw,hl,uw,hl obtained by Scheme 2 as the approximate eigenpair of (7).


## 4. Theoretical Analysis

Next, we shall discuss the error estimates ofSchemes 1 and 2. In our analysis, we introduce an auxiliary grid πhiΩ which is defined globally and denote the corresponding finite element space of degree ≤r by Vhi0Ω (i=1,2,…,l). We also assume that πhiΩi and Vhi0Ωi are the restrictions of πhiΩ and a subspace of Vhi0Ω to Ωi, respectively, and(50)VH0Ω⊂Vw0Ω⊂Vh10Ω⊂Vh20Ω⊂⋯⊂Vhl0Ω.ForD and Ωi stated at the beginning of Section 3, let Gi⊂Ω and F⊂Ω satisfy D⊂⊂F⊂⊂Gi⊂⊂Ωi (i=1,2,…,l).Theorem 1.
Assume thatMλ⊂Hr+sΩ∩Hr+1Ω/D¯ and (1<r+s, 0≤s<1), and H is properly small. Then, there exists u∈Mλ such that(51)uw−ua≲Hr+s−1+γ+wr+s−1,(52)uw−ub≲Hr+s−1+γ,(53)uw−ua,Ω∖F¯≲Hr+s−1+γ+wr,(54)λw−λ≲H2r+2s−2+2γ+w2r+2s−2.Proof.
Letu∈Mλ such that u−uH satisfies Lemma 2. From (12) and (13), Step 2 of Scheme 1, (14), and Lemmas 2 and 4, we derive that(55)uw−ua=λHTwuH−λTua≤λHTwuH−λTwua+λTwu−λTua≲λHuH−λub+λPwTu−Tua≲Hr+s−1+γ+wr+s−1,and then, (51) follows. By Lemmas 2 and 4,(56)uw−ua,Ω/D¯≲λHuH−λub+λPwTu−Tua,Ω/D¯≲Hr+s−1+γ+wr,and then, (53) follows. By calculation,(57)uw−ub=λHTwuH−λTub≤λHTwuH−λTwub+λTwu−λTub≲λHuH−λub+λPwTu−Tub≲Hr+s−1+γ+wr+s−1+γ≲Hr+s−1+γ,and then, (52) follows. From (19), we have(58)λw−λ=auw−u,uw−ubuw,uw−λbuw−u,uw−ubuw,uw.
Note thatuH and uw just approximate the same eigenfuntion u. The combination of (51), (52), and (58) yields (54).
Theorem2 is a critical result in this paper, which develops the results of Theorem 3.3 in [22].Theorem 2.
Assume thatRΩi holds (i=1,2,…,l), u∈Mλ. Then,(59)uw,hl−Phlua≲u−Phlub,Ωl+hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1uw,hl−1−λub+uw,hl−1−Phlua,Ω\Gl¯+uw,hl−1−ua,Ωl\F¯,l≥1.Proof.
Due to the inequality(60)uw,hl−Phlua,Ω≲uw,hl−Phlua,D+uw,hl−Phlua,Gl\D¯+uw,hl−Phlua,Ω\Gl¯,we shall estimate uw,hl−Phlua,D, uw,hl−Phlua,Gl\D¯, and uw,hl−Phlua,Ω\Gl¯, respectively.
First, we proceed to estimateuw,hl−Phlua,D. From (18), (47), and (48), we derive(61)auw,hl−Phlu,v=auw,hl,v−aPhlu,v=auw,hl−1+ehl,v−au,v=λw,hl−1buw,hl−1,v−λbu,v,∀v∈Vhl0Ωl.
It is obvious that(62)λw,hl−1buw,hl−1,v−λbu,v=λw,hl−1−λbu,v+λw,hl−1buw,hl−1−u,v,∀v∈H01Ω,which together with (61) yields(63)auw,hl−Phlu,v=λw,hl−1−λbu,v+λw,hl−1buw,hl−1−u,v,∀v∈Vhl0Ωl.
Sinceuw,hl−Phlu|Ωl∈VhlΩl and V0hlΩl⊂Vhl0Ωl, from the above formula and Lemma 3, we deduce that(64)uw,hl−Phlua,D≲uw,hl−Phlub,Ωl+λw,hl−1−λ+uw,hl−ub,Ωl.
By calculation, we have(65)uw,hl−Phlub,Ωl≤uw,hl−1−Phlub,Ωl+ehlb,Ωl≤u−Phlub,Ωl+u−uw,hl−1b,Ωl+ehlb,Ωl.
Substituting the above relation in (64), we obtain(66)uw,hl−Phlua,D≲λw,hl−1−λ+uw,hl−1−ub,Ωl+u−Phlub,Ωl+ehlb,Ωl.
To estimateehlb,Ωl, we use the Aubin–Nitsche duality argument. For any given f∈L2Ωl, consider the boundary value problem; find φ∈HΓ1Ωl:=v∈Hω1Ωl:v|∂Ωl\−1=0 such that(67)aφ,v=bf,v,∀v∈HΓ1Ωl.
Letφ be the generalized solution of (67) and φhl and φhl−1 be finite element solutions of (67) in Vhl0Ωl and Vhl−10Ωl, respectively. Then,(68)φ−φhla,Ωl≲hlγfb,Ωl,φ−φhl−1a,Ωl≲hl−1γfb,Ωl.
From (47) and (48), we obtain(69)auw,hl,φhl=λw,hl−1buw,hl−1,φhl.
Then, by the definitions ofφ, φhl, and ehl, we deduce that(70)behl,f=aehl,φ=aehl,φhl=auw,hl−uw,hl−1,φhl=aPhlu−uw,hl−1,φhl+auw,hl,φhl−aPhlu,φhl=aPhlu−uw,hl−1,φhl+λw,hl−1buw,hl−1,φhl−λbu,φhl=aPhlu−uw,hl−1,φhl−φ+aPhlu−uw,hl−1,φ−φhl−1+aPhlu−uw,hl−1,φhl−1+λw,hl−1buw,hl−1,φhl−λbu,φhl≲hl−1γPhlu−uw,hl−1a,Ωlf0,Ωl+aPhlu−uw,hl−1,φhl−1+λw,hl−1buw,hl−1,φhl−λbu,φhl.
Step 2 ofScheme 2 shows that(71)auw,h0,φh0=λw,h−1buw,h−1,φh0,namely, for l=1,(72)auw,hl−1,φhl−1=λw,hl−2buw,hl−2,φhl−1,for l>1, the above formula follows from (47) and (48). Therefore,(73)aPhlu−uw,hl−1,φhl−1=au−uw,hl−1,φhl−1=λbu,φhl−1−auw,hl−1,φhl−1=λbu,φhl−1−λw,hl−2buw,hl−2,φhl−1≲λu−λw,hl−2uw,hl−2b,Ωlfb,Ωl.
It is clear that(74)λw,hl−1buw,hl−1,φhl−λbu,φhl≲λw,hl−1uw,hl−1−λub,Ωlfb,Ωl.
Substituting the above two formulae in (70), we derive(75)behl,f≲hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1uw,hl−1−λubfb,Ωl.
Thus, we obtain(76)ehla,Ωl≲hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1uw,hl−1−λub.
Substituting (76) in (66), we obtain(77)uw,hl−Phlua,D≲u−Phlub,Ωl+hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1−λ+uw,hl−1−ub.
Similarly, sinceGl/D¯⊂⊂Ωl, we deduce(78)uw,hl−Phlua,D≲u−Phlub,Ωl+hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1−λ+uw,hl−1−ub.
The remainder is to analyzeuw,hl−Phlua,Ω\G¯. From (48), we see that(79)uw,hl−Phlua,Ω\Ω¯l=uw,hl−1−Phlua,Ω\Ω¯l,which leads to(80)uw,hl−Phlua,Ω\Gl¯≤uw,hl−Phlua,Ω\Ω¯l+uw,hl−1−Phlua,ΩlGl¯+ehla,Ωl\Gl¯≲uw,hl−1−Phlua,Ω\Gl¯+ehla,Ωl\Gl¯.
It follows from (7), (47), and (62) that(81)aehl,v=λw,hl−1buw,hl−1,v−auw,hl−1,v−λbu,v+au,v=λw,hl−1−λbu,v+λw,hl−1buw,hl−1−u,v−auw,hl−1−u,v,∀v∈Vh0Ωl.
Then, by Lemma3, we have(82)ehla,Ωl/G¯l≲ehlb,Ωl/F¯+λw,hl−1−λ+uw,hl−1−ua,Ωl/F¯,where F⊂Ω satisfies D⊂⊂F⊂⊂Gl. Substituting (82) in (80), we obtain(83)uw,hl−Phlua,Ω/Gl¯≲uw,hl−1−Phlua,Ω/Gl¯+ehlb,Ω/F¯+λw,hl−1−λ+uw,hl−1−ua,Ω/F¯.
It follows from substituting (76) in the above inequality that(84)uw,hl−Phlua,Ω/Gl¯≲uw,hl−1−Phlua,Ω/Gl¯+hl−1γPhlu−uw,hl−1a,Ωl+λu−λw,hl−2uw,hl−2b,Ωl+λw,hl−1uw,hl−1−λub+λw,hl−1−λ+uw,hl−1−ub,Ω/F¯.
Combining (60), (77), (78), and (84), finally, we obtain (59).Theorem 3.
Assume that the conditions of Theorem1 hold. Then, there exists u∈Mλ such that(85)uw,h1−ua,Ω≲h1r+s−1+wr+Hr+s−1+γ,(86)uw,h1−ub,Ω≲wr+Hr+s−1+γ,(87)uw,h1−ua,Ω/F¯≲wr+Hr+s−1+γ,(88)λw,h1−λ≲h12r+2s−2+w2r+H2r+2s−2+2γ.Proof.
Letu∈Mλ such that u−uH satisfies Lemma 2. From Theorem 2, we know l=1, h−1=H,h0=w, uw,h0=uw, λw,h0=λw, uw,h−1=uH, and λw,h−1=λH; thus, we obtain(89)uw,h1−Ph1ua,Ω≲u−Ph1u0,Ω1+wγPh1u−uwa,Ω1+λu−λHuHb,Ω1+λwuw−λub+uw−Ph1ua,Ω\G1¯+uw−ua,Ω1\F¯.
Using Lemma4, Theorem 1, and Lemma 2 to estimate the terms at the right-hand side of the above formula gives(90)uw,h1−Ph1ua,Ω≲h1r+s−1+γ+wγwr+s−1+Hr+s−1+γ+wr+s−1+γ+wr+s−1+γ+wr+wr+s−1+γ+wr≲Hr+s−1+γ+wr.
Combining (35) and (39) yields (85), (86), and (87). From (19), we have(91)λw,h1−λ=auw,h1−u,uw,h1−ubuw,h1,uw,h1−λbuw,h1−u,uw,h1−ubuw,h1,uw,h1.
Combining (85), (86), and (91) yields (88).Theorem 4.
Under the conditions of Theorem1, we further assume that RΩi holds (i=1,2,…,l), and(92)wr=OHr+s−1+γ,hlr+s−1>˜Hr+s−1+γ.Then, there existsu∈Mλ such that(93)uw,hl−ua,Ω≲hlr+s−1,(94)uw,hl−ub,Ω≲Hr+s−1+γ,(95)uw,hl−ua,Ω∖F¯≲Hr+s−1+γ,(96)λw,hl−λ≲hl2r+s+s−2.Proof.
Letu∈Mλ such that u−uH satisfies Lemma 2. The proof of (93)–(96) is completed by induction. When l=1, Scheme 2 is actually Scheme 1. Hence, from Theorems 1 and 3 and (92), we know that (93)–(96) hold for l=0,1.
Suppose (93)–(96) hold for l−2 and l−1, i.e.,(97)uw,hl−2−ua,Ω≲hl−2r+s−1,uw,hl−2−ub,Ω≲Hr+s−1+γ,uw,hl−2−ua,Ω∖F¯≲Hr+s−1+γ,λw,hl−2−λ≲hl−22r+s+s−2,uw,hl−1−ua,Ω≲hl−1r+s−1,uw,hl−1−ub,Ω≲Hr+s−1+γ,uw,hl−1−ua,Ω∖F¯≲Hr+s−1+γ,λw,hl−1−λ≲hl−12r+s+s−2.
Next, we shall prove that (93)–(96) hold for l. Using the above formula and Lemma 4 to estimate the terms at the right-hand side of (59) gives(98)uw,hl−Phlua,Ω≲hlr+s−1+γ+hl−1γhlr+s−1+hl−1r+s−1+Hr+s−1+γ+Hr+s−1+γ+Hr+s−1+γ+hlr+Hr+s−1+γ≲Hr+s−1+γ.
The combination of (35), (39), and (98) yields (93), (94), and (95). From (19), we have(99)λw,hl−λ=auw,hl−u,uw,hl−ubuw,hl,uw,hl−λbuw,hl−u,uw,hl−ubuw,hl,uw,hl.
The combination of (93), (94), and (99) yields (96).


## 5. Numerical Experiments

We will report some numerical experiments by using linear finite element and quadratic spectral element on uniform meshes. In our numerical experiments, we useScheme 2 to solve the problem such that Ωi=−1,−1+1/2i×3/2, i=0,1,2,…, and locally fine grids have the same degree of freedom as that of globally mesoscopic grid (see Tables 1–3).Table 1
V=−1/r.
DOFHDOFwλHλwλw,h1λw,h2λw,h3λw,h4416−0.3558419−0.4662707−0.4903221−0.4963517−0.4975584−0.4976819864−0.4320788−0.4974509−0.4993601−0.4997939−0.4998737−0.499881616256−0.4735824−0.4998515−0.4999630−0.4999897−0.4999946−0.4999951321024−0.4918086−0.4999910−0.4999978−0.4999994−0.4999997−0.4999997416−0.10798797−0.12015585−0.12343262−0.12435602−0.12457107−0.12462204864−0.11060517−0.12445755−0.12476411−0.12484341−0.12486019−0.1248633416256−0.12023285−0.12497717−0.12499349−0.12499748−0.12499831−0.12499849321024−0.12372913−0.12499867−0.12499966−0.12499990−0.12499995−0.12499996Table 2
V=r2/2.
DOFHDOFwλHλwλw,h1λw,h2λw,h3λw,h44162.35680871.78075191.56737551.51762951.51456511.51424578641.68710721.50410811.50102341.50025691.50017621.5001727162561.55887581.50025401.50006341.50001591.50001091.50001073210241.51553541.50001571.50000391.50000101.50000071.50000074163.07449592.60469122.52841622.50698412.50398412.50366258643.07227702.54184242.51423382.50540252.50458112.5045358162562.58908162.50040652.50010182.50002602.50001492.50001373210242.52452632.50002472.50000622.50000162.50000092.5000008Table 3
V=r.
DOFHDOFwλHλwλw,h1λw,h2λw,h3λw,h44162.23050591.92092111.87295011.86111241.86014971.86009378641.97854151.85848701.85644501.85593691.85588941.8558874162561.89391591.85592261.85579871.85576781.85576481.85576463210241.86580881.85576731.85575961.85575771.85575751.85575754162.92970732.70990462.68018142.67149832.67009072.66996438642.85823832.67352852.66993392.66885372.66867112.6686594162562.70939092.66801562.66787662.66784302.66783752.66783703210242.67922842.66784092.66783232.66783032.66782992.6678299In our experiments, the parameterμ is taken to be 1. We set R=40 for the eigenvalue problem with V=−1/r and l=1 and R=15 for the other cases. The coarse mesh size and the mesoscopic mesh size satisfy ω=H2 which means DOFw=DOFH2.We use MATLAB 2011b under the package of Chen (see [34]) to solve the problem, and the numerical results are shown in Tables 1–3. This tables corresponds to the results of different potential energy V. From these tables, we can see that, without increasing degree of freedom on locally fine grids, the first local defect correction can largely improve the accuracy of the eigenvalue, and the local defect corrections that follows can gradually improve the accuracy of the eigenvalue by overcoming the singularity at the origin. Here, we set(100)DOFw=DOFΩi,i=1,2,….In Figure2, we also plot the error curve of numerical eigenvalues obtained by multiscale Scheme 2. It can be seen that using coarse finite element space with the mesh size H, the error of the finite element eigenvalues is around poor accuracy 10−2. After performing the two-grid iteration, the error of the finite element eigenvalues can be increased from 10−2 to 10−5. The final mutiscale iteration can improve the error up to 10−6∼10−7. These figures show the accuracy and effectiveness of our numerical scheme.Figure 2
Error curve of numerical eigenvalues obtained by multiscaleScheme 2. (a) V=−1/r,λ=−0.5. (b) V=−1/r,λ=−0.125. (c) V=−r2/2,λ=1.5. (d) V=r2/2,λ=2.5. (e) V=r,λ=1.855757081489. (f) V=r,λ=2.667829482852.
(a)(b)(c)(d)(e)(f)


## 6. Conclusion

In this paper, we developed a efficient multiscale finite element method for solving the Schrödinger eigenvalue problem with three-dimensional domain. Our scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. The error estimates of eigenvalues and eigenfunctions are proved. Some numerical examples are presented to verify the effectiveness of our numerical method.



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