FINITE ELEMENT APPROXIMATIONS TO THE DISCRETE ... - LSEC

FINITE ELEMENT APPROXIMATIONS TO THE DISCRETE ¨ SPECTRUM OF THE SCHRODINGER OPERATOR WITH THE COULOMB POTENTIAL WEIYING ZHENG∗ AND LUNG-AN YING† Abstract. In the present paper, the authors consider the Schr¨ odinger operator H with the Coulomb potential defined in R3m , where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain approximation to the discrete spectrum of H is obtained. Since numerical solution is difficult for a higher-dimensional problem of dimension more than three, the finite element analyses in this paper are restricted to the S-state of the helium atom. The authors transform the six-dimensional Schr¨ odinger equation of the helium S-state into a three-dimensional form. Optimal error estimates for the finite element approximation to the three-dimensional equation, for all eigenvalues and eigenfunctions of the three-dimensional equation, are obtained by means of local regularization. Numerical results are shown in the last section. Key words. spectrum approximation, Schr¨ odinger equation, weighted norm, local regularization, finite element method AMS subject classifications. 65N30, 65N25, 81Q05

1. Introduction. The multielectron Coulomb problem in quantum mechanics cannot be solved in a finite form. Nevertheless it challenges and stimulates many mathematicians and physicists to devote themselves to developing efficient methods for solving the system. Several successful approximation techniques in quantum physics/chemistry have been developed for this problem. They include the Hartree–Fock method [15], the finite difference method [19], [35], the correlation-function hyperspherical-harmonic method [18], [24], and various variational approximations. For the Hartree–Fock method, every electron is considered independently to be in a central electric field formed by the nucleus and other electrons. The finite difference method needs a rectangular domain in RN and uniform grids. The double and triple basis set methods (which are variational methods indeed) are very powerful for the eigenvalue problem of the helium atom. Kono and Hattori (see [21], [22]) used two sets of basis funck −ξr1 −ηr2 k −ζ(r1 +r2 ) tions r1i r2j r12 A (“ξ terms”) and r1i r2j r12 e e A (“ζ terms”) to calculate the energy levels for the S, P , and D states of the helium atom. (A is an appropriate angular factor.) The former set of functions is expected to describe the whole wave function roughly, while the latter is expected to describe the short- and middle-range correlation effects. Their calculations yield 9–10 significant digits for S states. Kleindienst, L¨ uchow, and Merckens [20] and Drake and Yan [12] applied the double basis k −ξ1 r1 −η1 r2 e and set method to S-states of helium. Their basis functions are r1i r2j r12 i j k −ξ2 r1 −η2 r2 . Drake and Yan employed truncations to ensure numerical stabilr1 r2 r12 e ity and convergence. By complete optimization of the exponential scale factors ξ1 , η1 , ξ2 , and η2 , they achieved more than 15 significant digits. Recently, Drake, Cassar, and Nistor [13] obtained 21 significant digits for the ground state of helium by the triple basis set method. Korobov [23] even obtained 25 significant digits for the ground state of helium. That work can be used as a benchmark for other approaches ∗ School of Mathematical Sciences, Peking University, Beijing, 100871 China (zhengweiying @yahoo.com, [email protected]).

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WEIYING ZHENG AND LUNG-AN YING

for three-body systems. All three of these excellent works in variational methods promote the development of few-body problems in quantum mechanics. The finite element method (FEM) is used initially in elastic mechanics and fluid mechanics. It uses local interpolation functions to approximate the unknown function (see [8] and [36]), and thus can describe the local properties of wave functions. Therefore, we can expect to obtain good approximations to the energy. Important works on FEM applied to atomic and molecular problems first appeared in 1975 (see [3]). They were devoted to one- or two-dimensional problems [3], [4], [14], [16], [30]. In 1985, Levin and Schertzer [25] published the first work applying FEM to three-dimensional problems. They calculated the ground state of the helium atom. Most of the works applying FEM to three-body problems have appeared since 1990 (i.e., [1], [6], [31], [37]). All of the cited works obtained very good results. To apply FEM to quantum mechanics, we should consider three aspects of the problem. The first is the spectrum approximation of the whole space Schr¨odinger operator by the operator defined on some bounded domain. The second is the error estimate for FEM approximation. The third is the real computation of approximate solutions. To the best of our knowledge, we have not found any work analyzing the first two aspects. In this paper, we consider the first aspect for the system of an arbitrary atom. But as for the finite element aspect, since any problem of dimension more than three is a great challenge for both modern numerical methods and computers, we restrict the finite element analysis and computation to the S-state of the helium atom. In fact, we can transform the 3m-dimensional Schr¨odinger equation (see [38] for m = 2, 3) into a 3(m − 1)-dimensional form rigorously, and theoretical analysis of the FEM applied to the simplified equation can be obtained similarly, in view of the argument of sections 3, 4, and 5 in the present paper. However, real computations are very difficult to carry out because of numerous degrees of freedom. Our numerical results on the lithium atom (the Schr¨ odinger equation is nine-dimensional) will appear in another paper [39]. The present paper consists of three parts. First, we consider the bounded domain approximation of the 3m-dimensional Schr¨odinger equation (m is the number of electrons in an ion). The spectrum of the Schr¨odinger operator H consists of the discrete spectrum included in (−a, 0)(for some a > 0) and the continuous spectrum [0, +∞). We show that the spectrum of H becomes completely discrete if it is restricted to bounded domains. In section 2, we show that for any eigenvalue of the whole space problem and for any ² > 0, assuming the bounded domain large enough, there is an eigenpair of the bounded domain problem such that the errors of both the eigenvalue and the eigenfunction are smaller than ². Secondly, we analyze the finite element approximation of the S-state of the helium atom. In section 3, the six-dimensional Schr¨odinger equation is transformed into a three-dimensional form, and some Hilbert spaces with weighted inner products and norms are defined. Because we cannot say that the solutions of both the three-dimensional equation and the six-dimensional one are continuous, the technique of local regularization [25] is used to prove the convergency of the finite element scheme. In section 4, we describe the three-dimensional local regularization operator in detail. In section 5, an equivalent variational equation of the three-dimensional equation and its FEM approximation are given for the helium atom of the S-state. The optimal order error estimate of the finite element scheme is obtained. Thirdly, we have calculated approximate solutions by the finite element scheme. In section 6, we give the numerical results from two kinds of FEM approximations to the three-dimensional energy equation. The results are better than

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

51

existing finite element results. Furthermore, from the figures we can see that our approximate wave functions coincide very well with many physical properties well known to physicists, and with many essential physical assumptions in quantum mechanics which are not added into our computations a priori. The difficulties appear in three aspects: 1. the proof of the continuity and coercivity of the bilinear forms in the variational equations with the presence of the singularities in the Coulomb potential, 2. the proof of the convergency of the finite element scheme while the variational spaces are not standard Sobolev spaces, and 3. obtaining precise results in presence of numerous unknowns and singular integrals. Through the paper, C represents the generic constant independent of minded parameters; the symbol “⇐⇒” means “be equivalent to.” We use atomic units except where explicitly explained, i.e., Bohr radius a0 for length, Rydberg (Hartree only in section 6) for energy. We consider the nonrelativistic and spin-independent case. 2. Discrete spectrum approximations of the Schr¨ odinger operator in bounded domains. Let m > 0 be an integer, N = 3m. The Schr¨odinger equation of an m-electron ion is (2.1)

Hψ = Eψ

in RN ,

where Hψ = −∆ψ + V ψ, ¶ m µ 2 X ∂2 ∂2 ∂ + + , ∆ψ = ∂x2i ∂yi2 ∂zi2 i=1 m X 2Z

X

2 ; r ij i=1 1≤i 0 large enough and define ¯q ½ ¾ ¯ Bi (0, R) = (xi , yi , zi ) ∈ R3 ¯¯ x2i + yi2 + zi2 ≤ R , ¯q ½ ¾ ¯ ΩR = (x(1) , . . . , x(m) ) ∈ RN ¯¯ x2i + yi2 + zi2 ≤ R, i = 1, 2, . . . , m . Assuming supp v ⊂ ΩR , by H¨older’s inequality there exists a positive constant C, depending on R and p, such that µ ¶p Z µ ¶p Z v v (i) (2.8) dx = dx(i) ≤ Ckvkp0,6,Bi (0,R) ≤ Ckvkp1,Bi (0,R) . ri ri R3 Bi (0,R) Integrating (2.8) with respect to the rest of the variables produces the following: Z µ ¶p Z v d˜ x(i) ≤ C kvkp1,Bi (0,R) d˜ x(i) ≤ Ckvkp1,RN . ri RN RN −3

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

53

(2) For any 1 ≤ i < j ≤ m, let ξ (i) = x(i) − x(j) , ξ (i) = (ξi , ηi , ζi ), x(i) = (xi , yi , zi ). For any v = v(x(1) , . . . , x(m) ) ∈ H 1 (RN ), define vij (x(1) , . . . , ξ (i) , . . . , x(m) ) = v(x(1) , . . . , ξ (i) + x(j) . . . , x(m) ), By Tonelli’s theorem [34], we have Z Z v 2 (x(1) , . . . , x(i) , . . . , x(m) )dx(i) d˜ x(i) RN −3 R3 Z Z 2 = vij (x(1) , . . . , ξ (i) , . . . , x(m) )dξ (i) d˜ x(i) N −3 3 R ZR 2 = vij dx ∀v ∈ H 1 (RN ). RN

Thus vij ∈ L2 (RN ). In the same way, we have ∂vij , ∂xk Since

∂vij ∂ξi

Z

∂v ∂vij ∂xi , ∂xj

=

∂v ∂xi

∂vij ∈ L2 (RN ), ∂zk +

∂v ∂xj ,

1 ≤ k ≤ N,

k 6= i, j.

we have

¯ ¯ ¯ ¯ Z Z ¯ Z Z ¯ ¯ ∂v ¯2 (i) (i) ¯ ∂v ¯2 (i) (i) ¯ ∂v ¯2 (i) (i) ¯ ¯ ¯ dξ d˜ ¯ ¯ dx d˜ ¯ x = x = x , ¯ ¯ ¯ ¯ ¯ dξ d˜ ¯ R3 ∂ξi RN −3 R3 ∂xi RN −3 R3 ∂xi

Z

RN −3

i.e.,

=

∂vij , ∂yk

∂vij ∂ξi

∈ L2 (RN ). Similarly, we have ∂vij , ∂ηi

∂vij , ∂ζi

∂vij , ∂xj

∂vij , ∂yj

∂vij ∈ L2 (RN ). ∂zj

Therefore vij ∈ H 1 (RN ). In the same way, if u ∈ W 2,q (RN ), we have uij ∈ W 2,q (RN ), 1 ≤ i, j, ≤ m. By (1), v/rij = vij /|ξ(i) |, u/rij = uij /|ξ(i) | satisfy (2.3)–(2.5). ˆ x) = Let σ > 0 be a constant, x ˆ = (ˆ x(1) , . . . , x ˆ(m) ) = σx, x ˆ(i) = (ˆ xi , yˆi , zˆi ), ψ(ˆ N ψ(σx) for any x ∈ R . Then (2.1) becomes (2.9)

ˆ ψˆ − σ −σ 2 ∆

m X 2Z ψˆ i=1

rˆi

X

+σ

1≤i 0, since ψk ∈ H 1 (RN ), there exists a φk ∈ C0∞ (RN ) such that kψk − φk k1,RN < ²/l. Set R large enough such that ∪lk=1 suppφk ⊂ B = B(0, R); then (2.21)

l X

εB (λi ) ≤

kψk − φk k1,RN < ².

k=1

In view of (2.18)–(2.21), we have (2.16) and (2.17). 3. Weighted norms and Hilbert spaces. We consider the eigenvalue problem of the S-state of the helium atom, i.e., Z = m = 2. The eigenvalue equations of the Hamiltonian and the square of the angular momentum are µ

¶ 4 4 2 − − ψ = Eψ, −∆1 ψ − ∆2 ψ + r12 r1 r2 " ¶# 2 " X ¶# 2 2 µ 2 µ  X ∂ ∂ ∂ ∂ yi − zi + zi − xi  ∂zi ∂yi ∂xi ∂zi i=1 i=1

(3.1)

(3.2)

" +

2 µ X i=1

xi

∂ ∂ − yi ∂yi ∂xi

¶# 2

  + l(l + 1) ψ = 0, 

where l = 0, 1, . . . . Let θ0 , φ, φ0 be three Euler angles such that (r1 , θ0 , φ0 ) are the spherical coordinates of the first electron in the fixed system o − xyz, φ is the interfractial angle between the r1 −z plane and the r1 −r2 plane, and θ is the interelectronic angle. We introduce the Hylleraas–Breit transform [7]:  x1 = r1 sin θ0 cos φ0 ,      y1 = r1 sin θ0 sin φ0 ,     z = r cos θ0 , 1 1 0 0 0 0 0  x = r 2 (sin θ cos θ cos φ cos φ − sin θ sin φ sin φ + cos θ sin θ cos φ ),  2     y2 = r2 (sin θ cos φ cos θ0 sin φ0 + sin θ sin φ cos φ0 + cos θ sin θ0 sin φ0 ),    z2 = r2 (cos θ cos θ0 − sin θ sin θ0 cos φ).

(3.3)

We can transform (3.1) and (3.2) into the following forms by (3.3): (3.4)

L(ψ) − ·

(3.5)

∂2 1 0 ∂ + 0 2 + ctgθ 0 ∂θ ∂θ sin2 θ0

µ

A1 (ψ) A2 (ψ) − = Eψ, r12 r22

∂2 ∂2 + ∂φ2 ∂φ0 2

¶

¸ 2 cos θ0 ∂ 2 + l(l + 1) ψ = 0, − sin2 θ0 ∂φ∂φ0

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

57

where µ ¶ µ ¶ 1 ∂ 1 ∂ 2 ∂ψ 2 ∂ψ r − r 1 2 r12 ∂r1 ∂r1 r22 ∂r2 ∂r2 µ ¶ µ ¶ µ ¶ 1 1 ∂ ∂ψ 2 4 4 1 − + 2 sin θ + − − ψ, r12 r2 sin θ ∂θ ∂θ r12 r1 r2 µ ¶ 1 ∂ ∂2ψ 0 ∂ψ A1 (ψ) = sin θ + (ctg2 θ + ctg2 θ0 + 2ctgθctgθ0 cos φ) 2 0 0 0 sin θ ∂θ ∂θ ∂φ L(ψ) = −

+

∂2ψ sin φ ∂ 2 ψ ∂2ψ 1 ∂2ψ − 2 cos φ − 2 + 2 sin φctgθ 0 sin θ0 ∂φ 2 ∂θ∂θ0 sin θ0 ∂θ∂φ0 ∂φ∂θ0

+ 2ctgθ0 sin φ A2 (ψ) =

∂2ψ 2 ∂2ψ 0 − (ctgθ + ctgθ cos φ) , ∂θ∂φ sin θ0 ∂φ∂φ0

1 ∂2ψ . sin2 θ ∂φ2

For the S-state of the helium atom, l = 0; then (3.5) has only constant solutions. Thus any wave function u of the S-state depends on only three variables r1 , r2 , and θ. Therefore, we can transform (3.4) of the S-state into a three-dimensional form (3.6)

L(u) = Eu.

Assume that Ω ⊂ X = [0, +∞) × [0, +∞) × [0, π] is a bounded domain. u = u(r1 , r2 , θ), v = v(r1 , r2 , θ). We define inner products, norms, and Hilbert spaces as follows: Z Z (u, v)0 = uvr12 r22 sin θdr1 dr2 dθ, kuk20,r,Ω = u2 r12 r22 sin θdr1 dr2 dθ, Ω

Ω

¸ Z · ∂u ∂v ∂u ∂v ∂u ∂v (u, v)1 = (u, v)0 + r12 r22 sin θdr1 dr2 dθ, +r12 r22 +(r12 + r22 ) ∂r1 ∂r1 ∂r2 ∂r2 ∂θ ∂θ Ω ¶2 ¶2 µ µ µ ¶2 # Z " ∂u ∂u ∂u 2 2 2 2 2 2 2 sin θdr1 dr2 dθ, |u|1,r,Ω = r1 r2 + r1 r2 + (r1 + r2 ) ∂r ∂r ∂θ 1 2 Ω |u|22,r,Ω

¯ ¯ ¯ ∂u ¯2 ¯ ¯ =¯ ∂r1 ¯

1,r,Ω

¯ ¯ ¯ ∂u ¯2 ¯ ¯ +¯ ∂r2 ¯

1,r,Ω

¯ ¯2 ¯ ∂u ¯ + ¯¯ ¯¯ ∂θ

kuk2i+1,r,Ω = kuk2i,r,Ω + |u|2i+1,r,Ω , © ª Hri (Ω) = v| kvk2i,r,Ω < +∞ , Hr2 (Ω)

,

1,r,Ω

i = 0, 1, i = 0, 1,

½ ¾ Z 2 2 2 2 = v|kvk2,r,Ω < +∞, u (r1 + r2 ) sin θdr1 dr2 dθ < ∞ . Ω

Lemma 3.1. Assume that Ω ⊂ X is a bounded domain; then (3.7)

Hr1 (Ω) ,→,→ Hr0 (Ω).

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WEIYING ZHENG AND LUNG-AN YING

Furthermore, if there exists a constant dΩ > 0, such that for any (r1 , r2 , θ) ∈ Ω, r1 , r2 ≥ dΩ , then Hr2 (Ω) ,→,→ Hr1 (Ω).

(3.8)

Proof. (1): Proof of (3.7). Let x = (x1 , y1 , z1 , x2 , y2 , z2 ) = H(r1 , r2 , θ, θ0 , φ, φ0 ) ˆ ⊂ R6 be a bounded be the Hylleraas–Breit transform defined by (3.3), and let Ω domain defined by (3.9)

ˆ = {x | x = H(r1 , r2 , θ, θ0 , φ, φ0 ), (r1 , r2 , θ) ∈ Ω, 0 ≤ θ0 ≤ π, 0 ≤ φ, φ0 ≤ 2π}. Ω

The Jacobian determinant of (3.3) is µ ¶ ∂(x1 , y1 , z1 , x2 , y2 , z2 ) (3.10) det = r12 r22 sin θ sin θ0 . ∂(r1 , r2 , θ, θ0 , φ, φ0 ) By direct calculation, we have kuk2i,r,Ω = 8π1 2 kuk2i,Ωˆ , where k · ki,Ωˆ is the norm of the ˆ i = 0, 1. It is easy to show that Hr0 (Ω) and Hr1 (Ω) standard Sobolev space H i (Ω), 1 are Hilbert spaces, and Hr (Ω) ,→,→ Hr0 (Ω). n (2): Proof of (3.8). Let {vn } be a bounded sequence in Hr2 (Ω); then {vn }, { ∂v ∂r1 }, ∂vn 1 n { ∂v ∂r2 }, { ∂θ } are bounded uniformly in Hr (Ω). By (1), we can choose (successively) ∂vn ∂vn n a subsequence denoted as {vn } too, such that {vn }, { ∂v ∂r1 }, { ∂r2 }, { ∂θ } are Cauchy sequences in Hr0 (Ω). Thus {vn } is a Cauchy sequence in the measure ( kvn k20,r,Ω

)1/2 ¯ ¯ ¯ ¯ ¯ ! Z ï ¯ ∂vn ¯2 ¯ ∂vn ¯2 ¯ ∂vn ¯2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ + . ¯ ∂r1 ¯ + ¯ ∂r2 ¯ + ¯ ∂θ ¯ r1 r2 sin θdr1 dr2 dθ Ω

It is clear that ¯ ¯ Z ¯ Z ¯ ¯ ∂vn ¯2 2 2 ¯ ∂vn ¯2 2 ¯ ¯ (r1 + r22 ) sin θdr1 dr2 dθ ≤ 2 ¯ r r sin θdr1 dr2 dθ, ¯ ¯ ¯ d2Ω Ω ¯ ∂θ ¯ 1 2 Ω ∂θ and so we have ©

ª 2

kvn k21,r,Ω ≤ max 1, 2/dΩ

Ã

° ° ° °2 °2 °2 ! ° ° ° ° ° ° ∂v ∂v ∂v n n n ° ° ° kvn k20,r,Ω + ° +° +° . ° ∂r1 ° ° ∂r2 ° ° ∂θ ° 0,r,Ω 0,r,Ω 0,r,Ω

Therefore {vn } is a Cauchy sequence in Hr1 (Ω) and converges. Remark 3.2. Suppose that Ω is bounded. For any integer k > 0, define    ¯  X ¯ Pk (Ω) = p = αlmn r1l r2m θn ¯ αlmn ∈ R1 , (r1 , r2 , θ) ∈ Ω ;   0≤l+m+n≤k

then Pk (Ω) ⊂ H 2 (Ω) ⊂ Hr2 (Ω). Lemma 3.3. Supposing R > 0 is a constant, there exists a constant C independent of R such that, for any f ∈ H 1 ([0, R]), ÃZ ! Z Z R

(3.11) 0

R

f 2 dx ≤ C max{R2 , R−2 } 0

R

x2 f 2 dx + 0

0

f 2 dx .

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

59

Proof. For any f ∈ C 1 ([R/2, R]) there exists ξ ∈ [R/2, R] such that Z 2 R f (x)dx. f (ξ) = R R/2 Then for any x ∈ [R/2, R], by H¨older’s inequality we have ¯ ¯ Z x Z Z R ¯ ¯ 2 R |f (x)| = ¯¯f (ξ) + f 0 (t)dt¯¯ ≤ |f (t)|dt + |f 0 (t)|dt R ξ R/2 R/2 !1/2 à Z !1/2 à Z R R R 2 2 0 2 |f (t)| dt + |f (t)| dt . ≤ R R/2 2 R/2 Thus there exists a positive constant C independent of R such that √ √ (3.12) |f |0,∞,[R/2,R] ≤ C max{ R, 1/ R}kf k1,[R/2,R] . Set x0 ∈ [R/2, R]. By (3.12), we have, for any x ∈ [0, R] and f ∈ C ∞ ([0, R]), ( ( ) ·Z x ¸2 ) Z R 0 (3.13) f 2 (x) ≤ 2 f 2 (x0 ) + f 0 (x)dx ≤ C f 2 (x0 ) + R f 2 (x)dx x0

ÃZ

≤ C max{R, R−3 }

0

R

Z x2 f 2 (x)dx +

R/2

R

! 0

f 2 (x)dx

∀x ∈ [0, R],

0

where C is a constant independent of R. We obtain (3.11) for all functions in C ∞ ([0, R]) by integrating both sides of (3.13) over [0, R]. Therefore (3.11) is true for all functions in H 1 ([0, R]) by the density of C ∞ ([0, R]) in H 1 ([0, R]). Now, we expand each function u(r1 , r2 , θ) in some Banach space defined on Ω to R3 \ Ω. Thus r1 , r2 , θ are not the distances and the interelectronic angle in the previous sense. For any u = u(r1 , r2 , θ), define Z (3.14) |ku|k20,Ω = u2 r12 r22 | sin θ|dr1 dr2 dθ, Ω ¶2 µ ¶2 µ ¶2 # Z "µ ∂u ∂u ∂u (3.15) |ku|k21,Ω = (r12 + r22 )| sin θ|dr1 dr2 dθ, + + ∂r ∂r ∂θ 1 2 Ω H(Ω) = {u | |ku|k20,Ω + |ku|k21,Ω < ∞}. Theorem 3.4. Suppose that Ω ⊂ X is a bounded open domain satisfying C 1 regularity [2]. There exists a linear operator E:

H(Ω) → H(R3 )

such that, for any u ∈ H(Ω), (3.16) (3.17)

Eu(r1 , r2 , θ) = u(r1 , r2 , θ), almost everywhere in (r1 , r2 , θ) ∈ Ω, |kEu|ki,R3 ≤ C|ku|ki,Ω , i = 0, 1,

where C is a constant depending on Ω. Proof. Let B(0, 1) be the open unit ball in R3 . Since Ω is bounded and C 1 regular, there exist a finite number of bounded open sets O0 , . . . , OM such that O0 ⊂⊂

60

WEIYING ZHENG AND LUNG-AN YING

M Ω, ∂Ω ⊂ ∪M i=1 Oi , and Ω ⊂ ∪i=0 Oi , and there exist m + 1 transforms (ξ, η, ζ) = ϕi (r1 , r2 , θ) such that

ϕi (Oi ) = B(0, 1), ϕi (Oi ∩ ∂Ω) = Σ = B(0, 1) ∩ {ζ = 0}, ϕi (Oi ∩ Ω) = B + (0, 1) = {(ξ, η, ζ) ∈ B(0, 1) | ζ > 0}, ¯ ϕi ∈ C 1 (Oi ∩ ∂Ω), ϕ−1 ∈ C 1 (Σ). i ˆ = (ξ, η, ζ). We choose a For the convenience of notation, define x = (r1 , r2 , θ) and x partition of unity {αi } associated with {Oi } satisfying αi ∈ C0∞ (Oi ), 0 ≤ αi ≤ 1, 0 ≤ i ≤ M,

M X

αi (x) = 1

∀x ∈ Ω.

i=0

¯ Set ui = uαi ; then u = PM ui . Let (1): Proof for functions u ∈ C 1 (Ω). i=0 %i (ˆ x) = %(ϕ−1 x)), %(x) = r12 r22 | sin θ|, i (ˆ (3.18) ωi (ˆ x) = ω(ϕ−1 x)), ω(x) = (r12 + r22 )| sin θ|, i (ˆ vi (ˆ x) = ui (ϕ−1 x)) = ui (x). i (ˆ P3 P3 ∂vi ∂ xˆ k ∂ui i ∂xl For any 0 ≤ i ≤ M , since ∂∂vxˆ ki = l=1 ∂u k=1 ∂ x ˆ k , ∂xl = ˆ k ∂xl , there exists a ∂xl ∂ x constant C depending only on ϕi and ϕ−1 such that i Z Z 2 (3.19) |vi |2 %i dˆ x= |ui |2 %Ji dx ≤ C |kui |k0,Ω T Oi , T B + (0,1)

Ω

Oi

¯ ¯ ¯ ¯ ¯ ¯ 3 Z X ¯ ∂vi ¯2 ¯ ∂ui ¯2 ¯ ∂xl ¯2 ¯ ¯ ¯ ¯ ωi dˆ ¯ ¯ ωi dˆ x≤C x ¯ ˆk ¯ ¯ ¯ ¯ ∂x ˆk ¯ + (0,1) ∂xl B + (0,1) ∂ x B l=1 ¯° 3 ¯° X ¯° ∂ui ¯°2 ¯ ° ¯° ≤C ¯° ∂xl ¯° T , 0,Ω Oi

Z (3.20)

l=1

where Ji =

i det( ∂ϕ ∂x )

(3.21)

is the Jacobian determinant. Similarly, we have Z 2 |vi |2 %i dˆ x, |kui |k0,Ω T Oi ≤ C

¯° ¯° ¯° ∂ui ¯°2 ¯° ¯° ¯° ∂xl ¯°

(3.22)

0,Ω

B + (0,1)

T

≤C Oi

¯ ¯ ¯ ∂vi ¯2 ¯ ¯ x. ¯ ˆ k ¯ ωi dˆ B + (0,1) ∂ x

3 Z X k=1

We expand u0 by zero to the exterior of O0 and denote the extension as u ˜0 ; then u ˜0 ∈ C01 (R3 ). We expand vi (ˆ x) as follows: ½ ˆ ∈ B + (0, 1) ∪ Σ, vi (ˆ x), x (3.23) v˜i (ˆ x) = ˆ ∈ B − (0, 1), 4vi (ξ, η, − 12 ζ) − 3vi (ξ, η, −ζ), x where B − (0, 1) = {ˆ x ∈ B(0, 1) | ζ < 0}. Obviously, v˜i ∈ C 1 (B + (0, 1) ∪ B − (0, 1)). For ˆ 0 ∈ Σ, we have any multiple index α ∈ {(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)} and x lim

Dα v˜i (ˆ x) = Dα vi (ˆ x0 ),

lim −

Dα v˜i (ˆ x) =

ˆ ∈B + (0,1) x ˆ →ˆ x x0

ˆ ∈B (0,1) x ˆ →ˆ x x0

µ ¶ · µ ¶α3 ¸ 1 1 Dα vi ξ, η,− ζ +3(−1)α3 +1 Dα vi (ξ, η,−ζ) 4 − 2 2 (0,1)

lim −

ˆ ∈B x ˆ →ˆ x x0 α = D vi (ˆ x0 ).

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

61

Thus u ˜i (x) = v˜i (ϕi (x)) ∈ C01 (Oi ). Expand u ˜i by zero to the exterior of Oi and PM denote the extension by u ˜i too; then u ˜i ∈ C01 (R3 ). Define Eu = ˜i ; then i=0 u PM (Eu)(x) = i=0 ui (x) = u(x) ∀x ∈ Ω. Combing (3.19)–(3.23) yields (3.16) and (3.17). ˆ be defined as in (3.9). In view of (2): Proof for functions u ∈ H(Ω). Let Ω kuk2i,r,Ω =

1 kuk2i,Ωˆ , 8π 2

ˆ is ˆ i = 0, 1, in the sense of isomorphism. Since C 1 (Ω) we have Hri (Ω) ,→ H i (Ω), i ˆ ˆ there exists {vn (r1 , r2 , θ)} ⊂ dense in H (Ω), for any v(r1 , r2 , θ) ∈ Hri (Ω) ,→ H i (Ω) ˆ such that vn (r1 , r2 , θ) converge to v(r1 , r2 , θ) in H i (Ω), ˆ and hence in Hri (Ω), C 1 (Ω) P6 ∂vn ∂xi ∂vn ∂vn ∂xi i = 0, 1. Since ∂ξ = i=1 ∂xi ∂ξ , ∂xi , and ∂ξ are continuous and bounded, we ¯ where ξ = r1 , r2 , θ, θ0 , φ, φ0 . Thus C 1 (Ω) ¯ is dense in Hri (Ω), have vn (r1 , r2 , θ) ∈ C 1 (Ω), i = 0, 1. ¯ is dense in H(Ω) in view of H(Ω) ,→ Hr1 (Ω). There exists Since Ω ⊂ X, C 1 (Ω) 1 ¯ a sequence {un } ⊂ C (Ω) converging to u in H(Ω). By (3.18), {Eun } is a Cauchy sequence in H(R3 ) and hence converges to some w ∈ H(R3 ). Set Eu = w. Since |kun − Eu|ki,Ω = |kEun − Eu|ki,Ω → 0, we have Eu = u, a.e. in Ω. Furthermore, |kEu|ki,R3 = lim |kEun |ki,R3 ≤ C lim |kun |ki,Ω = C|ku|ki,Ω , n→∞

n→∞

i = 0, 1.

The proof is complete. We define Ωr1 ,θ as the projection of Ω onto the r1 −θ plane. For any (r1 , θ) ∈ Ωr1 ,θ , define Ωr2 (r1 , θ) = {r2 | (r1 , r2 , θ) ∈ Ω}. Ωr2 ,θ and Ωr1 (r2 , θ) are defined in the same way. Theorem 3.5. Let Ω ⊂ X be a bounded domain satisfying one of the following two conditions: (a) Ω is C 1 −regular; (b) The boundary ∂Ω is Lipschitz continuous, and there exists a constant d > 0 such that, for almost every (r2 , θ) ∈ Ωr2 ,θ (respectively, (r1 , θ) ∈ Ωr1 ,θ ), if Ωr ⊂ Ωr1 (r2 , θ) (respectively, Ωr2 (r1 , θ)) is a maximal simply connected set, then meas(Ωr ) ≥ d. Furthermore, we assume that there are f1 , f2 , . . . , fM ∈ (Hrm (Ω))0 satisfying (3.24)

∀p ∈ Pm−1 (Ω),

M X

fi (p) = 0 ⇐⇒ p = 0.

i=1

Then there exists a constant C(Ω) such that ¯M ¯) ¯X ¯ ¯ ¯ ≤ C(Ω) |v|m,r,Ω + ¯ fi (v)¯ ∀v ∈ Hrm (Ω), m = 1, 2. ¯ ¯ (

(3.25)

kvkm,r,Ω

i=1

Proof. (1) In view of (3.7), we can prove (3.25) in the case of m = 1 by the argument of Theorem 3.1.1 in [9, p. 115].

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(2) Proof of (3.25) in the case of m = 2. If (3.25) were false, then for any integer n > 0 there should exist vn ∈ Hr2 (Ω) such that kvn k2,r,Ω = 1 and ¯ ¯ M ¯X ¯ 1 ¯ ¯ (3.26) |vn |2,r,Ω + ¯ fi (vn )¯ < . ¯ ¯ n i=1

In view of Theorem 3.4, the definition of k · k2,r,Ω , and (3.25) for m = 1, there exists a subsequence of {vn } (also denoted as {vn }), which is a Cauchy sequence under the following measures: ° ° ° °  °∂ ·° ° ∂· °  ° ° ° °  , ° , i = 1, 2,  k · k0,r,Ω , ° ∂θ ° °  ∂r i 0,r,Ω 0,r,Ω )1/2 (Z "µ (3.27) ¶2 µ 2 ¶ 2 µ 2 ¶2 #  ∂ · ∂ · ∂2 ·  2 2  + + (r1 + r2 ) sin θdr1 dr2 dθ .  ∂r1 ∂θ ∂r2 ∂θ ∂θ2 Ω n (i) Suppose that Ω satisfies the condition (a). We set un = ∂v ∂θ ; then un ∈ H(Ω). Choose R > 0 to be sufficiently large such that Ω ⊂ [0, R] × [0, R] × [0, π]. By Lemma 3.3 and Theorem 3.4, we have µ ¶2 Z Z R ∂vn 2 r1 sin θdr2 ≤ (Eun )2 r12 sin θdr2 ∂θ Ωr2 (r1 ,θ) 0 ) (Z ¶2 Z Rµ R ∂Eun 2 2 2 2 r1 sin θdr2 ∀(r1 , θ) ∈ Ωr1 ,θ , ≤C (Eun ) r1 r2 sin θdr2 + ∂r2 0 0

where C depends on R. By Lemma 3.3, integrating both sides of the above inequality over Ωr1 ,θ produces Z µ (3.28) Ω

∂vn ∂θ

¶2 r12 sin θdr1 dr2 dθ ≤ C

1 X i=0

|kEun |k2i,[0,R]×[0,R]×[0,π] ≤ C

1 X

|kun |k2i,Ω .

i=0

R ∂· 2 2 ) r1 sin θdr1 dr2 dθ]1/2 . Similarly, {vn } Thus {vn } is a Cauchy sequence under [ Ω ( ∂θ R ∂· 2 2 is a Cauchy sequence under [ Ω ( ∂θ ) r2 sin θdr1 dr2 dθ]1/2 . Consequently, {vn } is a Cauchy sequence in Hr1 (Ω) by (3.27), and so in Hr2 (Ω) by (3.26). Suppose vn → PM v ∈ Hr2 (Ω); then |v|2,r,Ω = 0 and i=1 fi (v) = 0 by (3.26). Thus v ∈ P1 (Ω), and v = 0 by (3.24). Therefore, v = 0 contradicts the following identities: kvk2,r,Ω = limn→∞ kvn k2,r,Ω = 1. Thus (3.25) is true for m = 2. (ii) Suppose that Ω satisfies the condition (b). Without loss of generality, we may suppose that Ωr2 (r1 , θ) is simply connected for any (r1 , θ) ∈ Ωr1 ,θ . By Lemma 3.3, we have "µ µ ¶2 ¶2 µ 2 ¶2 # Z Z ∂vn ∂ vn ∂v n r12 sin θdr2 ≤ C r22 + r12 sin θdr2 ∂θ ∂θ ∂r 2 ∂θ Ωr2 (r1 ,θ) Ωr2 (r1 ,θ) ∀(r1 , θ) ∈ Ωr1 ,θ , where C depends on d but is independent of (r1 , θ). Thus we can get (3.25) by the argument of (i). Remark 3.6. Assume that Ω satisfies the conditions in Theorem 3.5. Define Z Z 1 |Ω| = r12 r22 sin θdr1 dr2 dθ, f (v) = vr2 r2 sin θdr1 dr2 dθ. |Ω| Ω 1 2 Ω

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

63

By H¨older’s inequality, we have |f (v)| ≤ |Ω|−2 kvk0,r,Ω . Thus f ∈ (Hr1 (Ω))0 . Now (3.26) implies (3.29)

kv − f (v)k1,r,Ω ≤ |v|1,r,Ω

∀v ∈ Hr1 (Ω).

Remark 3.7. Assume R > 0 and Ω = [0, R] × [0, R] × [0, π]. Define 1 H0r (Ω) = {v ∈ Hr1 (Ω)| v|r1 =R = v|r2 =R = 0}. 1 ˆ in the sense of isomorphism, and, by Poinc´are’s Then we have H0r (Ω) ,→ H01 (Ω) inequality, √ √ 2 kvk1,Ωˆ ≤ C(Ω)|v|1,Ωˆ = 8πC(Ω)|v|1,r,Ω . (3.30) kvk1,r,Ω = 4π

Equations (3.29) and (3.30) are the so-called Friedrichs-type inequality and Poinc´are-type inequality, respectively. Theorem 3.8. Let Ω ⊂ X be the domain in Theorem 3.5 and T : Hrk (Ω) → m Hr (Ω) be a linear and continuous mapping satisfying (3.31)

Tp = p

∀p ∈ Pk−1 (Ω).

Then there exists a constant C(Ω) such that (3.32)

ku − T ukm,r,Ω ≤ C(Ω)|u|k,r,Ω

∀u ∈ Hrk (Ω),

where 0 ≤ m ≤ 2, 1 ≤ k ≤ 2, m ≤ k. Proof. (3.32) can be proved by virtue of Theorem 3.5 and the argument of Theorem 3.1.4 in [9, p. 121]. 4. Local regularization operator. Because we cannot say that the solutions of (2.13) and (5.1) are continuous, difficulties appear in proving the convergency of the finite element scheme. The technique of local regularization (Cl´ement’s interpolation [11]) will be used. Thus we are in the position of describing the construction of the three-dimensional local regularization operator. Set R > 0 be large enough, and define Ω = [0, R] × [0, R] × [0, π], ∂ Ω = {(r1 , r2 , θ) ∈ Ω| r1 = R or r2 = R}, Γ1 = {(r1 , r2 , θ) ∈ ∂ Ω | r1 = 0}, Γ2 = {(r1 , r2 , θ) ∈ ∂ Ω | r2 = 0}, Γ3 = {(r1 , r2 , θ) ∈ ∂ Ω | θ = 0}, Γ4 = {(r1 , r2 , θ) ∈ ∂ Ω | θ = π}. Suppose that Th is a regular subdivision of Ω. Each element in Th is a cuboid. (We can also obtain similar results for regular hexahedrons, but the analysis is very tedious.) h is the maximal diameter of all elements. The regularity of K means that there exists a constant σ independent of K such that ∀K ∈ Th , hK ≤ σ|e|; here hK is ¯ by A1 , A2 , . . . , AI , and the diameter of K, e is any edge of K. Denote all nodes of Ω define 4i = ∪K∈Th ,Ai ∈K K as the macro element associated with the node Ai .   k   X Qk (K) = p | p = αlmn r1l r2m θn , (r1 , r2 , θ) ∈ K .   l,m,n=0

Since the behaviors of the weights (see (3.18)) on an inner element differ from those on a boundary element, different kinds of elements or macro elements must be affine equivalent to different reference elements or macro elements, respectively. Each macro element must be one of the following four cases:

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1. An element in Th . Let l10 = [0, 1], l11 = [1, 2]; its affine equivalent reference macro element must be one of l1i × l1j × l1k , 0 ≤ i, j, k ≤ 1. 2. The combination of two elements with a common face. Let l20 = [0, 2], l21 = [1, 3]; its affine equivalent reference macro element must be one of l2i × l1j × l1k , l1i × l2j × l1k , or l1i × l1j × l2k , 0 ≤ i, j, k ≤ 1. 3. The combination of four elements with a common edge. Its affine equivalent reference macro element must be one of l1i × l2j × l2k , l2i × l1j × l2k , or l2i × l2j × l1k , 0 ≤ i, j, k ≤ 1. 4. The combination of eight elements with a common vertex. Its affine equivalent reference macro element must be one of l2i × l2j × l2k , 0 ≤ i, j, k ≤ 1. Each reference element is a cube with unit volume and is included in some reference macro element. Clearly, the total number of reference elements and reference macro elements is finite. Suppose h∆ < σhK , for any K ⊂ ∆, where h∆ is the diameter of ∆. For any ˆ → K is the affine transform from some reference element K ˆ to K ∈ Th , FK : K −1 ˆ K. For any macro element ∆, assume that ∆ = ∪K⊂∆ FK (K) is a macro reference −1 −1 −1 element defined in cases 1, 2, 3, or 4. Define F∆ : F∆ |K = FK and F∆ : F∆ |Kˆ = FK , ˆ = K. with FK (K) ˆ denote vˆ := v ◦ F∆ and If v is a function defined on ∆ and u ˆ is defined on ∆, −1 u := u ˆ ◦ F∆ , respectively. Without ambiguity, we also use piecewise-defined norms on macro elements: X X −1 2 (4.1) |ˆ v |2m,r,∆ |v ◦ F∆ |2m,r,Kˆ , |u|2m,r,∆ = |ˆ u ◦ F∆ |m,r,K . ˆ = K⊂∆

ˆ ∆ ˆ K⊂

ˆ m = 0, 1, 2. In view of (4.1), it is easy to prove v ∈ Hrm (∆) ⇐⇒ vˆ ∈ Hrm (∆), 0 ˆ ˆ ˆ ( ∆) → P ( ∆) as follows: ∀v ∈ Hr0 (∆), : H We define the Hr0 -projection P∆ ˆ k r (4.2)

(P∆ ˆ v, p)0 = (v, p)0

ˆ ∀p ∈ Pk (∆).

Since the weights vanish on some boundary elements K ∩ (∪4i=1 Γi ) 6= ∅, we need to deal with their transformations under the affine transforms by detailed analysis. To do so, we first need the following estimate for the transformation of sin θ. Define  π/2 − h ≤ θ ≤ π/2,  1, sin θ, θ ≥ π/2, (4.3) Λ(θ, h) =  sin(θ + h), θ ≤ π/2 − h. Lemma 4.1. Let σ > 0 be a constant, h1 , h2 ≤ h ≤ σ min{h1 , h2 }, θ ≥ h1 , and θ+ h1 +h2 +h/σ ≤ π. When h is sufficiently small, there exists a constant C independent of h and θ such that ½ ¾ 1 1 1 (4.4) Λ(θ, h1 ) · max ≤ C. , , sin θ sin(θ + h1 ) sin(θ + h1 + h2 ) Proof. Let M = max{1/ sin θ, 1/ sin(θ + h1 ), 1/ sin(θ + h1 + h2 )}. We consider (4.4) in three cases: 1. When π/2 − h1 − h2 ≤ θ ≤ π/2, Λ · M ≤ M , since h is small enough, (4.4) is true obviously. 2. When θ ≥ π/2, it is clear that ΛM ≤

sin θ sin(π − θ) = . sin(θ + h1 + h2 ) sin(π − θ − h1 − h2 )

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

65

If θ ≤ π − 2(h1 + h2 ), then ΛM ≤

sin(π − θ) ≤ 2. sin((π − θ)/2)

If θ ≥ π − 2(h1 + h2 ), we need only to choose h such that 2 sin(h/σ) ≥ h/σ; then ΛM ≤

sin 2(h1 + h2 ) 4h 8 ≤ ≤ . sin(h/σ) h/(2σ) σ

2θ 1) 3. When θ ≤ π/2 − h1 − h2 , ΛM ≤ sin(θ+h ≤ sin sin θ sin θ ≤ 2. The proof is complete. ˆ is a macro element associated with Theorem 4.2. Suppose that ∆ = F∆ (∆) −1 m some node of Th , u ∈ Hr (∆). Define P∆ u = (P∆ ˆ) ◦ F∆ . Then ˆu

(4.5)

|u − P∆ u|l,r,∆ ≤ Chm−l ∆ |u|m,r,∆ ,

0 ≤ l ≤ m ≤ 2,

where C is a constant independent of h. Proof. For the sake of simplicity, without loss of generality, we may suppose ∆ = ∪4i=1 Ki and analyze Ki in two representative cases. (1) K1 , . . . , K4 are boundary elements where the weights degenerate. Suppose h∆ is small enough and 2 K1 = [0, h1 ]×[0, h2 ]×[0, h3 ], K2 = [0, h1 ]×[0, h2 ]×[h3 , h3 + hK 3 ], K3 K3 2 K3 = [0, h1 ]×[h2 , h2 + h2 ]×[0, h3 ], K4 = [0, h1 ]×[h2 , h2 + h2 ]×[h3 , h3 + hK 3 ].

The reference macro element and reference elements are defined as ˆ = [0, 1] × [0, 2] × [0, 2], K ˆ 1 = [0, 1] × [0, 1] × [0, 1], ∆ ˆ 2 = [0, 1] × [0, 1] × [1, 2], K ˆ 3 = [0, 1] × [1, 2] × [0, 1], K ˆ 4 = [0, 1] × [1, 2] × [1, 2]. K On any finite-dimensional space, all norms are equivalent, and so we have (4.6)

kP∆ ˆk2i,r,∆ ˆk20,r,∆ uk20,r,∆ ˆu ˆu ˆ ≤ CkP∆ ˆ ≤ Ckˆ ˆ.

Hence the projection P∆ ˆ is stable on k · ki,r,∆ ˆ , i = 1, 2. By (4.2) and Theorem 3.8, we have Z 2 3 3 (4.7) ˆ|2 ξ 2 η 2 sin(h3 ζ)dξdηdζ ku − P∆ uk0,r,K1 ≤ h1 h2 h3 |ˆ u − P∆ ˆu ˆ1 K

≤ max{2, 1/ sin ζ0 }h31 h32 h23 kˆ u − P∆ ˆk20,r,Kˆ ˆu

1

≤ Ch8∆ |ˆ u|2m,r,∆ ˆ,

m = 1, 2,

where ζ0 ∈ (0, 1) satisfies ζ0 ≤ 2 sin ζ0 . Similarly, we have Z K3 2 K2 2 3 K3 K2 (4.8) ku − P∆ uk0,r,K4 ≤ h1 h2 h3 (h2 + h2 ) Λ(h3 , h3 )

ˆ4 K

|ˆ u − P∆ ˆ|2 ξ 2 dξdηdζ ˆu

K3 2 K2 3 K2 ≤ Ch31 hK u − P∆ ˆk20,r,Kˆ ˆu 2 h3 (h2 + h2 ) Λ(h3 , h3 )kˆ

1

≤ Ch8∆ |ˆ u|2m,r,∆ ˆ,

m = 1, 2;

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WEIYING ZHENG AND LUNG-AN YING

(4.9) (4.10)

ku − P∆ uk20,r,K2 ∪K3 ≤ Ch8∆ |ˆ u|2m,r,∆ ˆ,

m = 1, 2,

|u − P∆ u|21,r,∆ ≤ Ch6∆ |ˆ u|2m,r,∆ ˆ,

m = 1, 2.

Set h∆ small enough such that h∆ < sin(2h∆ ); by detailed analysis similar to (4.7) and (4.8), we have (4.11)

2m−8 |ˆ u|2m,r,∆ |u|2m,r,∆ , ˆ ≤ Ch∆

m = 1, 2,

where C depends only on R and σ. Combining (4.7)–(4.11) yields (4.5). (2) K1 , . . . , K4 are inner elements where the weights are strictly positive: K1 = [r10 , r10 + h1 ] × [r20 , r20 + h2 ] × [θ0 , θ0 + h3 ], 2 K2 = [r10 , r10 + h1 ] × [r20 + h2 , r20 + h2 + hK 2 ] × [θ0 , θ0 + h3 ], 3 K3 = [r10 , r10 + h1 ] × [r20 , r20 + h2 ] × [θ0 + h3 , θ0 + h3 + hK 3 ],

K3 2 K4 = [r10 , r10 + h1 ] × [r20 + h2 , r20 + h2 + hK 2 ] × [θ0 + h3 , θ0 + h3 + h3 ],

where r10 , r20 , θ0 ≥ σh∆ . The reference macro element and reference elements are defined as ˆ = [1, 2] × [1, 3] × [1, 3], K ˆ 1 = [1, 2] × [1, 2] × [1, 2], ∆ ˆ 2 = [1, 2] × [2, 3] × [1, 2], K ˆ 3 = [1, 2] × [1, 2] × [2, 3], K ˆ 4 = [1, 2] × [2, 3] × [2, 3]. K Then for m = 1, 2, by affine transforms, there exists a generic constant C independent of h, such that (4.12) ku − P∆ uk20,r,K1 ≤ Ch1 h2 h3 Λ(θ0 , h3 )(r10 + h1 )2 (r20 + h2 )2 kˆ u − P∆ ˆk20,Kˆ ˆu

1

≤ Ch1 h2 h3 Λ(θ0 , h3 )(r10 + h1 )2 (r20 + h2 )2 kˆ u − P∆ ˆk20,r,Kˆ ˆu

1

≤

Ch3∆ (r10

2

2

+ h∆ ) (r20 + h∆ )

Λ(θ0 , h3 )|ˆ u|2m,r,∆ ˆ.

Similarly, for i = 2, 3, 4 we have (4.13)

ku − P∆ uk20,r,Ki ≤ Ch3∆ (r10 + h∆ )2 (r20 + h∆ )2 3 × [Λ(θ0 , h3 ) + Λ(θ0 + h3 , h3 + hK u|2m,r,∆ ˆ, 3 )]|ˆ

(4.14)

|u − P∆ u|21,r,∆ ≤ Ch∆ (r10 + h∆ )2 (r20 + h∆ )2 3 × [Λ(θ0 , h3 ) + Λ(θ0 + h3 , h3 + hK u|2m,r,∆ ˆ. 3 )]|ˆ

Set h∆ small enough such that h∆ < sin(2h∆ ); by affine transforms and detailed analysis similar to (4.13), we have (4.15)

o n K3 ) |ˆ u|2m,r,∆ ≤ C max 1/ sin θ , 1/ sin(θ + h ), 1/ sin(θ + h + h 0 0 3 0 3 ˆ 3 −2 × r10 (r20 + h2 )−2 h2m−3 |u|2m,r,∆ . ∆

Combining (4.12)–(4.15), we obtain (4.5) by Lemma 4.1. We can prove (4.5) for other macro elements similarly.

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

67

5. Finite element approximations. The equivalent weak form of (3.6) is the 1 following: Find (λ, u) ∈ R1 × H0r (Ω) and u 6= 0 such that (5.1)

ar (u, v) = λ (u, v)0

1 ∀v ∈ H0r (Ω),

where λ = K + E, K is the constant in (2.10), and ¶ µ Z " ∂u ∂v ∂u ∂v ∂u ∂v 2 2 ar (u, v) = + + (r12 + r22 ) r1 r2 ∂r1 ∂r1 ∂r2 ∂r2 ∂θ ∂θ Ω Ã ! # 2r12 r22 2 2 + p 2 − 4r1 r2 − 4r1 r2 uv sin θdr1 dr2 dθ. r1 + r22 − 2r1 r2 cos θ ˆ where Ω ˆ is defined as that in (3.9), Since a(·, ·) is continuous and coercive on H01 (Ω), 1 (Ω) by the proof of Lemma we know that ar (·, ·) is continuous and coercive on H 0r p 3.1. We define k · k1,r,Ω = ar (·, ·) for the sake of simplicity in notation. We consider the Lagrangian finite element approximation to (5.1). For any K ∈ Th , denote the set of nodes in K as V(K) = {8 vertices and (k + 1)3 − 8 k-section points of K}. ∪K∈Th V(K) is the set of nodes of Th . Define the finite element space as Vh = {v(r1 , r2 , θ) ∈ C 0 (Ω) | v|∂Ω = 0, v|K ∈ Qk (K) ∀K ∈ =h }. The discrete approximation of (5.1) is the following: Find (λh , uh ) ∈ R1 × Vh and uh 6= 0 such that (5.2)

ar (uh , vh ) = λh (uh , vh )0

∀vh ∈ Vh .

1 Obviously, Vh ⊂ C 0 (Ω) ∩ H0r (Ω), and so (5.2) is the Galerkin approximation of (5.1). We drop the subscript “B” in (2.13) (or (5.1)) and suppose that 0 < λ1 ≤ λ2 ≤ · · · are the eigenvalues of (5.1), 0 < λh1 ≤ λh2 ≤ · · · < λNh are the eigenvalues of (5.2), Nh = dim(Vh ). Denote the eigenspaces associated with λi and λhj as Vi and Vhj , respectively, 1 ≤ i ≤ · · · , 1 ≤ j ≤ Nh . By the minmax theorem [10], λi ≤ λhi , 1 ≤ i ≤ Nh . ˆ be a reference macro element and K ˆ ⊂∆ ˆ be a reference element. Define Let ∆ ˆ

ˆ = {aK | 1 ≤ i ≤ (k + 1)3 } V(K) i ˆ

3 ˆ associated with a basis {ϕˆK i | 1 ≤ i ≤ (k + 1) } of Qk (K). Assume ˆ

ˆ

K ϕˆK i (aj ) = δij ,

1 ≤ i, j ≤ (k + 1)3 ,

ˆ we define the finite where δij = 1 if i = j, δij = 0 if i 6= j. For any vˆ ∈ Hr0 (K), element interpolation operator by means of the local regularization as follows: (k+1)3

(5.3)

πKˆ vˆ =

X

ˆ

ˆ

(P∆ ˆ)(aK ˆK ˆv i )ϕ i .

i=1

ˆ ∈ Th , define the finite element interpolation Suppose v ∈ Hr0 (Ω). For any K = FK (K) −1 ˆ operator on ∆, K, and Ω as follows: π∆ ˆ|Kˆ = πKˆ vˆ, πK v = (πKˆ vˆ)◦FK , πv|K = πK v, ˆv

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ˆ ⊂∆ ˆ is some reference element. On any space of finite dimension, all norms where K are equivalent. There exists a constant C such that (5.4)

kπKˆ vˆkm,r,Kˆ ≤ CkP∆ ˆk0,∞,Kˆ ≤ CkP∆ ˆk0,r,Kˆ ≤ kˆ uk0,r,∆ ˆv ˆv ˆ.

ˆ → Hrm (K) ˆ is linear and continuous, m = 0, 1. Thus the operator πKˆ : Hr0 (∆) Theorem 5.1. There exists a constant C independent of h such that for any v ∈ Hr2 (Ω), (5.5)

kv − πvkm,r,Ω ≤ Ch2−m |v|2,r,Ω ,

m = 0, 1.

ˆ is a reference macro element and K ˆ ⊂∆ ˆ is a reference Proof. Suppose that ∆ element. By the definition of the operators πKˆ , π∆ ˆ , and P∆ ˆ , we have (5.6)

π∆ ˆp = p

ˆ ∀p ∈ Pk (∆).

ˆ we have By (5.4), (5.6), and Theorem 3.8, for any vˆ ∈ Hr2 (∆) (5.7)

kˆ v − π∆ ˆkm,r,∆ v |2,r,∆ ˆv ˆ ≤ C|ˆ ˆ,

m = 0, 1.

For any macro element ∆, by affine transforms, (5.7), and the argument in the proof of Theorem 4.2, we know that there exists a constant C independent of h such that (5.8)

kv − πvkm,r,∆ ≤ Ch2−m |v|2,r,∆ , ∆

m = 0, 1.

Summing up each side of (5.8) over all macro elements, we get (5.5). Theorem 5.2. For any 1 ≤ i ≤ Nh , suppose (λi , ui ) is an eigenpair of (5.1) with kui k0,r,Ω = 1. There exist a constant C independent of h and an eigenfunction uhi ∈ Vhi with kuhi k0,r,Ω = 1 such that |λi − λhi | < Ch2 , kui − uhi km,r,Ω < Ch2−m ,

(5.9) (5.10)

m = 0, 1.

Proof. By the theory of abstract spectrum approximation (p. 699 of [10]), we know that, for any 1 ≤ i ≤ Nh , there exists a constant C independent of h such that |λi − λhi | < Cε(λi )2 ,

(5.11) where (5.12)

ε(λi ) =

sup

inf kv − vh k1,r,Ω .

v∈Vi ,kvk1,r,Ω =1 vh ∈Vh

Let {uij , 1 ≤ j ≤ Ni } be a basis of Vi , kuij k1,r,Ω = 1, Ni = dimVi . By Theorem 5.1, we have (5.13)

ε(λi ) ≤

Ni X j=1

kuij − πuij k1,r,Ω ≤ Ch

Ni X

|uij |2,r,Ω ≤ Ch.

j=1

Thus (5.9) is true. We can prove (5.10) by Theorem 5.2 and the argument of Theorem 6.2 in [32, p. 235], but we do not give the tedious description here.

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

69

Remark 5.3. In real computation, we have introduced a variational equation equivalent to (5.1). Define µ = cos θ; then Ω = [0, R] × [0, R] × [−1, 1], and the corresponding bilinear form, inner products, and norms are Z Z 2 2 2 (u, v)0 = uvr1 r2 dr1 dr2 dµ, kuk0,r,Ω = u2 r12 r22 dr1 dr2 dµ, Ω Ω Z · ∂u ∂v ∂u ∂v (u, v)1 = (u, v)0 + r12 r22 + r12 r22 ∂r ∂r ∂r 1 1 2 ∂r2 Ω ¸ 2 2 2 ∂u ∂v dr1 dr2 dµ, + (r1 + r2 )(1 − µ ) ∂µ ∂µ " µ ¶2 µ ¶2 µ ¶2 # Z ∂u ∂u ∂u 2 2 2 2 2 2 2 2 |u|1,r,Ω = r1 r2 + r1 r2 + (r1 + r2 )(1 − µ ) dr1 dr2 dµ, ∂r ∂r ∂µ 1 2 Ω kuk21,r,Ω = kuk20,r,Ω + |u|21,r,Ω , Hri (Ω) = {v| kvk2i,r,Ω < +∞}, i = 0, 1, " ¶ µ Z ∂u ∂v ∂u ∂v ∂u ∂v ar (u, v) = r12 r22 + + (r12 + r22 )(1 − µ2 ) ∂r1 ∂r1 ∂r2 ∂r2 ∂µ ∂µ Ω Ã ! # 2r12 r22 + p 2 − 4r12 r2 − 4r1 r22 uv dr1 dr2 dµ. 2 r1 + r2 − 2r1 r2 µ 1 We define H0r (Ω) as in Remark 3.7. A variational equation equivalent to (5.1) is the 1 following: Find (λ, u) ∈ R1 × H0r (Ω) and u 6= 0 such that

(5.14)

ar (u, v) = λ (u, v)0

1 ∀v ∈ H0r (Ω).

The partitions Th for Ω = [0, R] × [0, R] × [−1, 1] are similar to those in section 4. ∀K ∈ Th define   k   ¯ X ¯ αlmn r1l r2m µn , (r1 , r2 , µ) ∈ K . Qk (K) = p ¯ p =   l,m,n=0

The finite element approximation to (5.14) is: Find (λh , uh ) ∈ R1 × Vh and uh 6= 0 such that (5.15)

ar (uh , vh ) = λh (uh , vh )0

∀v ∈ Vh ,

where Vh = {v(r1 , r2 , µ) ∈ C(Ω)| v|K ∈ Qk (K) ∀K ∈ Th ; v|∂Ω = 0} is the finite element space. Comparing (5.15) with (5.2) in real computation, we have found that (5.15) gives more precise results with the same number of unknowns. The analysis for (5.15) will be the subject of our future research. 6. Numerical results. Since Vh is a finite-dimensional space, define N = dim(Vh ). We can choose a basis {Φ1 , . . . , ΦN } of Vh such that suppΦi = ∪K∈=h ,ai ∈ΣK K,

70

WEIYING ZHENG AND LUNG-AN YING

PN where ai is a node of Th . Let uh = i=1 αi Φi and vh = Φi , 1 ≤ i ≤ N , in (5.2) and (5.15). Then we obtain an equivalent generalized eigenvalue problem: (6.1)

A X = λh M X, T

where X = (α1 , . . . , αN ) , A = (a(Φi , Φj ))N ×N , M = ((Φi , Φj ))N ×N . We use the inverse iteration method [5] to solve the generalized eigenvalue problem (6.1). This method is convenient for computing the smallest (real) eigenvalue of an (unsymmetric) generalized eigenvalue problem with large and sparse matrices. In each step of iteration, the main computational cost is the solution of the following system of equations for Y : (6.2)

A Y = F.

However, in fact we need only solve (6.2) in the first step if using LU -factorization of A, since we can store the inverse matrix of A for all following steps. The computational cost of (6.2) is of order O(N 3 ) for a dense matrix. Since the finite element matrices are banded, and their band widths are bounded by some positive integer M ¿ N , the cost of (6.2) is not more than 2M N 2 . Thus the first iteration of our eigenvalue solver needs O(N 2 ) floating point operations, but each of the following iterations needs only O(N ) floating point operations. For the solution of large sparse generalized eigenvalue problems, improvements of this method have been developed rapidly. They devote themselves to reducing the cost of the first iteration; i.e., they solve (6.2) by efficient iterative methods instead of LU -factorization. Each iterative step of their eigenvalue solvers (such as preconditioned inverse iteration [26]) needs only O(N ) operations. For more detailed analyses, we refer to Neymeyr’s excellent work [26], or to the journal articles [27], [28], [29] and references therein. We consider the improvement of our eigenvalue solver as future work. We carried out our computation on a personal computer: Intel PIII750 with 1G SDRAM. The experiment shows that 1. the energy errors decrease with R or the number of nodes increasing; 2. with the considered state becoming more highly exited, R should be larger, and more nodes far from the nucleus are needed; 3. very large R makes no remarkable improvement in the precision. The main error concerns the potential V = − r21 − r22 + r112 . For the triplet, the wave function is antisymmetric with respect to the two electrons, so they cannot be very close to each other. That is to say, when r12 is very small, the wave function R 2 u tends to zero. When we calculate Ω ru12 dr1 dr2 dµ with a Gaussian integration formula [33], the error for the triplet is much smaller than that for the singlet with the same number of Gaussian points. Furthermore, from the figures below, we can see that the wave function |u| of the ground state is much larger than that of excited states in the domain where r12 is small and in the neighborhood of the nucleus containing the singularities. Thus we have used more and more Gaussian points and grid points along the µ-direction, when the state varies from the triplet, the singlet to the ground state. All matrix elements are computed by the standard Gaussian integration formula. With the number of Gaussian points increasing, the computing time becomes longer. Let Ne be the number of elements associated with some partition of Ω, Ng be the number of Gaussian points along one direction, and Te be the CPU time to compute a pair of element matrices by the one-point Gaussian formula. The CPU time to obtain the global stiffness matrix and mass matrix is (6.3)

T ≈ Ne × Te × Ng3 .

¨ SPECTRUM APPROXIMATION TO THE SCHRODINGER EQUATION

71

Table 6.1 Computational efforts for the S-states. State Number of DOF Number of GPs CPU time T (hours)

1s1s 1 S 57472 27 × 27 × 27 44.3

1s2s 1 S 65320 21 × 21 × 21 20.6

1s2s 3 S 68243 9×9×9 1.62

Table 6.2 FEM results for the helium atom (all values in a.u.). State Highly precise results [12], [23] Results given by (5.15) Results given by (5.2) FEM [6] FEM [25] FEM [31]

1s1s 1 S

1s2s 1 S

1s2s 3 S

−2.903724377034119 . . .

−2.1459740460544 . . .

−2.1752293782367 . . .

−2.903724106

−2.1459740042

−2.1752293277

−2.903715597 −2.9036118 −2.90326 −2.90324

−2.1459703835 −2.145960

−2.1752288326 −2.1752214

The number of degrees of freedom (DOF), number of Gaussian points (GPs), and the computational time T are listed in Table 6.1. We place grid points symmetrically along r1 and r2 for all states. The grid points are (for r1 , r2 , µ) 1. 1s1s 1 S: • 0.0, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.6, 2.0, 2.6, 3.2, 4.2, 6.0, 9.0, 15.0; • −1.0, −0.6, −0.2, 0.2, 0.6, 1.0; 2. 1s2s 1 S: • 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 3.0, 3.4, 3.8, 4.2, 4.8, 5.6, 8.0, 11.5, 15.0, 20.0; • −1.0, −0.5, 0.5, 1.0; 3. 1s2s 3 S: • 0.0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 4.0, 4.5, 5.5, 7.0, 10.0, 13.0, 16.0, 20.0, 25.0; • −1.0, 0.0, 1.0. The relative error of our approximate eigenvalue for the ground (1s1s-) state is 10−7 a.u., and those for the 1s2s-states are 10−8 a.u. by (5.15). The precisions of existing finite element eigenvalues are generally 10−4 − 10−6 a.u. (see Table 6.2). From the graphs of approximate wave functions (see Figures 6.1–6.2), we can get the following properties. 1. Although we add no physical assumptions to our computations a priori, such as the symmetry (for the singlets) and the antisymmetry (for the triplet), our approximate wave functions coincide with these properties very well. 2. Wave functions oscillate heavily in the neighborhood of the nucleus where the singularity of the Coulomb potential is very strange. This is well known by physicists and chemists. 3. In a sufficiently small neighborhood of the nucleus, absolute values |uh | of wave functions are very small. This implies that electrons seldom visit there. 4. With the distance between each electron and the nucleus increasing, wave functions decrease quickly. Thus it is reasonable to solve the Schr¨odinger equation in bounded

72

WEIYING ZHENG AND LUNG-AN YING

−8

x 10 18 16

−8

x 10

14

18

u of the ground state

12

16

10

u 14

6

of 12 the 10 ground

4

state 8

2

6

0

4

−2 0

2

8

0 −2 15

5 0 r1

15

r2

r2

10 15

10

10

5

10

r1

5

5 0

15

0

Fig. 6.1. Wave function of the 1s1s 1 S-state.

−5

x 10 2.5 −7

x 10

2 7

1.5 6

1

5

0.5 us

4 3

0 −0.5

2 r1

r2

1

−1

0

−1.5

−1 0

−2

0 5 0

5 5

10

−2.5 25

15

15 15

15 20

10 20

10

10

20

20 5

Fig. 6.2. Wave functions of the 1s2s 1 S-state (left) and the 1s2s 3 S-state (right).

domains. Acknowledgment. The authors would like to thank professor Peizhu Ding of Jilin University for his valuable suggestions and discussions. REFERENCES [1] J. Ackermann, Finite-element expectation values for correlated two-electron wave functions, Phys. Rev. A, 52 (1995), pp. 1968–1975. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1978. [3] A. Askar, Finite element method for bound state calculations in quantum mechanics, J. Chem. Phys., 62 (1975), pp. 732–734. [4] A. Askar and A. S. Cakmak, Finite element methods for reactive scattering, Chem. Phys., 33 (1978), pp. 267–286. [5] K. Bathe and E. Wilson, Numerical Methods in Finite Element Analysis, Prentice–Hall, Englewood Cliffs, NJ, 1973. [6] M. Braun, W. Schweizer, and H. Herold, Finite-element calculations for the S states of helium, Phys. Rev. A, 48 (1993), pp. 1916–1920. [7] G. Breit, Separation of angles in the two-electron problem, Phys. Rev., 35 (1930), pp. 569–578.

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[8] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Berlin, 1998. [9] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North–Holland, Amsterdam, New York, Oxford, 1978. [10] P. G. Ciarlet and J. L. Lions, Handbook of Numerical Analysis II, in Finite Element Methods, North–Holland, Amsterdam, 1989. ´ment, Approximation by finite element functions using local regularization, RAIRO [11] P. Cle Anal. Num´ er., 9 (1975), pp. 77–84. [12] G. W. F. Drake, and Z.-C. Yan, Variational eigenvalues for the S states of helium, Chem. Phys. Letters, 229 (1994), pp. 486–490. [13] G. W. F. Drake, M. M. Cassar, and R. A. Nistor, Ground-state energies for helium, H− , and Ps− , Phys. Rev. A, 65 (2002), paper 054501. [14] M. Duff, H. Rabitz, A. Askar, A. Cakmak, and M. Ablowitz, A comparison between finite element methods and spectral methods as applied to bound state problems, J. Chem. Phys., 73 (1980), pp. 1543–1559. [15] Ch. Froese–Fischer, The Hartree–Fock Method for Atoms, Wiley-Interscience, New York, 1977. [16] M. Fridman, Y. Rosenfeld, A. Rabinovitch, and R. Thieberger, Finite element method for solving the two-dimensional Schr¨ odinger equation, J. Comput. Phys., 26 (1978), pp. 169– 180. [17] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, New York, 1983. [18] M. I. Haftel and V. B. Mandelzweig, Precise nonvariation calculations on the helium atom, Phys. Rev. A, 38 (1988), pp. 5995–5999. [19] I. L. Hawk and D. L. Hardcastle, Finite-difference solution to the Schr¨ odinger equation for the helium isoelectronic sequence, Comput. Phys. Commun., 16 (1979), pp. 159–166. ¨chow, and H.-P.Merckens, Accurate upper and lower bounds for some [20] H. Kliendienst, A. Lu excited S states of the He atom, Chem. Phys. Lett., 218 (1994), pp. 441–444. [21] A. Kono and S. Hattori, Variational calculations for excited states in He I: Improved estimation of the ionization energy from accurate energies for the n 3 S, n 1 D, n 3 D series, Phys. Rev. A, 31 (1985), pp. 1199–1202. [22] A. Kono and S. Hattori, Energy levels for S, P, D states in He through precision variational calculations, Phys. Rev. A, 34 (1986), 1727–1735. [23] V. I. Korobov, Nonrelativistic ionization energy for the helium atom, Phys. Rev. A, 66 (2002), paper 024501. [24] R. Krivec, M. I. Haftel, and V. B. Mandelzweig, Precise nonvariation calculation of excited states of helium with the correlation-function hyperspherical-harmonic method, Phys. Rev. A, 44 (1991), pp. 7158–7164. [25] F. S. Levin and J. Shertzer, Finite-element solution of the Schr¨ odinger equation for the helium ground state, Phys. Rev. A, 32 (1985), pp. 3285–3290. [26] K. Neymeyr, A Hierarchy of Preconditioned Eigensolvers for Elliptic Differential Operators, research report, Mathematisches Instit¨ ut, Universit¨ at T¨ ubingen, T¨ ubingen, Germany, 2001; available online http://na.uni-tuebingen.de/klaus/papers.shtml. [27] K. Neymeyr, A geometric theory for preconditioned inverse iteration, I: Extrema of the Rayleigh quotient, Linear Algebra Appl., 332 (2001), pp. 61–85. [28] K. Neymeyr, A geometric theory for preconditioned inverse iteration, II: Convergence estimates, Linear Algebra Appl., 332 (2001), pp. 87–104. [29] A. V. Knyazev and K. Neymeyr, A geometric theory for preconditioned inverse iteration, III: A short and sharp convergence estimate for generalized eigenvalue problems, Linear Algebra Appl., 358 (2003), pp. 95–114. [30] S. Nordholm and G. Bacsky, Generalized finite element method applied to bound state calculation, Chem. Phys. Lett., 42 (1976), pp. 259–263. [31] A. Scrinzi, A 3-dimensional finite elements procedure for quantum mechanical applications, Comput. Phys. Commun., 86 (1995), pp. 67–80. [32] G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice–Hall, Englewood Cliffs, NJ, 1973. [33] A. H. Stroud, Approximate Calculation of Multipole Integrals, Prentice–Hall, Englewood Cliffs, NJ, 1971. [34] J. Weidmann, Linear Operators in Hilbert Spaces, Springer-Verlag, New York, 1980. [35] N. W. Winter, A. Laferriere, and V. McKoy, Numerical solution of the two-electron Schr¨ odinger equation, Phys. Rev. A, 2 (1970), pp. 49–59. [36] O. C. Zienkiewwicz and R. L. Taylor, The Finite Element Method, 4th ed., Vol. 1, McGrawHill, London, 1989.

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[37] W. Zheng and L. Ying, Finite element calculation for helium atom, Internat. J. Quantum Chem., (2004), to appear. [38] W. Zheng, The Finite Element Method for Atomic and Molecular Problems, Ph. D. thesis, Peking University, Beijing, P. R. China, 2002. [39] W. Zheng, Lung-An Ying, and Peizhu Ding, Numerical solutions of the Schr¨ odinger equation for the ground lithium by the finite element method, Appl. Math. Comput., to appear.