Applied Mathematics and Computation 234 (2014) 214–222

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

The finite element method for computing the ground statesof the dipolar Bose–Einstein condensates

http://dx.doi.org/10.1016/j.amc.2014.01.0850096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: lixg@bistu.edu.cn (X.-G. Li).

Dong-Ying Hua, Xiang-Gui Li ⇑School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, PR China

a r t i c l e i n f o

Keywords:Finite element methodBose–Einstein condensatesGeneralized solution

a b s t r a c t

A finite element approximation for computing the ground states of the dipolar Bose–Ein-stein condensates with a nonlocal nonlinear convolution term is presented in one dimen-sion. Following the idea of the imaginary time method, we compute the ground state finitemethod solution of the Bose–Einstein condensates by solving a nonlinear parabolic differ-ential–integral equation. Theoretical analysis is given to show the existence and conver-gence of this finite method solution. Numerical results are given to verify efficiency ofour numerical method.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Since 1995, experimental realization of Bose–Einstein condensation (BEC) of trapped atomic gases (mainly alkali atoms)has spurred a great deal of experimental, theoretical and numerical activities, all aimed at gaining a deeper insight into thefascinating behavior of this novel state of matter [1]. In 2005, the first realization of a dipolar BEC of 52Cr [2,3] was observedat Stuttgart University. Needless to say, this dramatic breakthrough on the experimental front has stimulated a correspond-ing wave of studies on both theoretical and numerical fronts. Computational methods have been proposed [4–11] to analyzethe ground states and dynamical behaviors of Bose–Einstein condensates, which can be described by the stationary andtime-dependent Gross–Pitaevskii equation (GPE), respectively. As we know, degenerate quantum gases are usually domi-nated by s-wave contact interactions which is isotropic and short range, while dipolar quantum gases are governed bythe d-wave symmetry of dipole–dipole interactions which is anisotropic and long range [12,13]. Though it has becomeapparent that the contact interactions between the condensed atoms govern most of the observed phenomena, this long-range difference gives rise to novel properties [14,15]. In fact, the long-range nature of dipolar interactions means thatthe GPE which governs the BEC is not only nonlinear but also nonlocal. At temperature T much smaller than the critical tem-perature, a dipolar BEC in the mean field theory is well described by the macroscopic wave function wðx; tÞ whose evolutionis governed by the three-dimensional (3D) GPE [16–18]

i�hwtðx; tÞ ¼ ��h2

2mr2 þ VðxÞ þ U0jwj2 þ Vdip � jwðx; tÞj2

" #wðx; tÞ; x 2 R3; t > 0; ð1:1Þ

with vanishing boundary conditions, where �h is the Planck constant, m the mass of the atom and VðxÞ is the external trappingpotential which is generally harmonic, that is, VðxÞ ¼ m

2 ðx21x2 þx2

2y2 þx23z2Þ with xi ði ¼ 1;2;3Þ representing the trap fre-

quency in x; y; z directions respectively. U0 ¼ 4p�h2as=m is the local interaction between dipoles in the condensate with as

D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222 215

the s-wave scattering length, and it can be tuned between positive for repulsive interaction and negative for attractive inter-action by using a Feshbach resonance. VdipðxÞ is the long-range anisotropic dipolar interaction potential between two dipolesgiven by

VdipðxÞ ¼Cdd

4p1� 3 cos2 h

jxj3; x 2 R3; ð1:2Þ

where h is the angle between the polarization axis n ¼ ðn1;n2;n3Þ and the vector x, i.e., cos h ¼ n�xjxj . For magnetic dipoles we

have Cdd ¼ l0l2d , where l0 is the magnetic vacuum permeability and ld the dipole moment; and for electric dipoles we have

Cdd ¼ d2=l0, where l0 is the vacuum permittivity and d the electric dipole moment. Both the amplitude and sign of Cdd can be

tuned by rotation of the polarization axis. The wave function wðx; tÞ is normalized according to

kwk2 ¼Z

R3jwðx; tÞj2 dx ¼ N ð1:3Þ

where N is the number of the atoms in the dipolar BEC. After the dimensionless transformation [7], that is, by rescaling

lengths with x ¼ a0~x; t ¼ ~tx0

, w ¼ffiffiffiNp

~w

a3=20

, where x0 ¼minfx1;x2;x3g; a0 ¼ffiffiffiffiffiffiffiffi

�hmx0

q, we can get

i~w~t ¼ �12

~r2 ~wþ 12

c21~x2 þ c2

2~y2 þ c23~z2� �

~wþ bj~wj2 ~wþ k Udip � j~wj2� �

~w

with b ¼ U0N�hx0a3

0¼ 4pasN

a0, k ¼ mN

a0�h2, ci ¼xix0ði ¼ 1;2;3Þ and Udip ¼ Cdd

4p1�3 cos2 hj~xj3

. It’s easy to see k ¼ b=U0. To simplify the notation, we

remove the tilde from the above equation, and get the following dimensionless nonlocal GPE:

iwtðx; tÞ ¼ �12r2 þ VðxÞ þ bjwðx; tÞj2 þ kUdip � jwðx; tÞj2

� �wðx; tÞ ð1:4Þ

under the constraint kwðx; tÞk ¼ 1, where VðxÞ ¼ 12 c2

1x2 þ c22y2 þ c2

3z2� �

and Udip ¼ Cdd4p

1�3 cos2 hjxj3

. It is well known that the aboveGPE has two conservation quantities, i.e, the number of the particles and the total energy. That is to say,

Z

R3jwðx; tÞj2 dx �

ZR3jwðx; 0Þj2 dx ¼

ZR3jw0ðxÞj

2 dx ¼ 1 ð1:5Þ

and

Eðwðx; tÞÞ ¼Z

R3

12jrwj2 þ VðxÞjwj2 þ b

2jwj4 þ k

2ðUdip � jwj2Þjwj2

� �dx � Eðwðx;0ÞÞ ð1:6Þ

In this paper, we shall be interested in effectively one-dimensional BECs. Despite their mathematical simplicity, lowerdimensional dipolar BECs [19–21] bear nonetheless a significant physical interest as well as a clear advantage for numericalcomputations. For example, in view of experiments on the transport behavior of condensates in elongated optical trapswhich are periodic in only one direction of space [22]; for strong trap anisotropic the time scales along the compressedand elongated axes are very different, which makes an accurate numerical treatment hard. Instead of solving the full 3Dproblem, it is hence desirable to find governing equations for lower dimensional dipolar BECs which are suitable for efficientnumerical methods. We focus on the computation of the ground state. Besides the intrinsic interest of computing the groundstate of BEC, the availability of ground state numerical solvers is also important for BEC dynamics because the dynamicbehavior of BEC can be very sensitive to initial conditions. In this paper we adopt a finite element method rather than apseudospectral method [7] successfully used in dipolar Bose–Einstein condensates in 3D because that technic used in 3Dcan’t be extended to 1D. In addition, the convergence analysis of the pseudospectral is not an easy task. So now considerthe following 1D nonlinear differential-integral GPE

iwtðx; tÞ ¼ �12r2wðx; tÞ þ VðxÞwþ bjwðx; tÞj2wðx; tÞ þ kðUdip � jwðx; tÞj2Þwðx; tÞ; x 2 R; t > 0; ð1:7Þ

where VðxÞ ¼ x2

2 , Udip ¼ jxj�a with 0 < a < 1. And the normalized conservation constraint of the number of atoms isRRjwj2 dx ¼ 1.This paper is organized as follows. In Section 2, we show the theoretical detail of computing the ground state using a finite

element method. In Section 3, numerical examples are shown to verify our results. In Section 4, conclusion is made.

2. The computation of the ground states

The existence and uniqueness of the ground states was discussed in [19]. To obtain the ground state from the numericalpoint of view, we use the imaginary time method [4], that is,

wtðx; tÞ ¼12r2w� VðxÞwðx; tÞ � bjwðx; tÞj2wðx; tÞ � kðUdip � jwðx; tÞj2Þwðx; tÞ ð2:1Þ

216 D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222

withR

Rjwðx; tÞj2 dx ¼ 1. The resulting equation is now a nonlinear parabolic equation. The generalized solution or the weak

form of (2.1) is defined as follows: to find wðx; tÞ 2 L1ð0; T;H10Þ, which satisfies

Z

R

wt/dx ¼Z

R

12r2 � VðxÞ � bjwj2 � kðUdip � jwj2

� �w/dx;Z

R

wðx;0Þ/dx ¼Z

R

w0ðxÞ/dx;8/ðxÞ 2 H1

0:

8>><>>: ð2:2Þ

The numerical solution of the corresponding problem (2.2) is: to find the real function whðx; tÞ 2 VhðXÞ, which satisfies

ZX

wht/dx ¼Z

X

12r2 � VðxÞ � bjwhj

2 � kðUdip � jwhj2Þ

� �wh/dx;Z

Xwhðx;0Þ/dx ¼

ZR

w0ðxÞ/dx;

8>><>>: 8/ðxÞ 2 VhðXÞ; ð2:3Þ

where w0ðxÞ is the initial condition. Here the finite element method is used to get a numerical solution. Supposewhðx; tÞ ¼

Pmi¼0 aiðtÞ/iðxÞ, where /iðxÞ is the shape function at the mesh grid xi in the continuous linear space VhðXÞ with X

being a bounded area ½�a; a� approximating the whole space R and aiðtÞ is continuous with respect to the time variable t.Eq. (2.3) results in

Xm

i¼0

a0iðtÞZ

X/iðxÞ/jðxÞdx ¼ � 1

2

Xm

i¼0

aiðtÞZ

Xr/iðxÞ � r/jðxÞdx�

Xm

i¼0

aiðtÞZ

XVðxÞ/iðxÞ/jðxÞdx

� bXm

i¼0

aiðtÞZ

X

Xm

k¼0

akðtÞ/kðxÞ

2

/iðxÞ/jðxÞdx

� kXm

i¼0

aiðtÞZ

XUdip �

Xm

k¼0

akðtÞ/kðxÞ

2

0@

1A/iðxÞ/jðxÞdx; 0 6 j 6 m:

For simplicity, we denote Aij ¼R

X /iðxÞ/jðxÞdx, Bij ¼R

X r/iðxÞ � r/jðxÞdx, Cij ¼R

X VðxÞ/iðxÞ/jðxÞdx, DðaÞij ¼R

X

Pmk¼0 akðtÞ

/kðxÞj

2/iðxÞ/jðxÞdx and KðaÞij ¼R

X Udip �Pm

k¼0 akðtÞ/kðxÞ 2� �

/iðxÞ/jðxÞdx. If the element Aij is looked upon as an element

of the matrix A, it’s easy to see that A is symmetric, as well as the corresponding matrix B;C;DðaÞ and KðaÞ. By simple cal-culation, for all 1 6 j 6 m, we get

Aj�1j ¼16

h; Bj�1j ¼ �1h; Cj�1j ¼

Z xj

xj�1

VðxÞ/j�1ðxÞ/jðxÞdx;

DðaÞj�1j ¼1

20ða2

j�1 þ a2j Þ þ

115

aj�1aj

�h;

and for all 1 6 j 6 m� 1,

Ajj ¼23

h; Bjj ¼2h; Cjj ¼

Z xjþ1

xj�1

VðxÞ/2j ðxÞdx;

DðaÞjj ¼1

30ða2

j�1 þ a2jþ1Þ þ

25a2

j þ1

10ajðaj�1 þ ajþ1Þ

�h:

Moreover, we have

A00 ¼ Amm ¼h3; B00 ¼ Bmm ¼

1h; C00 ¼

Z x1

x0

VðxÞ/20ðxÞdx; Cmm ¼

Z xm

xm�1

VðxÞ/2mðxÞdx;

D00 ¼a2

0

5þ a2

1

30þ a0a1

10

� �h; Dmm ¼

a2m�1

30þ a2

m

5þ am�1am

10

� �h:

And it’s easy to see Aij ¼ Bij ¼ Cij ¼ DðaÞij ¼ 0 for all ji� jjP 2. Now we need to handle with KðaÞ. First, the convolution termFðx; tÞ ¼ Udip � j

Pmi¼0 aiðtÞ/iðxÞj

2 is considered. Notice

Fðx; tÞ :¼Z xm

x0

Pmi¼0 aiðtÞ/iðyÞ

2jx� yja

dy ¼ a20ðtÞ

Z x1

x0

/20ðyÞ

jx� yjadyþ a2

mðtÞZ xm

xm�1

/2mðyÞjx� yja

dy

þXm�1

i¼1

a2i ðtÞ

Z xiþ1

xi�1

/2i ðyÞ

jx� yjadyþ 2

X06i<j6m

aiðtÞajðtÞZ xm

x0

/iðyÞ/jðyÞjx� yja

dy: ð2:4Þ

D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222 217

Now we only need to handle the following three kinds of terms. That is,

LiðxÞ :¼Z xi

xi�1

/2i ðyÞ

jx� yjady ¼ 1

h2

Z xi

xi�1

ðy� xþ x� xi�1Þ2

jx� yjady;

RiðxÞ :¼Z xiþ1

xi

/2i ðyÞ

jx� yjady ¼ 1

h2

Z xiþ1

xi

ðxiþ1 � xþ x� yÞ2

jx� yjady;

and

MiðxÞ :¼Z xiþ1

xi

/iðyÞ/iþ1ðyÞjx� yja

dy ¼ 1

h2

Z xiþ1

xi

ðxiþ1 � xþ x� yÞðy� xþ x� xiÞjx� yja

dy:

Through careful computation, and let C ¼ 13�a� 2

2�aþ 11�a, then we have

LiðxÞ ¼1

h2

�Cðxi�1 � xÞ3�a þ ðxi � xÞ3�a

3� a� 2ðxi�1 � xÞðxi � xÞ2�a

2� aþ ðxi�1 � xÞ2ðxi � xÞ1�a

1� a; x 6 xi�1;

Cðx� xi�1Þ3�a þ ðxi � xÞ3�a

3� aþ 2ðx� xi�1Þðxi � xÞ2�a

2� aþ ðx� xi�1Þ2ðxi � xÞ1�a

1� a; xi�1 < x < xi;

Cðx� xi�1Þ3�a � ðx� xiÞ3�a

3� aþ 2ðx� xi�1Þðx� xiÞ2�a

2� a� ðx� xi�1Þ2ðx� xiÞ1�a

1� a; x P xi:

8>>>>>>>><>>>>>>>>:

RiðxÞ ¼1

h2

Cðxiþ1 � xÞ3�a � ðxi � xÞ3�a

3� aþ 2ðxiþ1 � xÞðxi � xÞ2�a

2� a� ðxiþ1 � xÞ2ðxi � xÞ1�a

1� a; x 6 xi;

Cðxiþ1 � xÞ3�a þ ðx� xiÞ3�a

3� aþ 2ðxiþ1 � xÞðx� xiÞ2�a

2� aþ ðxiþ1 � xÞ2ðx� xiÞ1�a

1� a; xi < x < xiþ1;

�Cðx� xiþ1Þ3�a þ ðx� xiÞ3�a

3� a� 2ðx� xiþ1Þðx� xiÞ2�a

2� aþ ðx� xiþ1Þ2ðx� xiÞ1�a

1� a; x P xiþ1:

8>>>>>>>><>>>>>>>>:

MiðxÞ ¼1

h2

�ðxiþ1 � xÞðxi � xÞ1� a

ðxiþ1 � xÞ1�a � ðxi � xÞ1�a� �

þ xiþ1 þ xi � 2x2� a

ðxiþ1 � xÞ2�a � ðxi � xÞ2�a� �

� 13� a

ðxiþ1 � xÞ3�a � ðxi � xÞ3�a� �

; x 6 xi;

ðxiþ1 � xÞðx� xiÞ1� a

ðxiþ1 � xÞ1�a þ ðx� xiÞ1�a� �

þ xiþ1 þ xi � 2x2� a

ðxiþ1 � xÞ2�a � ðx� xiÞ2�a� �

� 13� a

ðxiþ1 � xÞ3�a þ ðx� xiÞ3�a� �

; xi < x < xiþ1;

ðx� xiþ1Þðx� xiÞ1� a

ðx� xiþ1Þ1�a � ðx� xiÞ1�a� �

xiþ1 þ xi � 2x2� a

ðx� xiþ1Þ2�a � ðx� xiÞ2�a� �

þ 13� a

ðx� xiþ1Þ3�a � ðx� xiÞ3�a� �

; x P xiþ1:

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

So we have

Fðx; tÞ ¼ a20ðtÞR0ðxÞ þ

Xm�1

i¼1

a2i ðtÞ LiðxÞ þ RiðxÞð Þ þ a2

mðtÞLmðxÞ þ 2Xm�1

i¼0

aiðtÞaiþ1ðtÞMiðxÞ ð2:5Þ

and therefore KðaÞij ¼R

X Fðx; tÞ/iðxÞ/jðxÞdx. Then we have the following ordinary differential equations

A~a0ðtÞ ¼ � 12

Bþ C þ bDð~aÞ þ kKð~aÞ� �

~aðtÞ ð2:6Þ

with ~aðtÞ ¼ ða0ðtÞ;a1ðtÞ; . . . ;am�1ðtÞ;amðtÞÞT . The boundary condition wðx; tÞ ¼ 0 on @X gives a0ðtÞ ¼ amðtÞ ¼ 0 for all t > 0.The continuous initial condition wðx;0Þ ¼ w0ðxÞ shows that

Z

Xwhðx;0Þ/jðxÞdx ¼

ZX

w0ðxÞ/jðxÞdx; 0 6 j 6 m; ð2:7Þ

in the discrete level. That is,

A~að0Þ ¼~b ð2:8Þ

with the jth component of~b beingR

X w0ðxÞ/jðxÞdx. These equations can be solved by Runge–Kutta method or the finite dif-ference method. We solve the ODE system (2.6) with the semi-implicit backward Euler scheme

218 D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222

Aþ s 12

Bþ C þ bDð~akÞ þ kKð~akÞ� � �

~akþ1 ¼ A~ak ð2:9Þ

with the initial condition~a0 ¼ ~að0Þ in (2.8). As all the /jðxÞ ðj ¼ 0;1; . . . ;mÞ is independent, the above equations has a uniquelocal solution on ½0; t0�. Furthermore, it’s easy to see

Z

Xjwhðx; tÞj

2dx ¼ ~aT A~a

where ~a :¼ ~aðtÞ and after each discrete time level, whðx; tÞ should be normalized to satisfyR

X jwhðx; tÞj2 dx ¼ 1, that is,

whðx; tÞ :¼ whðx;tÞkwhðx;tÞk

. In addition, the total energy is

Eðwhðx; tÞÞ ¼12~aT B~aþ~aT C~aþ b

2~aT Dð~aÞ~aþ k

2~aT Kð~aÞ~a ¼ ~aT 1

2Bþ C þ b

2Dð~aÞ þ k

2Kð~aÞ

� �~a ð2:10Þ

Here, 12~aT B~a, ~aT C~a, b

2~aT Dð~aÞ~a and k

2~aT Kð~aÞ~a are the kinetic energy, the potential energy, the internal energy and the dipolar

energy, respectively.Next, we will prove that the numerical solution exists globally and is convergent to the generalized solution of (2.1) if

b P 4kð2aÞ1�a

1�a .

Proposition 1. Suppose the initial condition w0ðxÞ is bounded uniformly and b P 0, then we have

kwhðx; tÞkL2 6 C; ð2:11Þ

where the constant C is independent of h and t.

Proof. Set /ðxÞ ¼ wh in (2.3) and it is easy to see

12

ddt

ZXjwhj

2 dx ¼ �12

ZXjrwhj

2 dx�Z

XVðxÞjwhj

2 dx� bZ

Xjwhj

4 dx� kZ

XðUdip � jwhj

2Þjwhj2 dx;

which shows

ddtkwhk

2L2 6 0; ð2:12Þ

i.e.,

kwhðx; tÞkL2 6 kwhðx;0ÞkL2 :

Moreover, by (2.7) we have

Zwhðx;0Þwhðx;0Þdx ¼

Zw0ðxÞwhðx;0Þdx;

which results in

kwhðx;0ÞkL2 6 kw0ðxÞkL2

and the proof is completed. h

Proposition 2. Suppose kwhtðx;0Þk2L2 is bounded with respect to h uniformly and b P 4kð2aÞ1�a

1�a , then we have

kwhtðx; tÞkL2 6 C; ð2:13Þ

where C is independent of h and t.

Proof. Differentiate (2.3) with respect to t, then set / ¼ wht in the resulting equation, we obtain

ZX

whttwht dx ¼ 12

ZXr2whtwht dx�

ZX

VðxÞjwhtj2 dx� b

ZXðjwhj

2whÞtwht dx� kZ

XðUdip � jwhj

2Þwh

� �twht dx;

That is,

12

ddtkwhtk

2L2 ¼ � 1

2krwhtk

2L2 �

ZX

VðxÞjwhtj2 dx� b

ZX

12

@

@tjwhj

2� �2

þ jwhj2jwhtj

2

" #dx� k

ZXðUdip � jwhj

2Þtwhwht dx

� kZ

XðUdip � jwhj

2Þjwhtj2 dx 6 �b

ZXjwhj

2jwht j2 dx� k

ZXðUdip � jwhj

2Þtwhwht dx

¼ � bZ

Xjwhj

2jwht j2 dx� k

2

ZZX2

1jx� yja

@

@tjwhðy; tÞj

2 @

@tjwhðx; tÞj

2 dxdy;

D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222 219

Notice

ZZX2

1jx� yja

@

@tjwhðy; tÞj

2 @

@tjwhðx; tÞj

2 dxdy

6 4ZZ

X2

jwhðx; tÞwhtðx; tÞjjx� yj

a2

� jwhðy; tÞwhtðy; tÞjjx� yj

a2

dxdy

6 4ZZ

X2

jwhðx; tÞj2jwhtðx; tÞj

2

jx� yjadxdy

!12 ZZ

X2

jwhðy; tÞj2jwhtðy; tÞj

2

jx� yjadxdy

!12

¼ 4ZZ

X2

jwhðx; tÞj2jwhtðx; tÞj

2

jx� yjadxdy

and

ZZX2

jwhðx; tÞj2jwhtðx; tÞj

2

jx� yjadxdy ¼

Z a

�ajwhðx; tÞj

2jwhtðx; tÞj2 dx

Z a

�a

1jx� yja

dy

¼Z a

�ajwhðx; tÞj

2jwhtðx; tÞj2Z x

�a

1jx� yja

dyþZ a

x

1jx� yja

dy� �

dx

¼Z a

�ajwhðx; tÞj

2jwhtðx; tÞj2 1

1� aðaþ xÞ1�a þ ða� xÞ1�a� �

dx

62 � ð2aÞ1�a

1� a

ZXjwhðx; tÞj

2jwhtðx; tÞj2 dx;

we have

12

ddtkwhtk

2L2 6 �bþ 4kð2aÞ1�a

1� a

! ZXjwhðx; tÞj

2jwhtðx; tÞj2 dx:

Choosing b sufficiently large such that b P 4kð2aÞ1�a

1�a , we have

12

ddtkwhtk

2L2 6 0;

therefore,

kwhtðx; tÞk2L2 6 kwhtðx; 0Þk

2L2 6 C

which completes the proof. h

Proposition 3. Under the assumptions of Propositions 1 and 2, we have

krwhkL2 6 C; kVðxÞjwhj2kL1 6 C;

kwhkL4 6 C; kðUdip � jwhj2Þjwhj

2kL1 6 C:ð2:14Þ

Proof. By (2.3), we have

ZX

whtwh dx ¼ �12

ZXjrwhj

2 dx�Z

XVðxÞjwhj

2 dx� bZ

Xjwhj

4 dx� kZ

XðUdip � jwhj

2Þjwhj2 dx; ð2:15Þ

which shows

12

ZXjrwhj

2 dxþZ

XVðxÞjwhj

2 dxþ bZ

Xjwhj

4 dxþ kZ

XðUdip � jwhj

2Þjwhj2 dx 6 kwhtkL2kwhðx; tÞkL2 ;

Therefore,

12

ZXjrwhj

2 dxþZ

XVðxÞjwhj

2 dxþ bZ

Xjwhj

4 dxþ kZ

XðUdip � jwhj

2Þjwhj2 dx 6 C: ð2:16Þ

As the terms in the left hand side of (2.16) are all nonnegative, the proof is complete. h

Theorem 1. Under the assumptions of Propositions 1–3, there exists any T > 0 such that the finite element solution whðx; tÞ isconvergent to the generalized solution (2.2) on ½0; T�.

220 D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222

Proof. By Propositions 1–3, kwhkL2 , krwhkL2 , kwhtkL2 , kwhkL4 , and kðUdip � jwhj2Þjwhj

2kL1 are uniformly bounded, and therefore,the finite element solution is not only a local solution on ½0; t0�, but can be extended to any time interval ½0; T�. In addition, theboundedness of the above norms also implies there exists a subseries (still denoted by wh) for any fixed t, and wh has thefollowing properties: wh is strongly convergent to wðx; tÞ in L2,rwh;wht are weakly convergent torw;wt in L2, respectively, wh

is weakly convergent to w in L4 and ðUdip � jwhj2Þjwhj

2 is weakly convergent to ðUdip � jwj2Þjwj2 in L1. Let h! 0 in (2.3), and wehave

ZR

wt/dx ¼ 12

ZR

r2w/dx�Z

R

VðxÞw/dx� bZ

R

jwj2w/dx� kZ

R

ðUdip � jwj2Þw/dx; 8/ðxÞ 2 VhðXÞ: ð2:17Þ

As the space Vh composed by the basis /iðxÞ is condense in H10, which shows (2.17) is valid for all /ðxÞ 2 H1

0. That is, the lim-itation wðx; tÞ of whðx; tÞ as h! 0 is the generalized solution (2.2). That is to say, the generalized solution (2.2) existsglobally. h

3. Numerical examples

In this section, the ground states of a dipolar BEC with different parameters are given by the numerical scheme above. Theinitial condition is chosen to be w0ðxÞ ¼ 1

p1=4 e�x2=2. The ground state wg :¼ wkþ1h ¼

Pmj¼0 akþ1

j /jðxÞ is reached numerically in (2.6)when

k~akþ1 �~akk1 :¼ max06j6M

jakþ1j � ak

j j 6 e :¼ 10�6:

0.35

−20 −15 −10 −5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

x

ψg

Fig. 1. Ground state solution wg of a dipolar BEC with k ¼ 10 and b ¼ 500.

0 0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

7

8

9

t

Energy

EkinEpotEdip

0 0.5 1 1.5 2 2.5 3 3.510

20

30

40

50

60

70

80

90

100

110

t

Energy

EintEtotal

Fig. 2. The energy evolution of a dipolar BEC with k ¼ 10 and b ¼ 500.

−8 −6 −4 −2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

ψg

Fig. 3. Ground state solution wg of a dipolar BEC with k ¼ 0:8 and b ¼ 30.

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

t

Energy

EkinEpotEintEdipEtotal

Fig. 4. The energy evolution of a dipolar BEC with k ¼ 0:8 and b ¼ 30.

−8 −6 −4 −2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

ψg

Fig. 5. Ground state solution wg of a dipolar BEC with k ¼ 0:8 and b ¼ 80.

D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222 221

Example 1. We solve this problem on ½�16;16� with h ¼ 1=8 and s ¼ 0:01. Fig. 1 shows the ground state wgðxÞ of a dipolarBEC with a ¼ 0:5; b ¼ 500; k ¼ 10. And in Fig. 2, the energy evolution is given.

222 D.-Y. Hua, X.-G. Li / Applied Mathematics and Computation 234 (2014) 214–222

Example 2. We solve this problem on ½�8;8� with h ¼ 1=32 and s ¼ 0:005. Fig. 3 shows the ground state wgðxÞ of a dipolarBEC with a ¼ 0:5; b ¼ 30; k ¼ 0:8. And in Fig. 4, the energy evolution is given.

Example 3. In this example, we keep all the parameters the same as in Example 2 except b ¼ 80. The ground state is shownin Fig. 5, and it is short and stout than that in Fig. 3, which is consistent with the fact that the stronger the repulsive inter-action between the particles is, the bigger the cloud size will be.

In the above numerical examples, the kinetic energy, the internal energy, the dipolar energy and the total energy all de-crease, while the potential energy increases.

4. Conclusion

In this paper, a finite element method is presented to compute the ground states of the dipolar Bose–Einstein conden-sates. As the dipole–dipole interactions are anisotropic and long range, the governing equation described by a Gross–Pitaev-skii equation is nonlocal and nonlinear which brings difficulty in numerical simulation as well as in theoretical analysis. Thefinite element method rather than other methods is adopted in 1D to compute the ground states because in 3D, a dipolarBose–Einstein condensates can be rewritten as a coupled Gross–Pitaevskii–Poisson equations [7], where the time indepen-dent Poisson equation can be solve by the finite element method efficiently. Numerical examples shows our method is reli-able and efficient. And the extension of this method to 3D is in progress and will be presented in our future work.

Acknowledgement

This work was supported by National Natural Science Foundation of China (No. 11171032) and Beijing Municipal Educa-tion Commission (No. KM201110772017).

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