Finite element approximations of nonlinear eigenvalue problems in quantum physics

Comput. Methods Appl. Mech. Engrg. 200 (2011) 1846–1865

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Comput. Methods Appl. Mech. Engrg.

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Finite element approximations of nonlinear eigenvalue problemsin quantum physics q

Huajie Chen, Lianhua He, Aihui Zhou ⇑LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e i n f o

Article history:Received 14 April 2010Accepted 5 February 2011Available online 15 February 2011

Keywords:Adaptive computationConvergenceComplexityDensity functional theoryFinite elementNonlinear eigenvalue problem

0045-7825/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.02.008

q This work was partially supported by The Nationaunder Grants 10871198 and 10971059, The Funds forChina under Grant 11021101, The National Basic ReseGrant 2011CB309703, and The National High Technoment Program of China under Grant 2009AA01A134.⇑ Corresponding author. Tel.: +86 10 62625704; fax

E-mail addresses: [email protected] (H. [email protected] (A. Zhou).

a b s t r a c t

In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arisingfrom quantum physics. We derive both a priori and a posteriori finite element error estimates and obtainoptimal convergence rates for both linear and quadratic finite element approximations. In particular, weanalyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilizecertain relationship between the finite element eigenvalue problem and the associated finite elementboundary value approximations. We also present several numerical examples in quantum physics thatsupport our theory.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Nonlinear eigenvalue equations are typical mathematicalmodels in quantum physics. For instance, Schrödinger–Newtonequations model the quantum state reduction [1,2], the Gross–Pitaevskii equation (GPE) describe Bose–Einstein condensates(BEC) [3,4] and the Thomas–Fermi–von Weizsäcker (TFvW) typeequations and Kohn–Sham equations play a crucial role in densityfunctional theory [5–9].

Solving such kinds of nonlinear eigenvalue equations hasgreatly advanced our understanding of quantum physics and evenbeen used to explain and sometimes predict the behavior of com-plex systems in quantum physics. However, only a few results onnumerical analysis of nonlinear eigenvalue problems have beenobtained so far, see [10–14] for convergence of finite dimensionalapproximations and [15,16] for a priori convergence rates. We notethat numerical analysis given in [12,13,15] are only for problemswith convex energy functionals while [10] gives a rough a priori er-ror upper bound for a general case and [16] provides an a priori

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l Science Foundation of ChinaCreative Research Groups of

arch Program of China underlogy Research and Develop-

: +86 10 62542285.), [email protected] (L. He),

error estimate for planewave discretizations for the Kohn–ShamLDA model under a coercivity assumption.

In this paper, we focus on finite element approximations of thefollowing nonlinear eigenvalue problem: find k 2 R and u 2 H1

0ðXÞsuch that

RX u2 ¼ Z and

�,Duþ VuþNðu2Þu ¼ ku in X; ð1:1Þ

where X � R3; Z 2 N; , > 0; V : X ! R is a given function, and Nmaps a nonnegative function to some function on X. We are partic-ularly interested in adaptive finite element computations. We studyboth a priori and a posteriori error estimates and prove the conver-gence and complexity of adaptive finite element approximations.

Let us give a somewhat more detailed but informal descriptionof the main results in this paper. Let (k,u) be a solution of (1.1) and(kh,uh) be a finite element approximation to (k,u) on a shape-regu-lar mesh T h. Under some assumptions, which may allow to handlethe TFvW energy functional and the repulsive interaction in BEC,we prove that the following kind of a priori error estimates hold(see Theorem 3.1)

ku� uhk1;X K infv2Sh

0ðXÞku� vk1;X;

ku� uhk0;X þ jk� khjK rðhÞku� uhk1;X;

where Sh0ðXÞ is the finite element space associated with T h and

r(h) ? 0 (and even rðhÞ ¼ OðhÞ) as h ? 0. The above kind of a priorierror estimates are available in [15,16] for convex energy func-tional, but we generalize them to a more general energy functionaland prove them by a different approach.

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H. Chen et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1846–1865 1847

Let gh(uh,X) and osch(uh,X), which are computable, denote theestimator and oscillation respectively, we obtain the a posteriorierror estimates as follows (see Theorem 3.4)

ku� uhk2a;X 6 C1g2

hðuh;XÞ

and

C2g2hðuh;XÞ 6 ku� uhk2

a;X þ C3osc2hðuh;XÞ;

where k � ka;X ¼ ,1=2kr � k0;X; C1;C2 and C3 only depend on ,; ca andthe shape regularity constant c⁄.

Using the a posteriori error estimates, we then design an adap-tive finite element method for solving (1.1) and prove the conver-gence and complexity of adaptive finite element approximations.Let fT k; ðkk;ukÞ;gkðuk;XÞ; osckðuk;XÞgkP0 be the sequence ofmeshes, discrete solutions, estimators, and oscillations producedby the adaptive finite element method in the kth step. If the initialfinite element mesh is fine enough, then (see Theorems 4.1 and4.2)

ku� ukþ1k2a;X þ cg2

kþ1ðukþ1;XÞ 6 n2 ku� ukk2a;X þ cg2

kðuk;XÞ� �

and

ku� ukk2a;X þ cosc2

kðuk;XÞK ð#T k �#T 0Þ�2sjuj2s ;

where s, c > 0, n 2 (0,1), and #T k is the number of elements in T k.The crucial technical tool for deriving the a posteriori error esti-mates and analyzing the adaptive finite element approximationsare the perturbation arguments (c.f., e.g., [17,18]), which exploitthe relationship between the nonlinear eigenvalue problem andan associated linear boundary value problem (see Theorem 3.3and Lemma 4.1).

Numerical analysis of linear eigenvalue problems has beenthoroughly studied in the past decades (see, e.g., [17,19–22] andreferences cited therein). While adaptive finite element methodshave been extensively studied since Babuška and Vogelius [23]gave an analysis of an adaptive finite element method for linearsymmetric elliptic problems in one dimension. In particular,there are a number of work studying their convergence and com-plexity, for instances, [24–28,30,31] for linear boundary valueproblems, [18,32–35] for nonlinear boundary value problems,and [17,20,21,36,37] for linear eigenvalue problems. To our bestknowledge, however, there has been no work on either the conver-gence rate or the complexity of adaptive finite element approxima-tions for nonlinear eigenvalue problems except [11], in which onlyconvergence of a nonlinear eigenvalue problem has been studied.

The rest of this paper is organized as follows. In the next section,we formulate a nonlinear eigenvalue problem and give some rele-vant nonlinear analysis. In Section 3, we prove a priori error esti-mates of finite element approximations for both linear andquadratic elements and give a posteriori error estimates. In Section4, we present the adaptive finite element algorithm and analyzethe convergence and complexity of the adaptive finite elementapproximations. In Section 5, we present some numerical experi-ments in several typical quantum physics that support our theory.And we give some concluding remarks in Section 6. Finally, we pro-vide three appendices, including basic results for adaptive finiteelement approximations for a model problem that have used inour analysis, numerical analysis in H1-norm under a weak assump-tion, and a detailed proof of the minimal cardinality of the markedset in adaptive computation.

2. A nonlinear eigenvalue problem

Let X � R3 be a polytopic bounded domain. We shall use thestandard notation for Sobolev spaces Ws,p(X) and their associated

norms and seminorms, see, e.g., [38,39]. For p = 2, we denote Hs(X) =Ws,2(X) and H1

0ðXÞ ¼ fv 2 H1ðXÞ : v j@X ¼ 0g, where vjoX = 0 isunderstood in the sense of trace, k�ks,X = k�ks,2,X. The space H�1(X),the dual space of H1

0ðXÞ, will also be used. Let (�, �) be the standard in-ner product of L2(X) and h�, �i denote the dual inner product fromH1

0ðXÞ to H�1(X). Throughout this paper, we shall use C to denote ageneric positive constant which may stand for different values atits different occurrences. For convenience, the symbol [will be usedin this paper. The notation that A [ B means that A 6 CB for someconstant C that is independent of mesh parameters. We shall usePðp; ðc1; c2ÞÞ to denote a class of functions that satisfy the growthcondition:

Pðp; ðc1; c2ÞÞ ¼ f : 9a1; a2 2 R such that c1tp þ a1 6 f ðtÞ 6 c2tpfþ a2 8t P 0g;

with c1 2 R and c2, p 2 [0,1).

2.1. Problem setting

We study the case that nonlinear term N has the followingform:

NðqÞ ¼ N 1ðqÞ þ N 2ðqÞ;

where q ¼ u2; N 1 : ½0;1Þ ! R is a given function dominated bysome polynomial, and N 2 is the Hartree potential

N 2ðqÞ ¼ r�1 � q ¼Z

X

qðyÞjx� yjdy:

The energy functional associated with (1.1) is

EðuÞ ¼Z

Xð,jruðxÞj2 þ VðxÞu2ðxÞ þ Eðu2ðxÞÞÞdx

þ 12

Dðu2;u2Þ; ð2:1Þ

where E : ½0;1Þ ! R is defined by

EðsÞ ¼Z s

0N 1ðtÞdt

and D(�, �) is a bilinear form as follows

Dðf ; gÞ ¼Z

Xf ðxÞðr�1 � gÞðxÞdx:

The following assumptions are needed in our analysis:

A.I. V 2 L2(X).A.II. E 2 C1ð½0;1Þ;RÞ \ C2ðð0;1Þ;RÞ and E 2 Pðp; ðc1; c2ÞÞ satisfy-

ing one of the following conditions:1. c1 2 (0,1);2. p 2 [0,4/3];3. c1 2 (�1,0), p 2 (4/3,1) and

jc1j,

Zp�1 < infu2H1

0ðXÞ;kuk0;X¼1

ZXjruj2=

ZXjuj2p

� �:

A.III. N 1ðtÞ 2 Pðp1; ðc1; c2ÞÞ for some p1 2 [0,1], and there existq 2 (1,2], l 2 [0,5 � q] such that for all t1; t2 2 R, there holds

jN 1ðt21Þt1 �N 1ðt2

2Þt1 � 2N 01ðt22Þt2

2ðt1 � t2ÞjK ð1þmaxfjt1jl; jt2jlgÞjt1 � t2jq: ð2:2Þ

A.IV. N 01ðtÞt 2 Pðp2; ðc1; c2ÞÞ for some p2 2 [0,1].

We shall mention that these assumptions are satisfied by manytypical physical models, such as TFvW type orbital-free models andGPE mentioned above (see, e.g., [4,5,7,9,12]).

1848 H. Chen et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 1846–1865

The ground state solution of (1.1) is obtained by minimizing theenergy functional (2.1) in the admissible class

A ¼ u 2 H10ðXÞ : kuk2

0;X ¼ Z; u P 0n o

:

Namely, we solve the following minimization problem

inffEðvÞ : v 2 Ag: ð2:3Þ

Note that under Assumptions A.I and A.II, E(v) is bounded belowover A

EðvÞP C�1krvk20;X � b 8v 2 A ð2:4Þ

for some constants C and b (see, e.g., [7,10,13]). We see that underAssumptions A.I–A.III, there exists a minimizer u of (2.3). Thenonlinear eigenvalue problem (1.1) can be viewed as the Euler–Lagrange equation associated with (2.3). Any minimizer of (2.3)satisfies the following weak form of (1.1): find k 2 R and u 2 A suchthat

,ðru;rvÞ þ ðVuþNðu2Þu;vÞ ¼ kðu;vÞ 8v 2 H10ðXÞ: ð2:5Þ

The eigenvalue k can be computed from its correspondingeigenfunction u by

Zk ¼ EðuÞ þZ

XðN 1ðu2ðxÞÞu2ðxÞ � Eðu2ðxÞÞÞdxþ 1

2Dðu2;u2Þ: ð2:6Þ

For any ðk;uÞ 2 R� H10ðXÞ, we define F : R� H1

0ðXÞ ! H�1ðXÞby

hFðk;uÞ; vi ¼ ,ðru;rvÞ þ ðVuþNðu2Þu� ku; vÞ 8v 2 H10ðXÞ:

The Fréchet derivative of F with respect to u at (k,u) is denoted byF0uðk;uÞ : H1

0ðXÞ ! H�1ðXÞ, where

hF 0uðk;uÞv;wi ¼ ,ðrv ;rwÞ þ ððV þNðu2Þ � kÞv ;wÞþ 2ðN 01ðu2Þu2v ;wÞ þ 2Dðuv; uwÞ 8w 2 H1

0ðXÞ:

It is observed that as an operator from H10ðXÞ to H�1ðXÞ;F0uðk;uÞ is

Lipschitz continuous at u, i.e., there exit d > 0 and C > 0 such thatfor any w 2 H1

0ðXÞ satisfying kw � uk1,X 6 d, there holds

kF 0uðk;uÞ � F 0uðk;wÞk 6 Cku�wk1;X:

The following assumption is also involved in our discussion.

A.V. For the ground state solution (k,u) of (2.5), F0uðk;uÞ is an iso-morphism from H1

0ðXÞ to H�1(X), namely, there exists a con-stant b0 > 0 such that

infv2H1

0ðXÞsup

w2H10ðXÞ

hF 0uðk;uÞw;vikwk1;Xkvk1;X

P b0; ð2:7Þ

F 0uðk;uÞ is invertible on u? � fv 2 H10ðXÞ; ðu;vÞ ¼ 0g,

namely, there exists a constant b1 > 0 such that

infv2u?

supw2u?

hF 0uðk;uÞw;vikwk1;Xkvk1;X

P b1: ð2:8Þ

As a result of Assumption A.V, u is an isolated solution (c.f., e.g.,Appendix B or [34,40]). Namely, there exists a constant d > 0 suchthat u is the unique ground state solution in ball Bðu; dÞ � fw 2H1

0ðXÞ : kw� uk1;X 6 dg.

Remark 2.1. A sufficient condition of Assumption A.V being true isthat

hF 0uðk;uÞv; viP C�1krvk20;X 8v 2 H1

0ðXÞ ð2:9Þ

holds for some constant C > 0, which has been proved to be satisfiedby some TFvW models that are of convex functional, i.e., E00 > 0 [15].

Remark 2.2. Indeed, a priori error estimate in H1-norm can beobtained under assumption (2.8) instead of Assumption A.V (seeAppendix B).

2.2. Basic analysis

The following estimates concerning the nonlinear terms will beused in our analysis.

Using the Young’s inequality and the Hölder inequality, we havethe following.

Lemma 2.1. There exists a positive constant C such that

Dðv ;vÞ 6 Ckvk20;X 8v 2 L2ðXÞ;

jDðuv ;uwÞj 6 Ckvk0;Xkwk0;X 8v ;w 2 L2ðXÞ;jDðwv ;wvÞj 6 Ckwk2

1;Xkvk1;Xkvk1;X 8w;v;v 2 L2ðXÞ;kN 2ðvÞk0;1;X 6 Ckvk0;X 8v 2 L2ðXÞ:

Lemma 2.2. Let v ;w 2 H10ðXÞ satisfy kvk1;X þ kwk1;X 6 C for some

constant C. If Assumptions A.III and A.IV hold, then

ððN ðw2Þ � N ðv2ÞÞw;wÞK kw� vk0;X; ð2:10ÞkN ðw2Þw�Nðv2Þvk�1;X K kw� vk0;X; ð2:11Þ

and

kN ðw2Þw�Nðv2Þvk0;X K kw� vk1;X: ð2:12Þ

Proof. We first prove that (2.10) holds when N is replaced by N 1.Obviously

ðN 1ðw2Þ � N 1ðv2ÞÞw;w� �

K kw� vk0;X

is true when p2 = 0. Let p2 2 (0,1]. Then there exists d 2 [0,1] suchthat

ðN 1ðw2Þ � N 1ðv2ÞÞw;w� �

¼Z

X2N 01ðn

2Þn2 þN 1ðn2Þ� �

ðw� vÞw

þN 1ðv2Þðv �wÞw

KZ

Xn2p2 þ n2p1� �

ðw� vÞw

þ v2p1 ðv �wÞw K kn2p2k0;3=p2 ;Xkw

� vk0;Xkwk0;6=ð3�2p2Þ;X

þ kn2p1k0;3=p1 ;Xþ kv2p1k0;3=p1 ;X

� �� kw� vk0;Xkwk0;6=ð3�2p1Þ;X;

where n = w + d(v � w). Note that

knk0;6;X K kvk0;6;X þ kwk0;6;X K kvk1;X þ kwk1;X K C; ð2:13Þ

we have

ððN 1ðw2Þ � N 1ðv2ÞÞw;wÞK kw� vk0;X: ð2:14Þ

For Hartree potential term N 2, using the Hölder inequality andUncertainty Principle [41], we get

kr�1 � ðw2 � v2Þk0;1;X K krðwþ vÞk0;Xkw� vk0;X ð2:15Þ

and hence


jððr�1 � ðw2 � v2ÞÞw;wÞj 6 kr�1 � ðw2 � v2Þk0;1;Xkwk20;X

K krðwþ vÞk0;Xkw� vk0;Xkwk20;X

K kw� vk0;X; ð2:16Þ

which together with (2.14) leads to (2.10).We now turn to prove (2.11). Note that for any v 2 H1

0ðXÞ

ðN 1ðw2Þw�N 1ðv2Þv ;vÞK kw� vk0;Xkvk1;X 8v 2 H10ðXÞ:

We then have

kN 1ðw2Þw�N 1ðv2Þvk�1;X K kw� vk0;X:

For the Hartree potential term, we obtain from (2.15) that

kr�1 � ðw2 � v2Þwk0;X 6 kr�1 � ðw2 � v2Þk0;1;Xkwk0;X K kw� vk0;X:

Due to

N 2ðw2Þw�N 2ðv2Þv ¼ ðr�1 � ðw2 � v2ÞÞwþ ðr�1 � v2Þðw� vÞ;

we arrive at

kN 2ðw2Þw�N 2ðv2Þvk�1;X K kN 2ðw2Þw�N 2ðv2Þvk0;X

K kw� vk0;X: ð2:17Þ

Finally, we see from Assumption A.IV and (2.13) that

kN 1ðw2Þw�N 1ðv2Þvk0;X K kN 01ðn2Þn2k0;3=p2 ;X

kw� vk0;6=ð3�2p2Þ;X

þ kN 1ðn2Þk0;3=p1 ;Xkw� vk0;6=ð3�2p1Þ;X K kw� vk1;X;

which together with (2.17) completes the proof of (2.12). h

Lemma 2.3. Let v;w 2 H10ðXÞ satisfy kvk1;X þ kwk1;X 6 C for some

constant C. If Assumption A.III holds, then for any v 2 H10ðXÞ, there

holdZXN 1ðw2Þw�N 1ðv2Þw� 2N 01ðv2Þv2ðw� vÞ� �

v

K kv �wkq1;Xkvk1;X; ð2:18ÞZ

XN 2ðw2Þw�N 2ðv2Þw� 2N 2ðvðw� vÞÞv� �

v

K kv �wk21;Xkvk1;X: ð2:19Þ

Proof. Using Assumption A.III, the Hölder inequality and the Sobo-lev inequality, we haveZ

XN 1ðw2Þw�N 1ðv2Þw� 2N 01ðv2Þv2ðw� vÞ� �

v

K kmax jwjl; jv jln o

k0;6=l;Xkv �wkq0;6q=ð5�lÞ;Xkvk0;6;X

K kv �wkq1;Xkvk1;X;

which completes the proof of (2.18).To prove (2.19), we derive from Lemma 2.1 thatZ

X2N 2ðv2Þv � 2N 2ðvwÞv �N 2ðv2ÞwþN 2ðw2Þw� �

v

¼ D ðw� vÞðwþ vÞ; ðw� vÞvð Þ þ D ðw� vÞ2;vv� �

K kv �wk21;Xkvk1;X:

This completes the proof. h

3. Finite element analysis

Let dX be the diameter of X and fT hg be a shape regular familyof nested conforming meshes over X with size h 2 (0,dX): there ex-ists a constant c⁄ such that

hs

qs6 c� 8s 2 T h; ð3:1Þ

where, for each s 2 T h;hs is the diameter of s, and qs is the diam-eter of the biggest ball contained in s; h ¼maxfhs : s 2 T hg. LetEh denote the set of interior faces (edges or sides) of T h. We denotethe number of elements in T h by #T h, which will be used in theanalysis of complexity for adaptive finite element methods.

Let Sh,k(X) be a space of continuous functions on X such that forv 2 Sh,k(X), v restricted to each s is a polynomial of degree notgreater than k, namely,

Sh;kðXÞ ¼ fv 2 CðXÞ : v js 2 Pks 8s 2 T hg;

where Pks is the space of polynomials of degree not greater than a

positive integer k. Set Sh;k0 ðXÞ ¼ Sh;kðXÞ \ H1

0ðXÞ. We shall denoteSh;k

0 ðXÞ by Sh0ðXÞ for simplification of notation afterwards.

Let us now consider the finite element approximations for theground state solution of nonlinear eigenvalue problem (2.5). Theground state solution in finite element space Sh

0ðXÞ is obtained bysolving the minimization problem

inf EðvÞ : v 2 Sh0ðXÞ \ A

n o: ð3:2Þ

Under Assumptions A.I–A.III, we obtain the existence of the mini-mizer uh 2 Sh

0ðXÞ \ A (see, e.g., [7,10,13]).It is seen that any minimizer uh of (3.2) solves

,ðruh;rvÞ þ ðVuh þNðu2hÞuh; vÞ ¼ khðuh; vÞ 8v 2 Sh

0ðXÞ; ð3:3Þ

with the corresponding finite element eigenvalue kh 2 R satisfying

Zkh ¼ EðuhÞ þZ

XN 1ðu2

hðxÞÞu2hðxÞ � Eðu2

hðxÞÞ� �

dx

þ 12

D u2h;u

2h

� �: ð3:4Þ

Similar to (2.4), we have that under Assumptions A.I and A.II,kuhk1,X are uniformly bounded

supuh2A\Sh

0ðXÞ;h2ð0;dXÞkuhk1;X 6 C:

Under Assumption A.V, the ground state solution uh of (3.3) is lo-cally unique in the neighborhood of u when the mesh size of T h

is sufficiently fine (see Appendix B or [34]). Throughout this paper,we shall denote by (k,u) an isolated solution of (2.5) and (kh,uh) theunique finite element solution of (3.3) in the neighborhood of u.

It is shown in [10,13] that the finite element solutions convergeto the exact solution, namely

limh!0ku� uhk1;X ¼ 0; ð3:5Þ

limh!0jk� khj ¼ 0: ð3:6Þ

3.1. A priori error estimates

In this subsection, we shall derive optimal a priori error esti-mates for ku � uhk1,X, ku � uhk0,X and jk � khj, which will be usedin deriving the a posteriori error estimates in Section 3.2 and provethe convergence and complexity of an adaptive finite elementmethod in Section 4.

To give the a priori error estimates, we assume h0� 1 anddefine the Galerkin projection P0h : H1

0ðXÞ ! Sh0ðXÞ by


a0ðu; w� P0hw;vÞ ¼ 0 8v 2 Sh0ðXÞ;

where h 2 (0,h0] and

a0ðu; w;vÞ ¼ hF 0uðk;uÞw;vi 8w;v 2 H10ðXÞ: ð3:7Þ

We have from Assumption A.V that

kP0hwk1;X K kwk1;X 8w 2 H10ðXÞ ð3:8Þ

and

kw� P0hwk0;X þ hkw� P0hwk1;X K h infv2Sh

0ðXÞjw� vk1;X

8w 2 H10ðXÞ: ð3:9Þ

Let M : H�1ðXÞ ! H10ðXÞ be an operator defined by

a0ðu; Mw;vÞ ¼ ðw;vÞ 8v 2 H10ðXÞ;

i.e., M ¼ ðF0uðk;uÞÞ�1. It is seen from Assumption A.V that M is a

bounded operator from H�1(X) to H10ðXÞ:

kMwk1;X K kwk�1;X 8w 2 H�1ðXÞ: ð3:10Þ

Note that (2.5) and (3.3) can be rewritten as

u ¼ Mð2N 01ðu2Þu3 þ 2N 2ðu2ÞuÞ ð3:11Þ

and

uh ¼ P0hðkhM � kMÞuh þ P0hMnh þ P0hMvh; ð3:12Þ

respectively, where

nh ¼ N 1ðu2Þuh �N 1ðu2hÞuh þ 2N 01ðu2Þu2uh; ð3:13Þ

vh ¼ N 2ðu2Þuh �N 2ðu2hÞuh þ 2N 2ðuuhÞu: ð3:14Þ

We can now prove the main results of this section, which plays animportant role in our a posteriori error analysis in the next section.

Theorem 3.1. If h0� 1,h 2 (0,h0] and Assumptions A.I–A.V hold, then


0ðXÞku� vk1;X; ð3:15Þ

ku� uhk0;X K rðhÞku� uhk1;X; ð3:16Þ

and

jk� khjK rðhÞku� uhk1;X; ð3:17Þ

where rðhÞ ¼ hþ ku� uhkq�11;X and r(h) ? 0 as h ? 0.

Proof. Associated with eigenpair (k,u) that satisfies (2.5), we havea useful identity (see, e.g., Lemma 3.1 of [13])

,ðrv ;rvÞ þ ððV þNðv2ÞÞv ;vÞðv ;vÞ � k

¼ ,ðrðv � uÞ;rðv � uÞÞ þ ððV þNðu2ÞÞðv � uÞ;v � uÞðv; vÞ

þ ððN ðv2Þ � N ðu2ÞÞv; vÞðv ;vÞ � k

ðv � u;v � uÞðv ;vÞ

8v 2 H10ðXÞ; ð3:18Þ

which together with (2.10) implies

jkh � kjK kuh � uk21;X þ kuh � uk0;X: ð3:19Þ

Set u = u(u,uh) � Zuh, we have that u 2 u? and

u� uh ¼ Z�1 uþ 12ku� uhk2

0;Xu� �

: ð3:20Þ

Since (3.11) and (3.12) imply

u� uh ¼ ðI � P0hÞuþ ðk� khÞP0hMuh þ P0hMð2N 01ðu2Þu3 � nhÞþ P0hMð2N 2ðu2Þu� vhÞ; ð3:21Þ

we obtain that

Z�1u ¼ ðI � P0hÞuþ ðk� khÞP0hMuh þ P0hMð2N 01ðu2Þu3 � nhÞ

þ P0hMð2N 2ðu2Þu� vhÞ �12

Z�1ku� uhk20;Xu: ð3:22Þ

Due to (2.8), there is a unique solution wu 2 u? satisfying

a0ðu; wu;vÞ ¼ ðu;vÞ 8v 2 u?: ð3:23Þ

Note that

,ðrwu;rvÞþ ðV þNðu2Þ þ 2N 01ðu2Þu2 � kÞwu þ 2N 2ðuwuÞu;v� �

¼ 2Z�1ðN 01ðu2Þu2wu;uÞðu;vÞ þ 2Z�1Dðuwu;u2Þðu;vÞþ ðu� Z�1ðu;uÞu;vÞ8v 2 H1

0ðXÞ;

we have that wu 2 H2(X), kwuk2,X [ kuk0,X, and

Z�1kuk20;X ¼ Z�1a0ðu; wu;uÞ ¼ Z�1a0ðu; u;wuÞ¼ a0ðu; ðI � P0hÞu;wuÞ þ ðk� khÞa0ðu; P0hMuh;wuÞþ a0ðu; P0hMð2N 01ðu2Þu3 � nhÞ;wuÞþ a0ðu; P0hMð2N 2ðu2Þu� vhÞ;wuÞ

� 12

Z�1ku� uhk20;Xa0ðu; u;wuÞ: ð3:24Þ

Next, we analyze (3.24). It follows from (3.9) that

a0ðu; ðI � P0hÞu;wuÞ ¼ a0ðu; ðI � P0hÞu;wu � P0hwuÞK kðI � P0hÞuk1;Xkwu � P0hwuk1;X

K hku� uhk1;Xkuk0;X: ð3:25Þ

Using wu 2 u? and identity

a0ðu; P0hMuh;wuÞ ¼ a0ðu; Mu;wuÞ þ a0ðu; Mu; P0hwu �wuÞþ a0ðu; Mðuh � uÞ; P0hwuÞ;

we obtain

ðk� khÞa0ðu; P0hMuh;wuÞK ðhþ ku� uhk0;XÞjk� khjkuk0;X: ð3:26Þ

We deduce from Assumption A.III, (2.18) and (3.8) that

a0ðu; P0hMð2N 01ðu2Þu3 � nhÞ;wuÞ¼ a0ðu; P0hMðN 1ðu2

hÞuh �N 1ðu2Þuh

� 2N 01ðu2Þu2ðuh � uÞÞ;wuÞ¼ ðN 1ðu2

hÞuh �N 1ðu2Þuh � 2N 01ðu2Þu2ðuh � uÞ; P0hwuÞK ku� uhkq

1;Xkuk0;X: ð3:27Þ

By (2.19), we have that

a0ðu; P0hMð2N 2ðu2Þu� vhÞ;wuÞ ¼ ð2N 2ðu2Þu� vh; P0hwuÞ

¼ ðN 2ðu2hÞuh �N 2ðu2Þuh � 2N 2ðuðuh � uÞÞu; P0hwuÞ

K ku� uhk21;Xkuk0;X: ð3:28Þ

Taking into account (3.19), (3.25)–(3.28), we conclude that

kuk0;X K hku� uhk1;X þ jk� khjku� uhk0;X þ ku� uhkq1;X

K rðhÞku� uhk1;X;

where

rðhÞ ¼ hþ ku� uhkq�11;X ð3:29Þ

with r(h) ? 0 as h ? 0.


So we obtain from (3.20) that

ku� uhk0;X 6 Z�1ðkuk0;X þ12ku� uhk2

0;XÞK rðhÞku� uhk1;X:

Namely, we arrive at (3.16).Now we turn to prove (3.15). From (3.8), (3.10) and (3.21), we

may estimate ku � uhk1,X as follows

ku� uhk1;X K kðI � P0hÞuk1;X þ jk� khjkP0hMuhk1;X

þ kP0hMð2N 01ðu2Þu3 � nhÞk1;X

þ kP0hMð2N 2ðu2Þu� vhÞk1;X

K infv2Sh

0ðXÞku� vk1;X þ jk� khj þ k2N 01ðu2Þu3 � nhk�1;X

þ k2N 2ðu2Þu� vhk�1;X:

Note that

k2N 01ðu2Þu3 � nhk�1;X ¼ supv2H1

0ðXÞ;kvk1;X¼1ZXN 1ðu2


�� 2N 01ðu2Þu2ðuh � uÞ

�v

and

k2N 2ðu2Þu� vhk�1;X ¼ supv2H1

0ðXÞ;kvk1;X¼1ZXN 2ðu2


�� 2N 2ðuðuh � uÞÞuÞv ;

which together with (2.18) and (2.19) imply

k2N 01ðu2Þu3 � nhk�1;X þ k2N 2ðu2Þu� vhk�1;X K ku� uhkq1;X:

Using (3.16) and (3.19), we then have


0ðXÞku� vk1;X þ ku� uhk0;X

þ ku� uhkq1;X K inf

v2Sh0ðXÞku� vk1;X þ rðhÞku� uhk1;X;

which implies (3.15).Combining (3.15), (3.16) and (3.19), we obtain (3.17). This

completes the proof. h

Remark 3.1. We mention that Cancès et al. [15] have given a priorierror estimates for this type of nonlinear eigenvalue problemsunder the assumption that the energy functional is convex, i.e.,E00 > 0 (c.f., also, [16]). We derive here the optimal a priori errorestimates for more general problems from using a different argu-ments with the initial triangulation being fine enough. However,it is still open whether Assumption A.V is satisfied by ground statesolutions of general orbital-free models [5,7,9].

We can then derive optimal a priori error estimates for both lin-ear and quadratic finite element approximations under certainassumptions, which are generalizations of that in [15,16].

Theorem 3.2. Let (kh,k,uh,k) be the ground state solutions of (3.3) inSh,k(X) for k = 1, 2 and Assumptions A.I and A.V hold. If N 1 satisfies(2.2) for l 2 [0,3 � q], then there exists h0 > 0 such that for h 2 (0,h0],there holdsjk� kh;1j þ ku� uh;1k0;X þ hku� uh;1k1;X K h2

: ð3:30Þ

If in addition, X is a cuboid, V 2 H1ðXÞ; E 2 C3ðð0;1Þ;RÞ; E000ðtÞt3=2 islocally bounded, and Assumption A.III is true for q = 2, then there existsh0 > 0 such that for h 2 (0,h0], there holds

jk� kh;2j þ hku� uh;2k0;X þ h2ku� uh;2k1;X K h4: ð3:31Þ

Proof. Note that when l 2 [0,3 � q], we have

ku� uh;1kq0;6q=ð5�lÞ;X 6 ku� uh;1kð5�q�lÞ=2

0;X ku� uh;1kð3q�5þlÞ=20;6;X

6 ku� uh;1k0;Xku� uh;1kq�11;X ;

which implies that (3.27) becomes as

a0ðu; P0hMð2N 01ðu2Þu3 � nhÞ;wuÞK ku� uh;1k0;Xku� uh;1kq�11;X kuk0;X:

ð3:32Þ

Taking into account (3.19), (3.25), (3.26), (3.28) and (3.32), wehave

kuk0;X K rðhÞku� uh;1k1;X;

where

rðhÞ ¼ hþ ku� uh;1k1;X; ð3:33Þ

with r(h) ? 0 as h ? 0. Hence (3.30) is a consequence of Theorem3.1 together with (3.33) when h0� 1.

Under the assumptions that V 2 H1(X) and E 2 C3ðð0;1Þ;RÞ, weobtain by a standard elliptic regularity argument that u 2 H3(X)(see, e.g., [42,43]). So we have ku � uh,2k1,X [ h2, ku � uh,2k0,X [ h3

and jkh,2 � kj[ h3 immediately from Theorem 3.1.Note that (3.18) implies

kh;2 � k ¼ ,ðrðuh;2 � uÞ;rðuh;2 � uÞÞ þ ððV þNðu2Þ � kÞ

� ðuh;2 � uÞ;uh;2 � uÞ þZ

Xxhðuh;2 � uÞ;

where

xh ¼ u2h;2

Nðu2h;2Þ � N ðu2Þuh;2 � u

:

Using the standard argument, we have that kuh,2 � uk0,1,X ? 0 ash ? 0. Hence xh is uniformly bounded in H1(X). Consequently, forh 2 (0,h0] and h0� 1, there holds

jkh;2 � kjK ku� uh;2k21;X þ ku� uh;2k�1;X: ð3:34Þ

To estimate ku � uh,2k�1,X, we let v 2 H10ðXÞ and wv 2 u\ be the

solution to the adjoint problem

a0ðu; wv ;/Þ ¼ ðv ;/Þ 8/ 2 u?: ð3:35Þ

Note that

,ðrwv ;r/Þþ ðV þNðu2Þ þ 2N 01ðu2Þu2 � kÞwv þ 2N 2ðuwvÞu;/� �

¼ 2Z�1ðN 01ðu2Þu2wv ;uÞðu;/Þ þ 2Z�1Dðuwv ;u2Þðu;/Þþ ð/� Z�1ð/;uÞu;/Þ8/ 2 H1

0ðXÞ:

We deduce from V 2 H1ðXÞ; E 2 C3ðð0;1Þ;RÞ, E000ðtÞt3=2 is locallybounded, and Assumption A.IV that wv 2 H3(X) (see, e.g., [42,43])and

kwvk3;X K kvk1;X 8v 2 H10ðXÞ:

Consequently

kwv � P0hwvk1;X K h2kvk1;X:

Combing (3.24) and (3.35) and the fact u ¼ uðu;uh;2Þ � Zuh;2 2 u?,for all v 2 H1

0ðXÞ we have


Z�1Z

Xvu ¼ Z�1a0ðu; wv ;uÞ ¼ Z�1a0ðu;u;wvÞ

¼ a0ðu; ðI � P0hÞu;wvÞ þ ðk� khÞa0ðu; P0hMuh;2;wvÞþ a0ðu; P0hMð2N 01ðu2Þu3 � nhÞ;wvÞþ a0ðu; P0hMð2N 2ðu2Þu� vhÞ;wvÞ

� 12

Z�1ku� uh;2k20;Xa0ðu; u;wvÞ;

where nh and vh are defined by (3.13) and (3.14) by replacing uh byuh,2.

We obtain from the similar argument to that in Theorem 3.1 thatZX

vuK h4kvk1;X 8v 2 H10ðXÞ:

Therefore, we have

ku� uh;2k�1;X ¼ supv2H1

0ðXÞnf0g

RX vðu� uh;2Þkvk1;X

K h4;

which together with (3.34) completes the proof. h

Hereafter, we assume that h0� 1 and Assumptions A.I–A.Vhold.

3.2. A posteriori error estimates

In this subsection, we shall provide a posteriori error estimatesof finite element approximations of (3.3). We shall apply the per-turbation argument (see, e.g., [17,18]) to build certain relationshipsbetween nonlinear eigenvalue problem (2.5) and linear boundaryproblem (A.2) and use the a priori error estimates (3.16) and(3.17) derived in Section 3.1.

Let að�; �Þ ¼ ð,r�;r�Þ, one sees that there exists a constant0 < ca 6 , <1 such that

cakvk21;X 6 aðv ;vÞ 8v 2 H1

0ðXÞ:

Let K : H�1ðXÞ ! H10ðXÞ be the operator defined by

aðKw;vÞ ¼ ðw;vÞ 8v 2 H10ðXÞ;

for which there holds

kKvk1;X K kvk�1;X 8v 2 H�1ðXÞ: ð3:36Þ

Thus (2.5) and (3.3) can be rewritten as

u ¼ Kðku� Vu�Nðu2ÞuÞ

and

uh ¼ PhKðkhuh � Vuh �Nðu2hÞuhÞ;

respectively, where Ph : H10ðXÞ ! Sh

0ðXÞ is the Galerkin projectiondefined by

aðw� Phw;vÞ ¼ 0 8v 2 Sh0ðXÞ: ð3:37Þ

For any w 2 H10ðXÞ, there apparently hold

kPhwka;X K kwka;X and limh!0kw� Phwka;X ¼ 0; ð3:38Þ

where k � ka;X ¼ ,1=2kr � k0;X.We have for wh ¼ Kðkhuh � Vuh �Nðu2

hÞuhÞ that

uh ¼ Phwh: ð3:39Þ

Now we shall set up a relationship of a posteriori error esti-mates between the nonlinear eigenvalue problem and the associ-ated linear boundary value problem, from which various aposteriori error estimates for the nonlinear eigenvalue problemcan be obtained since the results for the linear boundary valueproblem have been well-constructed (see Appendix A).

Theorem 3.3. There exists j(h) 2 (0,1) such that j(h) ? 0 as h ? 0and

ku� uhka;X ¼ kwh � Phwhka;X þOðjðhÞÞku� uhka;X: ð3:40Þ

Proof. Since (3.39) implies

u� uh ¼ wh � Phwh þ u�wh;

it is sufficient to estimate ku � whka,X.By definition, we have

u�wh ¼ Kðku� khuhÞ þ KVðuh � uÞþ KðN ðu2

hÞuh �Nðu2ÞuÞ: ð3:41Þ

From (3.36) and Theorem 3.1, we get for the first term of (3.41) that

kKðku� khuhÞka;X 6 kkKðu� uhÞka;X þ jk� khjkKuhka;X

Kku� uhk0;X þ jk� khjK rðhÞku� uhka;X: ð3:42Þ

For the second term of (3.41), since there holds

kKVðu� uhÞka;X K kVðu� uhÞk�1;X K kVk0;Xku� uhk0;3;X; ð3:43Þ

it is only necessary for us to prove that there exists jðhÞ 2 ð0;1Þsuch that jðhÞ ! 0 as h ? 0 and

ku� uhk0;3;X ¼ OðjðhÞÞku� uhk1;X: ð3:44Þ

Indeed, we obtain from (3.43) and (3.44) that

kKVðu� uhÞka;X K jðhÞku� uhka;X: ð3:45Þ

Since for any e > 0, there exists a positive constant Ce satisfying

kwk0;3;X 6 Cekwk0;X þe2kwk1;X 8w 2 H1

0ðXÞ; ð3:46Þ

we get from (3.16) that there exists a positive constant C suchthat

ku� uhk0;3;X 6 CCerðhÞ þe2

� �ku� uhk1;X 8h 2 ð0; h0;

which leads to (3.44) for some kðhÞ 2 ð0;1Þ satisfying kðhÞ ! 0 ash ? 0.

Using (2.11), (3.16) and (3.36), we have for the last term of(3.41) that

kKðN ðu2hÞuh �Nðu2ÞuÞka;X K kN ðu2

hÞuh �Nðu2Þuk�1;X

K ku� uhk0;X K rðhÞku� uhka;X: ð3:47Þ

Set jðhÞ ¼ rðhÞ þ jðhÞ, we obtain from (3.41), (3.42), (3.45), and(3.47) that

ku�whka;X KjðhÞku� uhka;X: ð3:48Þ


In fact, Theorem 3.3 implies that up to a high order term, the er-ror of the nonlinear eigenvalue problem is equivalent to that ofboundary value problem (A.1) with khuh � Vuh �Nðu2

hÞuh as thesource term. Define

~jðh0Þ ¼ suph2ð0;h0

jðhÞ:

Obviously, ~jðh0Þ � 1 if h0� 1.

Remark 3.2. Either to ensure that the discrete problem is well-posed or to provide a structure-preserving approximation, we shallrequire that h0 is small enough for a finite element approximationof (3.3) (c.f., e.g., [44,45]).


Let T 0 be the initial mesh with size h0 2 (0,1) and denote T theclass of all conforming refinements by bisection of T 0 with shaperegularity constant c⁄ defined by (3.1). For T h 2 T, we define theelement residual RsðuhÞ and the jump residual Je(uh) for (3.3) asfollows:

RsðuhÞ ¼ khuh �Nðu2hÞuh � Vuh þ ,Duh in s 2 T h;

JeðuhÞ ¼ �,ruþh � mþ � ,ru�h � m� ¼ ,½½ruhe � me on e 2 Eh;

where e is the common side of elements s+ and s� with unit out-ward normals m+ and m�, respectively, and me = m�.

For s 2 T h, we define the local error indicator gh(uh,s) by

g2hðuh; sÞ ¼ h2

skRsðuhÞk20;s þ

Xe2Eh ;e�@s

hekJeðuhÞk20;e ð3:49Þ

and the oscillation osch(uh,s) by

osc2hðuh; sÞ ¼ h2

skRsðuhÞ � RsðuhÞk20;s; ð3:50Þ

where RsðuhÞ is the L2-projection of RsðuhÞ to polynomials of somedegree on s.

Given a subset x �X, we define the error estimator gh(uh,x)and the oscillation osch(uh,x) by

g2hðuh;xÞ ¼

Xs2T h ;s�x

g2hðuh; sÞ and

osc2hðuh;xÞ ¼

Xs2T h ;s�x

osc2hðuh; sÞ:

Theorem 3.4. Let h0� 1 and h 2 (0,h0]. There exist positive con-stants C1,C2 and C3, which only depend on ,; ca and the shaperegularity constant c⁄ such that

ku� uhk2a;X 6 C1g2

hðuh;XÞ ð3:51Þ

and

C2g2hðuh;XÞ 6 ku� uhk2

a;X þ C3osc2hðuh;XÞ: ð3:52Þ

Proof. Recall that Lwh ¼ khuh � Vuh �Nðu2hÞuh, where L is defined

by (A.1). From (A.6) and (A.7) we have

kwh � Phwhk2a;X 6

eC 1~g2hðPhwh;XÞ ð3:53Þ

andeC 2 ~g2hðPhwh;XÞ 6 kwh � Phwhk2

a;X þ eC3gosc2hðPhwh;XÞ; ð3:54Þ

where ~g2hðPhwh;XÞ and gosc2

hðPhwh;XÞ are defined by (A.5) when uh isreplaced by Phwh and eC1; eC2 and eC3 are defined in Theorem A.1.

Note that (3.40) means

ku� uhka;X 6 ð1þ eC ~jðh0ÞÞkwh � Phwhka;X: ð3:55Þ

We may choose

C1 ¼ eC1ð1þ eC ~jðh0ÞÞ2 ð3:56Þ

and get (3.51) from combing (3.53) and (3.55).Similar to the proof of (3.51), we obtain (3.52) from (3.39),

(3.40) and (3.54). In particular, we may choose C2 and C3 satisfying

C2 ¼ eC2ð1� eC ~jðh0ÞÞ2; C3 ¼ eC3ð1� eC ~jðh0ÞÞ2: ð3:57Þ


1 Like the most work on numerical study of convergence of adaptive finite elementapproximations, we have ignored two important theoretical and practical issues: theinexact solution of the resulting algebraic system and the numerical integration.

Remark 3.3. We may derive a posteriori error estimates for finiteelement eigenvalue approximations from (3.17), but they are notoptimal. We mention that an a posteriori error estimate for thistype of nonlinear eigenvalue problems has been given in [11].

However, we note that upper bound estimate (3.51), which isrequired in our proof of the convergence rate and complexity ofadaptive finite element methods, does not involve any oscillationterm.

4. Adaptive approximations

In this section, we shall study numerical approximations of(2.5) using adaptive finite element methods. We shall design anadaptive finite element algorithm and prove the convergence andoptimal complexity of adaptive finite element approximations,for which certain relationship between nonlinear eigenvalue prob-lem (2.5) and linear boundary problem (A.2) will be built by theperturbation argument (see, e.g., [17,18]), too. We shall mentionthat the a priori error estimates (3.16) and (3.17) derived in Section3.1 will also play an important role here.

To describe the adaptive finite element algorithm for (1.1), weshall replace the subscript h by an iteration counter k of the adap-tive algorithm afterwards for convenience. Given an initial triangu-lation T 0 with size h0, we can generate a sequence of nestedconforming triangulations T k using the following loop:

Solve ! Estimate ! Mark ! Refine:

More precisely, to get T kþ1 from T k we first solve the discrete equa-tion to get (kk,uk) on T k. The error indicator is estimated by (kk,uk)and then used to mark a set of elements that are to be refined. Ele-ments are refined in such a way that the triangulations are stillshape regular and conforming.

Here, we shall not discuss the step ‘‘Solve’’, which deserves aseparate investigation. We assume that we have the exact solu-tions of finite-dimensional problems at hand.1 The procedure ‘‘Esti-mate’’ determines the element indicators for all elements s 2 T k. Theonly requirement on ‘‘Mark’’ is the so-called Dörfler Strategy, whichis stated as follows:

Marking Strategy Given a parameter 0 < h < 1:

1. Construct a minimal subset Mk of T k by selecting some ele-ments in T k such that
Xs2Mk
g2kðuk; sÞP h

Xs2T k

g2kðuk; sÞ: ð4:1Þ

2. Mark all the elements in Mk.

As shown in [25], the procedure ‘‘Refine’’ here is not required tosatisfy the Interior Node Property of [27,28]. For any T k 2 T and asubset Mk � T k of marked elements at the kth step, the ‘‘Refine’’procedure outputs a conforming triangulation T kþ1 2 T, where allelements of Mk are bisected at least once. We define

RT k!T kþ1¼ T k n ðT k \ T kþ1Þ

as the set of refined elements, thus Mk � RT k!T kþ1. Note that more

than the marked elements in Mk are refined in order to keep themesh conforming.

Now we address the adaptive finite element algorithm for (1.1)as follows.

Algorithm 4.1. Adaptive finite element algorithm

1. Pick an initial mesh T 0, and let k = 0.2. Solve (3.3) on T k and get the finite element ground state solu-

tion (kk,uk).


3. Compute local error indicators gkðuk; sÞ 8s 2 T k.4. Construct Mk � T k by a marking strategy that satisfies (4.1).5. Refine T k to get a new conforming mesh T kþ1.6. Let k = k + 1 and go to 2.

It is shown in [11] that the adaptive finite element approxima-tions produced by the above adaptive finite element algorithm isconvergent. Although we are able to give some analysis of adaptivefinite element eigenvalue approximations by using (3.17), they arenot optimal. In the following discuss, we focus our attention on theanalysis of adaptive finite element eigenfunction approximationsonly.

4.1. Convergence rate

Now we shall establish some relationships between two levelapproximations. We use T H to denote a coarse mesh and T h to de-note a refined mesh of T H .

Lemma 4.1. Let h;H 2 ð0;h0; wh ¼ Kðkhuh � Vuh �Nðu2hÞuhÞ and

wH ¼ KðkHuH � VuH �Nðu2HÞuHÞ. Then

ku�uhka;X ¼ kwH�PhwHka;XþOð~jðh0ÞÞ ku�uhka;Xþku�uHka;X

� �;

ð4:2Þ

oschðuh;XÞ ¼goschðPhwH;XÞ þ Oð~jðh0ÞÞ ku� uhka;X þ ku� uHka;X

� �;

ð4:3Þ

and

ghðuh;XÞ ¼ ~ghðPhwH;XÞ þ Oð~jðh0ÞÞ ku� uhka;X þ ku� uHka;X

� �:

ð4:4Þ

Proof. First, combining (3.38) and (3.48) and the following fact

u� uh ¼ wH � PhwH þ PhðwH �whÞ þ u�wH;

we obtain (4.2).Due to uh = PhwH + Ph(wh � wH), we obtain from the definition of

oscillation that

goschðPhwh;XÞ ¼goschðPhwH þ Phðwh �wHÞ;XÞ: ð4:5Þ

Note that goschðPhwh;XÞ ¼ oschðuh;XÞ, (3.39) and (4.5), we only needto estimate goschðPhðwh �wHÞ;XÞ.

Since Lwh ¼ khuh � Vuh �Nðu2hÞuh and LwH ¼ kHuH�

VuH �Nðu2HÞuH , we know that wh � wH is the solution of typical

boundary value problem (A.1) with khuh � kHuH þ VðuH � uhÞþN ðu2

HÞuH �Nðu2hÞuh as the source term. Let G = Ph(wh � wH), we

have

eRsðGÞ ¼ khuh � kHuH þ VðuH � uhÞ þ N ðu2HÞuH �Nðu2

hÞuh � LG

and

gosc2hðPhðwh �wHÞ;XÞ ¼

Xs2T h

gosc2hðG; sÞ

¼Xs2T h

h2sk eRsðGÞ � eRsðGÞk2

0;s

6

Xs2T h

h2sk eRsðGÞ þ LG

� ð eRsðGÞ þ LGÞk20;s

þXs2T h

h2skLG� LGk2

0;s; ð4:6Þ

where eRsðGÞ is defined by (A.4) by replacing uh by G.

Next, we estimate the two terms in (4.6). Using the fact thatkhuh and kHuH are piecewise polynomials over T h and T H respec-tively, we get thatXs2T h

h2sk eRsðGÞ þ LG� ð eRsðGÞ þ LGÞk2

0;s

KXs2T h

h2s kVðuH � uhÞk2

0;s þ kN ðu2HÞuH

��Nðu2

hÞuhk20;s

�: ð4:7Þ

Due to the inverse inequality, we haveXs2T h

h2skVðuH � uhÞk2

0;s KXs2T h

h2skVk

20;skuH � uhk2

0;1;s;

which together with the Sobolev inequality yieldsXs2T h

h2skVðuH � uhÞk2

0;s KXs2T h

hskVk20;skuh � uHk2

0;6;s K hkuh � uHk21;X:

ð4:8Þ

It then follows from (2.12), (4.7) and (4.8) that

Xs2T h

h2sk eRsðGÞþLG�ð eRsðGÞþLGÞk2

0;s

!1=2

Kðh1=2þhÞkuh�uHk1;X

Kh1=20 ku�uhka;Xþku�uHka;X

� �:

Note that the inverse inequality, (3.38) and (3.48) imply

Xs2T h

h2skLG� LGk2

0;s

!1=2

K kPhðwh �wHÞka;X

K ~jðh0Þðku� uhka;X þ ku� uHka;XÞ: ð4:9Þ

We conclude from (4.6), (4.7) and (4.9) thatgoschðPhðwh �wHÞ;XÞK ~jðh0Þðku� uhka;X þ ku� uHka;XÞ; ð4:10Þ

here and hereafter we still denote h1=20 þ ~jðh0Þ by ~jðh0Þ. Combing

(4.5) and (4.10), we arrive at (4.3).Finally, we prove (4.4). We obtain from (A.7), (3.48) and (4.10)

that

~ghðPhðwh �wHÞ;XÞK kðwh �wHÞ � Phðwh �wHÞka;X

þgoschðPhðwh �wHÞ;XÞ

K ~jðh0Þ ku� uhka;X þ ku� uHka;X

� �;

which together with the fact

~ghðPhwh;XÞ ¼ ~ghðPhwH þ Phðwh �wHÞ;XÞ

leads to

~ghðPhwh;XÞ ¼ ~ghðPhwH;XÞ þ Oð~jðh0ÞÞ ku� uhka;X þ ku� uHka;X

� �:

This is nothing but (4.4) and completes the proof. h

Now we are able to get the convergence rate of the adaptivefinite element computations.

Theorem 4.1. Let h 2 (0,1) and fukgk2N0be a sequence of finite

element solutions corresponding to a sequence of nested finite elementspaces fSk

0ðXÞgk2N0produced by Algorithm 4.1. If h0� 1, then there

exist constants c > 0 and n 2 (0,1) depending only on ,; ca, the shaperegularity constant c⁄ and the marking parameter h such that

ku�ukþ1k2a;Xþcg2

kþ1ðukþ1;XÞ6 n2 ku�ukk2a;Xþcg2

kðuk;XÞ� �

; ð4:11Þ


where

c ¼~c

1� C4d�11 ~j2ðh0Þ

; ð4:12Þ

with C4 a positive constant.

Proof. For convenience, we use uh, uH to denote uk+1 and uk,respectively. So it is sufficient to prove that for uh and uH, thereholds,

ku� uhk2a;X þ cg2

hðuh;XÞ 6 n2 ku� uHk2a;X þ cg2

HðuH;XÞ� �

:

Due to wh ¼ Kðkhuh � Vuh �Nðu2hÞuhÞ, wH ¼ KðkHuH � VuH�

Nðu2HÞuHÞ and uH = PHwH, we obtain from Theorem A.2 that there ex-

ist constants ~c > 0 and ~n 2 ð0;1Þ satisfying

kwH � PhwHk2a;X þ ~c~g2

hðPhwH;XÞ

6 ~n2 kwH � uHk2a;X þ ~cg2

HðuH;XÞ� �

: ð4:13Þ

Using (4.2) and the Young’s inequality, we see that there exists aconstant bC > 0 such that

ku� uhk2a;X þ ~cg2

hðuh;XÞ 6 ð1þ d1ÞkwH � PhwHk2a;X

þ bCð1þ d�11 Þ~j2ðh0Þðku� uhk2

a;X

þ ku� uHk2a;XÞ þ ð1þ d1Þ~c~g2

hðPhwH;XÞ

þ bCð1þ d�11 Þ~j2ðh0Þ~cðku� uhk2

a;X

þ ku� uHk2a;XÞ;

where d1 2 (0,1) satisfies

ð1þ d1Þ~n2 < 1: ð4:14Þ

It then follows from (3.48), (4.13), and identity ~gHðPHwH;XÞ ¼gHðuH;XÞ that there exists a positive constant C⁄ depending on bCand ~c such that

ku�uhk2a;Xþ ~cg2

hðuh;XÞ6 ð1þ d1Þ~n2 kwH �uHk2a;Xþ ~cg2

HðuH ;XÞ� �

þC�d�11 ~j2ðh0Þðku�uhk2

a;Xþku�uHk2a;XÞ

6 ð1þ d1Þ~n2 1þ eC ~jðh0Þ� �2

ku�uHk2a;Xþ ~cg2

HðuH;XÞ� �

þC�d�11 ~j2ðh0Þ ku�uhk2

a;Xþku�uHk2a;X

� �:

Hence, if h0� 1, then there exists a positive constant C4 dependingon C⁄ and eC such that

ku� uhk2a;X þ ~cg2

hðuh;XÞ 6 ð1þ d1Þ~n2 ku� uHk2a;X þ ~cg2

HðuH;XÞ� �

þ C4 ~jðh0Þku� uHk2a;X

þ C4d�11 ~j2ðh0Þku� uhk2

a;X:

Consequently,

ku� uhk2a;X þ

~c1� C4d

�11 ~j2ðh0Þ

g2hðuh;XÞ

6ð1þ d1Þ~n2 þ C4 ~jðh0Þ

1� C4d�11 ~j2ðh0Þ

ku� uHk2a;X þ

ð1þ d1Þ~n2~c1� C4d

�11 ~j2ðh0Þ

g2HðuH;XÞ:

Since h0� 1 implies ~kðh0Þ � 1, we have that the constant ndefined by

n ¼ ð1þ d1Þ~n2 þ C4 ~jðh0Þ1� C4d

�11 ~j2ðh0Þ

!1=2

satisfies n 2 (0,1) if h0� 1.

Finally, we arrive at (4.11) by using the fact that

ð1þ d1Þ~n2~cð1þ d1Þ~n2 þ C4 ~jðh0Þ

< c:


4.2. Complexity

To investigate the complexity of Algorithm 4.1 for solving (3.3),we need to use the following lemma, which connects the total er-ror reduction with the Dörfler marking strategy.

Lemma 4.2. Let uH 2 SH0 ðXÞ be the solution of (3.3) over T H, and

T h 2 T be any refinement of T H such that the discrete solutionuh 2 Sh

0ðXÞ satisfies

ku� uhk2a;X þ c�osc2

hðuh;XÞ 6 b2� ku� uHk2

a;X þ c�osc2HðuH;XÞ

� �;

ð4:15Þ

with constants c⁄ > 0 and b� 2 0;ffiffi12

q� �. If h0� 1, then the following

inequality holdsXs2R

~g2HðuH; sÞP h

Xs2T H

~g2HðuH; sÞ;

with R ¼ RT H!T h; h ¼ eC 2ð1�2 ~b�

2ÞeC 0ðeC 1þð1þ2C2�eC 1Þ~c�Þ

, where eC0; ~b�; C� and ~c� are

constants defined in the proof.

Proof. Recall that wH ¼ KðkHuH � VuH �Nðu2HÞuHÞ and wh ¼

Kðkhuh � Vuh �Nðu2hÞuhÞ, we conclude from (4.2) and (4.3) that

kwH �PhwHka;X ¼ ku�uhka;XþOð~jðh0ÞÞ kwH �PHwHka;XþkwH �PhwHka;X

� �andgoschðPhwH;XÞ ¼ oschðuh;XÞ

þ Oð~jðh0ÞÞ kwH � PHwHka;X þ kwH � PhwHka;X

� �:

Using the similar arguments in the proof of Theorem 4.1, from(4.15) we obtain

kwH � PhwHk2a;X þ ~c�gosc2

hðPhwH;XÞ

6 ~b�2 kwH � PHwHk2

a;X þ ~c�gosc2HðPHwH;XÞ

� �ð4:16Þ

with

~b� ¼ð1þ d1Þb2

� þ C5 ~jðh0Þ1� C5d

�11 ~j2ðh0Þ

!1=2

; ~c� ¼c�

1� C5d�11 ~j2ðh0Þ

; ð4:17Þ

where C5 > 0 and d1 2 (0,1) are the same as that in the proof of The-orem 4.1.

Set eC0 ¼maxf1;eC 3~c� g, we get from (A.7) that

ð1� 2 ~b�2ÞeC2 ~g2

HðPHwH;XÞ

6 ð1� 2 ~b�2Þ kwH � PHwHk2

a;X þ eC3gosc2HðPHwH;XÞ

� �6eC0ð1� 2 ~b�

2Þ kwH � PHwHk2a;X þ ~c�gosc2

HðPHwH;XÞ� �

;

which together with (4.16) implieseC2eC0

ð1� 2 ~b�2ÞXs2T H

~g2HðPHwH; sÞ

6 kwH � PHwHk2a;X þ ~c�gosc2

HðPHwH;XÞ � kwH � PhwHk2a;X

� 2~c�gosc2hðPhwH;XÞ: ð4:18Þ


Thus using the orthogonality and Lemma A.1, we arrive at

kwH � PHwHk2a;X � kwH � PhwHk2

a;X ¼ kPHwH � PhwHk2a;X

6eC1

Xs2R

~g2HðPHwH; sÞ: ð4:19Þ

Note thatXs2T H\T h

gosc2HðPHwH; sÞ 6 2

Xs2T H\T h

gosc2hðPhwH; sÞ

þ 2C2�kPHwH � PhwHk2

a;X

andgosc2HðPHwH; sÞ 6 ~g2

HðPHwH; sÞ 8s 2 T H;

where C⁄ is a positive constant depending on , and the shape reg-ularity constant c⁄.

Thus we havegosc2HðPHwH;XÞ � 2gosc2

hðPhwH;XÞ6

Xs2R

~g2HðPHwH; sÞ þ

Xs2T H\T h

gosc2HðPHwH; sÞ

� 2X

s2T H\T h

gosc2hðPhwH; sÞ

6

Xs2R

~g2HðPHwH; sÞ þ 2C2

�kPHwH � PhwHk2a;X

6 ð1þ 2C2�eC1Þ

Xs2R

~g2HðPHwH; sÞ: ð4:20Þ

Combining (4.18)–(4.20), we obtaineC 2eC0

ð1�2 ~b�2ÞXs2T H

~g2HðPHwH;sÞ6 eC1þð1þ2C2

�eC 1Þ~c�

� �Xs2R

~g2HðPHwH;sÞ;

namely,Xs2R

~g2HðuH; sÞP h

Xs2T H

~g2HðuH; sÞ;

with

h ¼eC 2ð1� 2 ~b�

2ÞeC0ðeC1 þ ð1þ 2C2�eC1Þ~c�Þ

:


Now we analyze the complexity in a class of functions, which isintroduced as follows

Asc ¼ fv 2 H1

0ðXÞ : jv js;c <1g;

where s, c > 0,

jvjs;c ¼ supe>0

e inffT k�T 0 :infvk2T k

ðkv�vkk2a;Xþðcþ1Þosc2

kðvk ;T kÞÞ1=2

6egð#T k �#T 0Þs

and T k � T 0 means T k is a refinement of T 0. It is seen from the def-inition that, for all c > 0; As

c ¼ As1. For simplicity, here and hereaf-

ter, we use As to stand for As1, and use jvjs to denote jvjs,c. So As is

the class of functions that can be approximated within a given tol-erance e by continuous piecewise polynomial functions over a par-tition T k with number of degrees of freedom satisfying #T k �#T 0

K e�1=sjv j1=ss .

The key point relating the best mesh with adaptive finite ele-ment triangulations is the fact that Marking Strategy selects themarked setMk with minimal cardinality (see Appendix C for a de-tailed proof).

Lemma 4.3. Let u 2 As;uk be the discrete solution of (3.3) over a

conforming partition T k, and h 2 ð0; C2cC3ðC1þð1þ2C2

�C1ÞcÞÞ. If h0� 1, then

#Mk K ku� ukk2a;X þ cosc2

kðuk;XÞ� ��1=2s

juj1=ss ; ð4:21Þ

where the hidden constant depends on the discrepancy between the

marking parameter h and C2cC3ðC1þð1þ2C2

�C1ÞcÞ.

The proof of complexity of Algorithm 4.1 relies on the lowerbound (3.52), the linear convergence rate (4.11), and the cardinal-ity of Mk (4.21).

Theorem 4.2. Let u 2 As and fukgk2N0be a sequence of finite element

solutions corresponding to a sequence of nested finite element spacesfSk

0ðXÞgk2N0produced by Algorithm 4.1 . If h0� 1, then


kðuk;XÞK ð#T k �#T 0Þ�2sjuj2s ;

where the hidden constant depends on the exact solution u and the dis-crepancy between h and C2c

C3ðC1þð1þ2C2�C1ÞcÞ

.

Proof. It follows from Theorem 6.1 in [46] that

#T k �#T 0 KXk�1

j¼0

#Mj;

which together with (4.21) implies

#T k �#T 0 KXk�1

j¼0

ku� ujk2a;X þ cosc2

j ðuj;XÞ� ��1=2s

juj1=ss :

Note that (3.52) implies

ku� ujk2a;X þ cg2

j ðuj;XÞ 6 �C ku� ujk2a;X þ cosc2

j ðuj;XÞ� �

;

where �C ¼maxð1þ cC2; C3

C2Þ: It then turns out

#T k �#T 0 KXk�1

j¼0

ku� ujk2a;X þ cg2

j ðuj;XÞ� ��1=2s

juj1=ss :

Due to (4.11), we obtain for 0 6 j < k that

ku� ukk2a;X þ cg2

kðuk;XÞ 6 n2ðk�jÞ ku� ujk2a;X þ cg2

j ðuj;XÞ� �

:

Consequently,

#T k �#T 0 K juj1=ss ku� ukk2

a;X þ cg2kðuk;XÞ

� ��1=2sXk�1

j¼0

nk�j

s

K juj1=ss ku� ukk2

a;X þ cg2kðuk;XÞ

� ��1=2s;

the last inequality holds because of the fact n < 1.Since osck(uk,X) 6 gk(uk,X), we arrive at

#T k �#T 0 K ku� ukk2a;X þ cosc2

kðuk;XÞ� ��1=2s

juj1=ss :


5. Numerical examples

In this section, we shall report on some numerical experimentsin three dimensions to support our theory. Our numerical compu-tations are carried out on Shenteng 7000 supercomputer in theComputer Network Information Center, Chinese Academy of Sci-ences. Our codes are based on the toolbox PHG of the State KeyLaboratory of Scientific and Engineering Computing, Chinese Acad-emy of Sciences. All of the computational results are given in atom-ic unit (a.u.).


Example 1. Consider the ground state solution of GPE for BEC witha harmonic oscillator potential of a stirrer corresponding a far-bluedetuned Gaussian laser beam [3,47], i.e.,

Vðx; y; zÞ ¼ 12

c2x x2 þ c2

y y2 þ c2z z2

� �þxe�dððx�r0Þ2þy2Þ;

where cx = 1, cy = 1, cz = 2,x = 4, d = r0 = 1. We solve the followingnonlinear problem: find ðk;uÞ 2 R� H1

0ðXÞ such that kuk0,X = 1 and

� 12 Dþ V þ bjuj2

� �u ¼ ku in X;

u ¼ 0 on @X;

(

where X = [�8,8]3 and b = 200.We first carry out the computations on uniform meshes. The

numerical errors are plotted against the mesh size in Fig. 1. It isobserved that the convergence rates are consistent with thatTheorem 3.2 predicts for both linear finite elements and quadraticfinite elements.

We solve the problem by using Algorithm 4.1 and plot thenumerical errors with respect to the number of degrees of freedomin Fig. 2. It is shown that error ku � uhk1,X is proportional to the a

10−0.9 10−0.7 10−0.5 10−0.3 10−0.110−3

10−2

10−1

100

mesh size

erro

r

linear elements for BEC on uniform meshes

||u−uh||1,Ω

||u−uh||0,Ω

|λ−λh|

|E−Eh|

slope=1.0slope=2.0

Fig. 1. (Example 1) Convergence curves of the numerical erro

103 104

10−1

100

number of degrees of freedom

erro

r

linear elements for BEC on adaptive meshes

||u−uh||1,Ω

a posteriori error estimatorslope=−1/3

Fig. 2. (Example 1) The convergence curves of the a posteriori

posteriori error estimators, which implies the efficiency of the aposteriori error estimators given in Section 3.2. By using linearfinite elements, we observe that the convergence curves of errorku � uhk1,X and the ground state energy (also the eigenvalue) areapproximately parallel to the lines with slope �1/3 and �2/3,respectively. These mean that the approximations of ground statesolutions have the optimal convergence rate, which coincides withTheorem 4.2. The same conclusions can be obtained for quadraticelements from Fig. 3, too.

We present contour plots of the ground state solutions on theinterior slice z = 0 and the corresponding adaptive finite elementmeshes in Figs. 4 and 5 for linear and quadratic elementsrespectively. It is observed that with the a posteriori errorestimators, the refinement is carried out automatically at theregions where the computed functions vary rapidly. As a result, thecomputational accuracy can be controlled efficiently and thecomputational cost is reduced significantly.

Example 2. Consider lithium-helium molecules by TFvW orbital-free model. The external electrostatic potential is

10−0.6 10−0.5 10−0.4 10−0.3 10−0.2 10−0.1 100

10−4

10−3

10−2

10−1

100

mesh size

erro

rquadratic elements for BEC on uniform meshes

||u−uh||1,Ω

||u−uh||0,Ω

|λ−λh|

|E−Eh|

slope=2.0slope=3.0slope=4.0

rs for linear and quadratic elements on uniform meshes.

103 10410−3

10−2

10−1


erro

r

linear elements for BEC on adaptive meshes

|λ−λh|

|E−Eh|

slope=−2/3

error estimator and numerical errors for linear elements.

104 10510−2

10−1

100


erro

r

quadratic elements for BEC on adaptive meshes

||u−uh||1,Ω


104 105

10−5

10−4

10−3

10−2

10−1


erro

r

quadratic elements for BEC on adaptive meshes

|λ−λh|

|E−Eh|

slope=−4/3

Fig. 3. (Example 1) The convergence curves of the a posteriori error estimator and numerical errors for quadratic elements.

Fig. 4. (Example 1) Contour plot of the ground state solution u2h;1 on the interior slice z = 0 and the corresponding adaptive meshes of linear elements.


VðxÞ ¼ �X2

j¼1

zjjx� xjj�1; ð5:1Þ

where xj and zj are the location and the charge of the jth nucleus(j = 1,2), respectively. The nonlinear terms are given by

Nðu2Þ ¼Z

R3

u2ðyÞj � �yjdyþ 5

3CTFu4=3 þ txcðu2Þ;

tLDAc ðqÞ ¼

0:0311 ln rs � 0:0584þ 0:0013rs ln rs � 0:0084rs

�ð0:1423þ 0:0633rs þ 0:1748ffiffiffiffirsp Þ=ð1þ 1:0529

ffiffiffiffirsp þ 0:33

where CTF ¼ 310 ð3p2Þ

23 and vxc is the exchange-correction potential.

The exchange-correction potential used in our computations is gi-ven by

txcðqÞ ¼ tLDAx ðqÞ þ tLDA

c ðqÞ; ð5:2Þ

where

tLDAx ðqÞ ¼ �

3p

� �1=3

q1=3;

if rs < 134rsÞ2 if rs P 1

Fig. 5. (Example 1) Contour plot of the ground state solution u2h;2 on the interior slice z = 0 and the corresponding adaptive meshes of quadratic elements.

103 104

100

101


erro

r

linear elements for LiH on adaptive meshes

||u−uh||1,Ω


103 10410−2

10−1

100


erro

rlinear elements for LiH on adaptive meshes

|λ−λh|

|E−Eh|

slope=−2/3

Fig. 6. (Example 2) The convergence curves of the a posteriori error estimator and numerical errors for linear elements.

104

10−1

100

101


erro

r

quadratic elements for LiH on adaptive meshes

||u−uh||1,Ωa posteriori error estimatorslope=−2/3

104

10−3

10−2

10−1

100


erro

r

quadratic elements for LiH on adaptive meshes

|λ−λh|

|E−Eh|

slope=−4/3

2 × 104 2 × 104






and rs ¼ ð 34pq Þ

1=3. We solve the following nonlinear problem: findðk;uÞ 2 R� H1

0ðXÞ such that kuk20;X ¼ 4 and

� 110Du� z1

jx�x1 ju� z2

jx�x2 juþu

RXjuðyÞj2jx�yj dyþ 5

3 CTF u7=3þvxcðu2Þu¼ ku in X;

u¼ 0 on @X;

(

where x1 = (�1.5,0,0), z1 = 3, x2 = (1.5,0,0), z2 = 1, and X =(�10.0,10.0)3.

The numerical results obtained by Algorithm 4.1 are shown inFigs. 6 and 7 for both linear and quadratic elements, respectively. Itis observed that the a posteriori error estimators as well as theerrors of numerical approximations of ground state solution,energy, and eigenvalue have the optimal convergence rates. Thesevalidate our theory (c.f. Theorem 4.2). The cross-sections of the

contour plot and the adaptive meshes are displayed in Figs. 8 and9, from which we observed that more refined nodes appear in thearea where the nuclei are located.

Example 3. Finally, we consider an aluminum cluster in the facecentered cubic lattice consisting of 4 � 4 � 4 unit cells with avacancy in the center. There are totally 364 aluminum atoms inthe system, LDA (5.2) and the GHN pseudopotential [48] are used.We solve the following nonlinear problem: find ðk;uÞ 2 R� H1

0ðXÞsuch that kuk2

0;X ¼ 364 and

� 110 DuþVGHN

pseu uþ uR

XjuðyÞj2jx�yj dyþ 5

3 CTFu7=3 þ vxcðu2Þu¼ ku in X;

u¼ 0 on @X;

(

where X = (�50.0,50.0)3.

103 104 10510−1

100

101


erro

rlinear elements for aluminum cluster on adaptive meshes

||u−uh||1,Ω


103 104 10510−4

10−3

10−2

10−1

100

101


erro

r

linear elements for aluminum cluster on adaptive meshes

|λ−λh|

|E−Eh|

slope=−2/3

Fig. 10. (Example 3) The convergence curves of the a posteriori error estimator and numerical errors for linear elements.

104 105

100

101


erro

r

quadratic elements for aluminum cluster on adaptive meshes

||u−uh||1,Ω


104 105

10−2

10−1

100


erro

rquadratic elements for aluminum cluster on adaptive meshes

|λ−λh|

|E−Eh|

slope=−4/3






The numerical errors as well as the reductions of a posteriorierror estimators are shown in Figs. 10 and 11, which support ourresults of convergence and complexity. The cross-sections of theadaptive meshes are displayed in Figs. 12 and 13, respectively.We observe that with the a posteriori error estimators, therefinement is carried out automatically at the regions where thecomputed functions vary rapidly, especially near the nuclei. As aresult, the computational accuracy can be controlled efficientlyand the computational cost is reduced significantly.

6. Concluding remarks

We have analyzed finite element approximations of theground state solutions for a class of nonlinear eigenvalue prob-lems which are derived from quantum physics. We have giventhe optimal a priori error estimates and analyzed the conver-gence and complexity of adaptive finite element algorithms.We have also applied finite element discretizations to solve sev-eral typical models in quantum physics, which support ourtheory.

The model we studied in this paper is particularly interestingfrom a mathematical point of view (c.f., e.g., [7,49]) and is our startpoint of analyzing the more complex quantum models. We under-stand that Kohn–Sham model is powerful for electronic structurecalculations and is the most widely used model in Physics andChemistry. Indeed, it is our on-going project to study finite elementapproximations of a more general class of eigenvalue problems de-rived from Kohn–Sham density functional theory [8] in electronicstructure calculations, where multiple eigenvalues have to be con-sidered. We shall address the a priori and a posterior error esti-mates and design and analyze adaptive finite element algorithmsin our forthcoming work.

Acknowledgements

The authors thank Dr. Xiaoying Dai, Prof. Xingao Gong, Prof. Li-hua Shen, and Dr. Dier Zhang for their stimulating discussions andfruitful cooperations on electronic structure computations thathave motivated this work. The authors are grateful to Prof. LinboZhang and Mr. Hui Liu for their assistance on numericalcomputations.

Appendix A. A model problem

We provide basic results for adaptive finite element approxima-tions of a model problem that have been used in our analysis. Con-sider a homogeneous boundary value problem:

Lu � �,Du ¼ f in X;

u ¼ 0 on @X:

ðA:1Þ

The weak form of (A.1) reads as follows: find u 2 H10ðXÞ such that

aðu;vÞ ¼ ðf ;vÞ 8v 2 H10ðXÞ: ðA:2Þ

It is well known that (A.2) is uniquely solvable for any f 2 H�1(X).A standard finite element scheme for (A.2) is: find uh 2 Sh

0ðXÞsatisfying

aðuh;vÞ ¼ ðf ; vÞ 8v 2 Sh0ðXÞ: ðA:3Þ

Similar to the element residual RsðuhÞ for (3.3), we define the ele-ment residual for eRsðuhÞ for (A.3) by

eRsðuhÞ ¼ f � Luh ¼ f þ ,Duh in s 2 T h: ðA:4Þ

For s 2 T h, we define the local error indicator ~ghðuh; sÞ and the oscil-lation goschðuh; sÞ by (3.49) and (3.50) when RsðuhÞ is replaced byeRsðuhÞ.

Given a subset x �X, we define the error estimator ~ghðuh;xÞand the oscillation goschðuh;xÞ by

~g2hðuh;xÞ ¼

Xs2T h ;s�x

~g2hðuh; sÞ and

gosc2hðuh;xÞ ¼

Xs2T h ;s�x

gosc2hðuh; sÞ: ðA:5Þ

We now recall the well-known upper and lower bounds for theenergy error in terms of the residual-type estimator (see, e.g.,[27–29]).

Theorem A.1. Let u 2 H10ðXÞ be the solution of (A.2) and uh 2 Sh

0ðXÞbe the solution of (A.3). Then there exist constants eC1; eC2 and eC3 > 0depending only on ,; ca and the shape regularity c⁄ such that

ku� uhk2a;X 6

eC 1~g2hðuh;XÞ ðA:6Þ


andeC 2 ~g2hðuh;XÞ 6 ku� uhk2

a;X þ eC3gosc2hðuh;XÞ: ðA:7Þ

Now we address the marking strategy of solving (A.3) in detail.Similar to the marking strategy for (3.3), we define a marking strat-egy for (A.3) to enforce error reduction as follows:

Given a parameter 0 < h < 1:

1. Construct a minimal subset Mk of T k by selecting some ele-ments in T k such that
Xs2Mk
~g2kðuk; sÞP h

Xs2T k

~g2kðuk; sÞ: ðA:8Þ

2. Mark all the elements in Mk.

An adaptive finite element algorithm for solving (A.2) is stated asfollows:

Algorithm A.1. Adaptive finite element algorithm

1. Pick an initial mesh T 0 and let k = 0.2. Solve (A.3) on T k and get the finite element approximation uk.3. Compute local error indicators ~gkðuk; sÞ 8s 2 T k.4. Construct Mk � T k by a marking strategy that satisfies (A.8).5. Refine T k to get a new conforming mesh T kþ1.6. Let k = k + 1 and go to 2.

Finally we show the convergence result of Algorithm A.1, whoseproof is given in [25].

Theorem A.2. Let fukgk2N0be a sequence of finite element solutions

corresponding to a sequence of nested finite element spacesfSk

0ðXÞgk2N0produced by Algorithm A.1. Then there exist constants

~c > 0 and ~n 2 ð0;1Þ depending only on ,, the shape regularity c⁄ andthe marking parameter h, such that for any two consecutive iterateswe have

ku� ukþ1k2a;X þ ~c~g2

kþ1ðukþ1;XÞ 6 ~n2 ku� ukk2a;X þ ~c~g2

kðuk;XÞ� �

:

Indeed, constant ~c has the following form

~c ¼ 1

ð1þ d�1ÞC2�

ðA:9Þ

with C⁄ > 0 depending on regularity constant c⁄ and d 2 (0,1) being aconstant.

For the distance between two nested solutions of (A.3), we havethe following localized upper bound estimate (see [25]).

Lemma A.1. Let uH 2 SH0 ðXÞ and uh 2 Sh

0ðXÞ be solutions of (A.2) overT H and its any refinement T h, respectively with marked elementMH

and refined elements R ¼ RT H ! T h . Then the following localizedupper bound is valid

kuH � uhk2a;X 6

eC1

Xs2R

~g2HðuH; sÞ: ðA:10Þ

Appendix B. A priori H1-norm error estimate

To give a priori error estimate in H1-norm, we need only to re-quire (2.8) rather than the whole of Assumption A.V.

First, we denote R� H10ðXÞ by X and R� H�1ðXÞ by X⁄ with the

associated norm

kðk;uÞkX ¼ jkj þ kuk1;X:

Note that eigenvalue problem (2.5) can be rewritten in the follow-ing compact form

Gðk;uÞ ¼ 0 2 X�; ðB:1Þ

where G : X ! X� is defined by

hGðk;uÞ; ðl;vÞi ¼ hFðk;uÞ; vi þ lðZ �Z

Xu2Þ 8ðl; vÞ 2 X: ðB:2Þ

The Fréchet derivative of G at (k,u) is given by G0ðk;uÞ : X ! X� for(c,w) 2 X,

hG0ðk;uÞðc;wÞ; ðl;vÞi ¼ hF 0uðk;uÞw;vi þ cðu; vÞþ lðu;wÞ 8ðl;vÞ 2 X: ðB:3Þ

Proposition B.1. Assume that (2.8) is satisfied. If (k,u) is the groundstate solution of (2.5) , then G0ðk;uÞ is an isomorphism from X to X⁄.

Proof. It is sufficient to prove that equation

G0ðk;uÞðc;wÞ ¼ ðg; f Þ ðB:4Þ

is uniquely solvable in X for each (g, f) 2 X⁄.We define a bilinear form bu : R� H1

0ðXÞ ! R by

buðl;vÞ ¼ lðu; vÞ:

It is obvious that bu(�, �) is continuous and u? ¼ ker bu �fv 2 H1

0ðXÞ : buðl;vÞ ¼ 0 8l 2 Rg. Using (B.3), Eq. (B.4) can berewritten as a saddle-point problem as follows

a0ðu; w; vÞ þ buðc;vÞ ¼ ðf ;vÞ 8v 2 H10ðXÞ;

buðl;wÞ ¼ gl 8l 2 R;

(ðB:5Þ

where a0(u; �, �) is defined by (3.7).

For saddle-point problem (B.5), there are well known condi-tions for solvability (see, e.g., Theorem 1.1 in [50, II]): bilinear formbu has to satisfy a so-called inf–sup condition

infl2R

supv2H1

0ðXÞ

buðl;vÞkvk1;Xjlj

P jb ðB:6Þ

for some constant jb > 0 and variational problem

a0ðu; w;vÞ ¼ ðf ;vÞ 8v 2 ker bu ðB:7Þ

is uniquely solvable for any f 2 H�1(X). Note that the unique solv-ability of (B.7) is equivalent to assumption (2.8). So we only needto prove (B.6).

For a given l 2 R, we choose v = lu and obtain

buðl;vÞ ¼ Zl2; ðB:8Þ

where kuk20;X ¼ Z is used.

Note that

kvk1;X ¼ kluk1;X K jljkuk1;X: ðB:9Þ

We derive (B.6) from combining 2.4, B.8 and B.9, where jb > 0 isindependent of l.

From Theorem 1.1 in [50, II], we conclude that G0ðk;uÞ is anisomorphism from X to X⁄. This completes the proof. h

Now we can prove that the ground state solutions are isolated.

Proposition B.2. Assume that (2.8) is satisfied. If (k,u) is the groundstate solution of (2.5) , then u is an isolated ground solution, i.e., thereexists a constant d > 0 such that u is the unique solution of (2.5) inBdðuÞ ¼ fv 2 H1

0ðXÞ : kv � uk 6 dg.


Proof. Proposition B.1 and the inverse function theorem implythat (k,u) is an isolated solution. We shall use a contradiction argu-ment to prove that u is isolated. Assume that u is not isolated inH1

0ðXÞ, i.e., for any d > 0, there exists ud such that kud � uk1,X 6 dand ud is a ground state solution of (2.5), too.

It is seen from (2.6) that there exists a constant C such thatjkd � kj 6 Ckud � uk1,X, which is a contradiction to the fact that(k,u) is an isolated solution. This completes the proof. h

Then we turn to study finite element discretizations of (B.1). Wedenote R� Sh

0ðXÞ by Xh and the dual space by X�h. Then (3.3) can berewritten as

hGðkh;uhÞ; ðl;vÞi ¼ 0 8ðl;vÞ 2 Xh: ðB:10Þ

Following the arguments in the proof of Lemma 2.4 in [34], we canprove the following result.

Theorem B.3. Assume that (2.8) is satisfied. If (k,u) is the groundstate solution of (2.5) , then for a sufficiently fine mesh T h, there existsa constant d > 0 and a locally unique solution uh of (3.3) in Bd(u).Moreover, there holds


0ðXÞku� vk1;X: ðB:11Þ

Proof. Define a bilinear form eA : X � X ! R by

eAððc;wÞ; ðl; vÞÞ ¼ hG0ðk;uÞðc;wÞ; ðl;vÞi 8ðc;wÞ; ðl; vÞ 2 X:

Since G0ðk;uÞ is an isomorphism from X to X⁄, we have for a given(ch,wh) 2 Xh that

supðl;vÞ2X

eAððch;whÞ; ðl;vÞÞkðch;whÞkXkðl; vÞkX

P a:

where a is a positive constant.Using the continuity of eA, we get for (lh,vh) 2 Xh that

eAððch;whÞ; ðl; vÞÞ � eAððch;whÞ; ðlh;vhÞÞ6 jeAððch;whÞ; ðl� lh;v � vhÞÞjK kðch;whÞkXðjl� lhj þ kv � vhk1;XÞ;

which together with the fact

limh!0

infðlh ;vhÞ2Xh

ðjl� lhj þ kv � vhk1;XÞ ¼ 0

implies that there exists h0 > 0 such that

infðc;wÞ2Xh

supðl;vÞ2Xh

eAððc;wÞ; ðl;vÞÞkðc;wÞkXkðl;vÞkX

Pa2> 0 for h 2 ð0;h0:

Moreover, we have that G0 is Lipschitz continuous at (k,u). Thereforewe can obtain from Theorem 2.6 in [34] that for sufficiently finemesh T h; ðkh; uhÞ is the unique solution of (B.10) in the neighbor-hood of (k,u). Using the same contradiction argument as that inthe proof of Proposition B.2, we have that uh is the unique groundstate solution in a neighborhood of u.

Finally, we obtain from Theorem 2.6 in [34] that

jk� khj þ ku� uhk1;X K infðl;vÞ2Xh

ðjk� lj þ ku� vk1;XÞ

¼ infv2Sh

0ðXÞku� vk1;X;

which leads to (B.11). This completes the proof. h

Note that to derive an optimal a priori error estimate in L2-norm, we need Assumption A.V., namely, not only (2.8) but also(2.7).

Appendix C. Proof of Lemma 4.3

Proof. Let a, a1 2 (0,1) satisfy a1 2 (0,a) and

h <C2c

C3ðC1 þ ð1þ 2C2�C1ÞcÞ

ð1� a2Þ:

Choose d1 2 (0,1) to satisfy (4.14) and

ð1þ d1Þ2a21 6 a2; ðC:1Þ

which implies

ð1þ d1Þa21 < 1: ðC:2Þ

Set

e ¼ 1ffiffiffi2p a1 ku� ukk2

a;X þ cosc2kðuk;XÞ

� �1=2

and let T e be a refinement of T 0 with minimal degrees of freedomsatisfying

ku� uek2a;X þ ðcþ 1Þosc2

e ðue;XÞ 6 e2: ðC:3Þ

It follows from u 2 As that

#T e �#T 0 K e�1=sjuj1=ss :

Let T � ¼ T e T k be the smallest common refinement of T k and T e.Note that we ¼ Kðkeue � Vue �Nðu2

e ÞueÞ satisfies

Lwe ¼ keue � Vue �Nðu2e Þue;

we get from the definition of oscillation and the Young’s inequalitythat

gosc2� ðP�we;XÞ 6 2gosc2

� ðPewe;XÞ þ 2C2�kPewe � P�wek2

a;X;

where Pe and P⁄ are Galerkin projections on T e and T � defined by(3.37).

Due to the orthogonality

kwe � P�wek2a;X ¼ kwe � Pewek2

a;X � kP�we � Pewek2a;X;

we arrive at

kwe � P�wek2a;X þ

12C2�

gosc2� ðP�we;XÞ

6 kwe � Pewek2a;X þ

1C2�

osc2e ðPewe;XÞ:

Since (A.9) implies ~c 6 12C2�, we obtain that

kwe � P�wek2a;X þ ~cgosc2

� ðP�we;XÞ 6 kwe � Pewek2a;X

þ 1

C2�

osc2e ðPewe;XÞ

6 kwe � Pewek2a;X

þ ð~cþ rÞosc2e ðPewe;XÞ;

where r ¼ 1C2�� ~c 2 ð0;1Þ. Applying a similar argument as that in the

proof of Theorem 4.1 when (4.4) is replaced by (4.3), we concludethat

ku� u�k2a;X þ cosc2

� ðu�;XÞ 6 a20 ku� uek2

a;X þ ðcþ rÞosc2e ðPewe;XÞ

� �6 a2

0 ku� uek2a;X þ ðcþ 1Þosc2

e ðPewe;XÞ� �

;

ðC:4Þ


where

a20 ¼ð1þ d1Þ þ C4 ~jðh0Þ

1� C4d�11 ~j2ðh0Þ

and C4 is the constant appearing in the proof of Theorem 4.1. Thus,by (C.3) and (C.4), it follows

ku� u�k2a;X þ cosc2

� ðu�; T �Þ 6 �a2 ku� ukk2a;X þ cosc2

kðuk; T kÞ� �

;

with �a ¼ 1ffiffi2p a0a1. In view of (C.2), we have �a2 2 0; 1

2

� �when h0� 1.

Set �h ¼ eC 2ð1�2a2ÞeC 0ðeC 1þð1þ2C2�eC 1ÞcÞ

; c ¼ c1�C5d�1

1~j2ðh0Þ

; eC0 ¼max 1;eC 3c

� �, and a2 ¼

ð1þd1Þ�a2þC5 ~jðh0Þ1�C5d�1

1~j2ðh0Þ

: We obtain from Lemma 4.2 that T � satisfiesXs2R

g2kðuk; sÞP �h

Xs2T k

g2kðuk; sÞ;

where R ¼ RT k!T � is the refined elements from T k to T �.It follows from the definition of c (4.12) and ~c (A.9) that c < 1

and hence eC0 ¼eC 3c . Since h0� 1, we get that c > c and a 2 0; 1ffiffi

2p a

� �from (C.1). It is easy to see from (3.56), (3.57) and c > c that

�h ¼eC 2ð1� 2a2ÞeC 3

c ðeC 1 þ ð1þ 2C2�eC 1ÞcÞ

PeC2eC3

eC 1c þ 1þ 2C2

�eC 1

� � ð1� a2Þ

¼

C2

ð1�eC ~jðh0ÞÞ2

C3

ð1�eC ~jðh0ÞÞ2C1

cðð1þeC ~jðh0ÞÞ2Þþ 1þ 2C2

�C1

ð1þeC ~jðh0ÞÞ2

! ð1� a2Þ

PC2

C3C1c þ ð1þ 2C2

�C1Þ� � ð1� a2Þ

¼ C2cC3ðC1 þ ð1þ 2C2

�C1ÞcÞð1� a2Þ > h;

when h0� 1. Thus

#Mk 6 #R 6 #T � �#T k 6 #T e �#T 0

61ffiffiffi2p a1

� ��1=s


kðuk; T kÞ� ��1=2s

juj1=ss ;

which is the desired estimate (4.21) with an explicit dependence onthe discrepancy between h and C2c

C3ðC1þð1þ2C2�C1ÞcÞ

via a1. This completes

the proof. h

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http://dx.doi.org/10.1093/imanum/drp055

http://dx.doi.org/10.1093/imanum/drp055

http://www.matheon.de/research/

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