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Numerical analysis of nonlinear eigenvalue problems
Eric Cancès, Rachida Chakir, Yvon Maday
To cite this version:
Eric Cancès, Rachida Chakir, Yvon Maday. Numerical analysis of nonlinear eigenvalue problems.
Journal of Scientific Computing, Springer Verlag, 2010, 45 (1-3), pp.90-117. �10.1007/s10915-010-
9358-1�. �hal-00392025�
Numerical analysis of nonlinear eigenvalue problems
Eric Canc` es
∗, Rachida Chakir
†and Yvon Maday
†‡May 13, 2009
Abstract
We providea priorierror estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue prob- lems of the form−div(A∇u) +V u+f(u2)u=λu,kukL2 = 1. We focus in particular on the Fourier spectral approximation (for periodic problems) and on theP1 andP2 finite-element discretizations. Denoting by (uδ, λδ) a vari- ational approximation of the ground state eigenpair (u, λ), we are interested in the convergence rates ofkuδ−ukH1, kuδ−ukL2 and |λδ−λ|, when the discretization parameterδgoes to zero. We prove that ifA,V andf satisfy certain conditions, |λδ−λ| goes to zero as kuδ−uk2H1 +kuδ −ukL2. We also show that under more restrictive assumptions on A, V andf, |λδ−λ|
converges to zero askuδ−uk2H1, thus recovering a standard result forlinear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the erroruδ−uin negative Sobolev norms.
1 Introduction
Many mathematical models in science and engineering give rise to nonlinear eigen- value problems. Let us mention for instance the calculation of the vibration modes of a mechanical structure in the framework of nonlinear elasticity, the Gross-Pitaevskii equation describing the steady states of Bose-Einstein condensates [9], or the Hartree- Fock and Kohn-Sham equations used to calculate ground state electronic structures of molecular systems in quantum chemistry and materials science (see [3] for a mathematical introduction).
The numerical analysis of lineareigenvalue problems has been thoroughly studied in the past decades (see e.g. [1]). On the other hand,nonlineareigenvalue problems seem to have received much less attention from numerical analysts.
In this article, we focus on a particular class of nonlinear eigenvalue problems arising in the study of variational models of the form
I= inf
E(v), v∈X, Z
Ω
v2= 1
(1) where
Ω is a regular bounded domain or a rectangular brick ofRd andX=H01(Ω) or
Ω is the unit cell of a periodic latticeRof Rd andX =H#1(Ω)
∗Universit´e Paris-Est, CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vall´ee Cedex 2, France.
†UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
‡Division of Applied Mathematics, Brown University, Providence, RI, USA
withd= 1, 2 or 3, and where the energy functionalE is of the form E(v) = 1
2a(v, v) +1 2 Z
Ω
F(v2(x))dx (2)
with
a(u, v) = Z
Ω
(A∇u)· ∇v+ Z
Ω
V uv. (3)
Recall that if Ω is the unit cell of a periodic latticeRofRd, then for alls∈Rand k∈N,
H#s(Ω) =
v|Ω, v∈Hlocs (Rd)|vR-periodic , C#k(Ω) =
v|Ω, v∈Ck(Rd)|vR-periodic . We assume in addition that
• A∈(L∞(Ω))d×d andA(x) is symmetric for almost allx∈Ω (4)
∃α >0 s.t. ξTA(x)ξ≥α|ξ|2for allξ∈Rd and almost allx∈Ω (5)
• V ∈Lp(Ω) for somep >2 (6)
• F∈C1([0,+∞),R)∩C2((0,∞),R) andF′′>0 on (0,+∞) (7)
∀R >0, ∃CR∈R+ s.t. ∀0< t1≤R, ∀t2∈R,
(F′(t22)−F′(t21))t22≤CR(1 +|t2|3)|t2−t1| and (8) F′(t22)t2−F′(t21)t2−2F′′(t21)t21(t2−t1)≤CR(1 +|t2|)|t2−t1|2. (9) In particular, the functionF(t) =ctq satisfies the assumptions (7)-(9) if and only ifc >0 and 32 ≤q≤2. This allows us to handle the Thomas-Fermi kinetic energy functional (q= 53) as well as the repulsive interaction in Bose-Einstein condensates (q= 2). In order to simplify the notation, we denote byf(t) =F′(t).
Making the change of variableρ=v2 and noticing thata(|v|,|v|) = a(v, v) for all v∈X, it is easy to check that
I= inf
E(ρ), ρ≥0, √ρ∈X, Z
Ω
ρ= 1
, (10)
where
E(ρ) = 1
2a(√ρ,√ρ) +1 2 Z
Ω
F(ρ).
Under assumptions (4)-(9), (10) has a unique solutionρ0 and (1) has exactly two solutions: u=√ρ0and−u. Besides,EisC1onX and for allv∈X,E′(v) =Avv where
Av=−div(A∇·) +V +f(v2).
Note thatAv defines a self-adjoint operator on L2(Ω), with form domainX. The functionutherefore is solution to the Euler equation
∀v∈X, hE′(u)−λu, viX′,X = 0 (11) for some λ ∈ R (the Lagrange multiplier of the constraint kuk2L2 = 1) and equa- tion (11), complemented with the constraint kukL2 = 1, takes the form of the nonlinear eigenvalue problem
Auu=λu
kukL2= 1. (12)
In addition, u ∈ C0(Ω), u > 0 in Ω and λ is the ground state eigenvalue of the linear operatorAu. An important result is thatλis a simpleeigenvalue ofAu. All these properties are classical. For the sake of completeness, their proofs are however given in the Appendix.
Let us now turn to the main topic of this article, namely the derivation of a priori error estimates for variational approximations of the ground state eigenpair (λ, u).
We denote by (Xδ)δ>0a family of finite-dimensional subspaces ofX such that min{ku−vδkH1, vδ ∈Xδ} −→
δ→0+ 0 (13)
and consider the variational approximation of (1) consisting in solving Iδ= inf
E(vδ), vδ ∈Xδ, Z
Ω
vδ2= 1
. (14)
Problem (14) has at least one minimizer uδ, which satisfies
∀vδ ∈Xδ, hE′(uδ)−λδuδ, vδiX′,X = 0 (15) for some λδ ∈ R. Obviously, −uδ also is a minimizer associated with the same eigenvalueλδ. On the other hand, it is not known whetheruδ and−uδ are the only minimizers of (14). This follows from the fact that the set
ρ| ∃uδ ∈Xδ s.t. kuδkL2 = 1, ρ=u2δ
is not convex in general. We will see however (cf. Theorem 1) that for any global minimumuδ of (14) such that (u, uδ)≥0, the following holds true
kuδ−ukH1 −→
δ→0+ 0.
In addition, a simple calculation leads to
λδ−λ=h(Au−λ)(uδ−u),(uδ−u)iX′,X + Z
Ω
wu,uδ(uδ−u) (16) where
wu,uδ =u2δf(u2δ)−f(u2) uδ−u .
The first term of the right-hand side of (16) is nonnegative and goes to zero as kuδ−uk2H1. We will prove in Theorem 1 that the second term goes to zero at least askuδ−ukL2. Therefore,|λδ−λ|converges to zero withδat least askuδ−uk2H1+ kuδ−ukL2.
The purpose of this article is to provide more precise a priori error bounds on kuδ−ukH1, kuδ−ukL2 and|λδ−λ|. In Section 2, we prove a series of estimates valid in the general framework described above. We then turn to more specific examples, where the analysis can be pushed further. In Section 3, we concentrate on the discretization of problem (1) with
Ω = (0,2π)d, X =H#1(0,2π)d, E(v) =1
2 Z
Ω|∇v|2+1 2
Z
Ω
V v2+1 2 Z
Ω
F(v2),
in Fourier modes. In Section 4, we deal with theP1andP2finite element discretiza- tions of problem (1) with
Ω rectangular brick ofRd, X=H01(Ω),
E(v) = 1 2
Z
Ω|∇v|2+1 2
Z
Ω
V v2+1 2
Z
Ω
F(v2).
Lastly, we discuss the issue of numerical integration in Section 5.
2 Basic error analysis
The aim of this section is to establish error bounds onkuδ−ukH1,kuδ−ukL2 and
|λδ−λ| in a general framework. In the whole section, we make the assumptions (4)-(9) and (13), and we denote byuthe unique positive solution of (1) and byuδ
a minimizer of the discretized problem (14) such that (uδ, u)L2≥0.
Lemma 1 The functionalE is twice differentiable at uand for all(v, w)∈X×X, hE′′(u)v, wi=hAuv, wiX′,X+ 2
Z
Ω
f′(u2)u2vw. (17) There exists β >0 andM ∈R+ such that for all v∈X,
0 ≤ h(Au−λ)v, viX′,X ≤Mkvk2H1 (18) βkvk2H1 ≤ h(E′′(u)−λ)v, viX′,X ≤Mkvk2H1. (19) There exists γ >0 such that for all δ >0,
γkuδ−uk2H1 ≤ h(Au−λ)(uδ−u),(uδ−u)iX′,X. (20) Proof The quadratic functionalv7→ 12a(v, v) is clearly twice differentiable onX. Using (8) and (9), it is easy to check that the functional Φ : v 7→ 12
R
ΩF(v2) is twice differentiable onX as well and that
Φ′(u) =f(u2)u, hΦ′′(u)v, wiX′,X = Z
Ω
(f(u2) + 2f′(u2)u2)vw.
This straightforwardly leads to (17). We have for allv∈X,
h(Au−λ)v, viX′,X ≤ kAkL∞k∇vk2L2+kVkL2kvk2L4+kf(u2)kL∞kvk2L2
and
h(E′′(u)−λ)v, viX′,X ≤ h(Au−λ)v, viX′,X+ 2kf′(u2)u2kL∞kvk2L2. Hence the upper bounds in (18) and (19). We now use the fact that λ, the lowest eigenvalue ofAu, is simple (see Lemma 2 in the Appendix). This implies that there existsη >0 such that
∀v∈X, h(Au−λ)v, viX′,X ≥η(kvk2L2− |(u, v)L2|2)≥0. (21) This provides on the one hand the lower bound (18), and leads on the other hand to the inequality
∀v∈X, h(E′′(u)−λ)v, viX′,X ≥2 Z
Ω
f′(u2)u2v2.
As f′=F′′>0 in (0,+∞) andu >0 in Ω, we therefore have
∀v∈X\ {0}, h(E′′(u)−λ)v, viX′,X >0.
Reasoning by contradiction, we deduce from the above inequality and the first inequality in (21) that there existsη >e 0 such that
∀v∈X, h(E′′(u)−λ)v, viX′,X ≥eηkvk2L2. (22) Besides, there exists a constantC∈R+ such that
∀v∈X, h(Au−λ)v, viX′,X ≥α
2k∇vk2L2−Ckuk2L2. (23) Let us establish this inequality for d = 3 (the case whend= 1 is straightforward and the case when d= 2 can be dealt with in the same way). For allx∈X,
h(Au−λ)v, viX′,X = Z
Ω
(A∇v)· ∇v+ Z
Ω
(V +f(v2)−λ)v2
≥ αk∇vk2L2− kVkL2kvk2L4+ (f(0)−λ)kvk2L2
≥ αk∇vk2L2− kVkL2kvk1/2L2 kvk3/2L6 + (f(0)−λ)kvk2L2
≥ αk∇vk2L2−C63/2kVkL2kvk1/2L2k∇vk3/2L6 + (f(0)−λ)kvk2L2
≥ α
2k∇vk2L2+
f(0)−λ−27C66 32α3kVk4L2
kvk2L2,
whereC6is the Sobolev constant such that∀v∈X,kvkL6≤C6k∇vkL2. The coer- civity of E′′(u)−λ(i.e. the lower bound in (19)) is a straightforward consequence of (22) and (23).
To prove (20), we notice that
kuδk2L2− |(u, uδ)L2|2≥1−(u, uδ)L2 =1
2kuδ−uk2L2. It therefore readily follows from (21) that
h(Au−λ)(uδ−u),(uδ−u)iX′,X ≥ η
2kuδ−uk2L2.
Combining with (23), we finally obtain (20).
Forw∈X′, we denote byψw the unique solution to the adjoint problem findψw∈u⊥ such that
∀v∈u⊥, h(E′′(u)−λ)ψw, viX′,X =hw, viX′,X, (24) where
u⊥=
v∈X | Z
Ω
uv= 0
.
The existence and uniqueness of the solution to (24) is a straightforward consequence of (19) and the Lax-Milgram lemma. Besides,
∀w∈X′, kψwkH1 ≤β−1MkwkX′ ≤β−1MkwkL2. (25)
We can now state the main result of this section.
Theorem 1 It holds
kuδ−ukH1 −→
δ→0+ 0. (26)
Besides, there exists δ0>0 andC∈R+ such that for all0< δ < δ0, kuδ−ukH1 ≤ C min
vδ∈Xδkvδ−ukH1 (27)
kuδ−uk2L2 ≤ C
kuδ−uk2H1kuδ−ukL2
+kuδ−ukH1 min
ψδ∈Xδ
kψuδ−u−ψδkH1
(28)
|λδ−λ| ≤ C kuδ−uk2H1+kuδ−ukL2
. (29)
Proof Using (20) and the convexity ofF, we get E(uδ)−E(u) = 1
2hAuuδ, uδiX′,X −1
2hAuu, uiX′,X
+1 2 Z
Ω
F u2δ
−F u2
−f u2
(u2δ−u2)
= 1
2h(Au−λ)(uδ−u),(uδ−u)iX′,X
+1 2 Z
Ω
F u2+ (u2δ−u2)
−F u2
−f u2
(u2δ−u2)
≥ γ
2kuδ−uk2H1. (30)
Let Πδu∈Xδ be such that
ku−ΠδukH1 = min{ku−vδkH1, vδ ∈Xδ}.
We deduce from (13) that (Πδu)δ>0 converges to u in X when δ goes to zero.
Denoting byueδ=kΠδuk−1L2Πδu(which is well defined, at least forδsmall enough), we also have
δ→0lim+keuδ−ukH1 = 0.
The functionalE being strongly continuous onX, we obtain kuδ−uk2H1 ≤ 2
γ(E(uδ)−E(u))≤ 2
γ(E(euδ)−E(u)) −→
δ→0+0. (31) It follows that there existsδ1>0 such that
∀0< δ≤δ1, kuδkH1≤2kukH1, kuδ−ukH1 ≤ 1 2.
Next, we remark that
λδ−λ = hE′(uδ), uδiX′,X− hE′(u), uiX′,X
= a(uδ, uδ)−a(u, u) + Z
Ω
f(u2δ)u2δ− Z
Ω
f(u2)u2
= a(uδ−u, uδ−u) + 2a(u, uδ−u) + Z
Ω
f(u2δ)u2δ− Z
Ω
f(u2)u2
= a(uδ−u, uδ−u) + 2λ Z
Ω
u(uδ−u)−2 Z
Ω
f(u2)u(uδ−u) +
Z
Ω
f(u2δ)u2δ− Z
Ω
f(u2)u2
= a(uδ−u, uδ−u)−λkuδ−uk2L2−2 Z
Ω
f(u2)u(uδ−u) +
Z
Ω
f(u2δ)u2δ− Z
Ω
f(u2)u2
= h(Au−λ)(uδ−u),(uδ−u)iX′,X+ Z
Ω
wu,uδ(uδ−u) (32) where
wu,uδ =u2δf(u2δ)−f(u2) uδ−u . Using (8) and (18), we obtain that for all 0< δ≤δ1,
|λδ−λ| ≤ Mkuδ−uk2H1+kwu,uδkL2ku−uδkL2
≤ Mkuδ−uk2H1+Ck1 +|uδ|3kL2ku−uδkL2
≤ Mkuδ−uk2H1+C(1 +kuδk3H1)ku−uδkL2
≤ C kuδ−uk2H1+ku−uδkL2
, (33)
where Cdenotes constants independent ofδ.
In order to evaluate the H1-norm of the erroruδ−u, we first notice that
∀vδ ∈Xδ, kuδ−ukH1 ≤ kuδ−vδkH1+kvδ−ukH1, (34) and that
kuδ−vδk2H1 ≤ β−1h(E′′(u)−λ)(uδ−vδ),(uδ−vδ)iX′,X
= β−1
h(E′′(u)−λ)(uδ−u),(uδ−vδ)iX′,X +h(E′′(u)−λ)(u−vδ),(uδ−vδ)iX′,X
. (35) For allwδ∈Xδ
h(E′′(u)−λ)(uδ−u), wδiX′,X
=− Z
Ω
f(u2δ)uδ−f(u2)uδ−2f′(u2)u2(uδ−u)
wδ+ (λδ−λ) Z
Ω
(uδ−u)wδ. Using (9) and (33), we therefore obtain that for all 0< δ≤δ1 and allwδ ∈Xδ,
|h(E′′(u)−λ)(uδ−u), wδiX′,X| ≤CkwδkH1kuδ−uk2H1. (36) It then follows from (19), (35) and (36) that for all 0< δ≤δ1and allvδ ∈Xδ,
kuδ−vδkH1 ≤C kvδ−ukH1+kuδ−uk2H1
.
Combining with (34) we obtain that there exists 0< δ2≤δ1andC∈R+ such that for all 0< δ≤δ2and allvδ ∈Xδ,
kuδ−ukH1 ≤Ckvδ−ukH1. (37) Thus (27) is proved.
Letu∗δ be the orthogonal projection, for the L2 inner product, of uδ on the affine space
v∈L2(Ω)| R
Ωuv= 1 . One has
u∗δ ∈X, u∗δ−u∈u⊥, u∗δ−uδ= 1
2kuδ−uk2L2u, from which we infer that
kuδ−uk2L2 = Z
Ω
(uδ−u)(u∗δ−u) + Z
Ω
(uδ−u)(uδ−u∗δ)
= Z
Ω
(uδ−u)(u∗δ−u)−1
2kuδ−uk2L2
Z
Ω
(uδ−u)u
= Z
Ω
(uδ−u)(u∗δ−u) +1
2kuδ−uk2L2
1−
Z
Ω
uδu
= Z
Ω
(uδ−u)(u∗δ−u) +1
4kuδ−uk4L2
= huδ−u, u∗δ−uiX′,X+1
4kuδ−uk4L2
= h(E′′(u)−λ)ψuδ−u, u∗δ−uiX′,X+1
4kuδ−uk4L2
= h(E′′(u)−λ)(uδ−u), ψuδ−uiX′,X
+1
2kuδ−uk2L2h(E′′(u)−λ)u, ψuδ−uiX′,X+1
4kuδ−uk4L2
= h(E′′(u)−λ)(uδ−u), ψuδ−uiX′,X +kuδ−uk2L2
Z
Ω
f′(u2)u3ψuδ−u+1
4kuδ−uk4L2. For allψδ ∈Xδ, it holds
kuδ−uk2L2 = h(E′′(u)−λ)(uδ−u), ψδiX′,X
+h(E′′(u)−λ)(uδ−u), ψuδ−u−ψδiX′,X
+kuδ−uk2L2
Z
Ω
f′(u2)u3ψuδ−u+1
4kuδ−uk4L2
= −
Z
Ω
f(u2δ)uδ−f(u2)uδ−2f′(u2)u2(uδ−u) ψδ
+(λδ−λ) Z
Ω
uδψδ
+h(E′′(u)−λ)(uδ−u), ψuδ−u−ψδiX′,X
+kuδ−uk2L2
Z
Ω
f′(u2)u3ψuδ−u+1
4kuδ−uk4L2
= −
Z
Ω
f(u2δ)uδ−f(u2)uδ−2f′(u2)u2(uδ−u) ψδ
+(λδ−λ) Z
Ω
uδ(ψδ−ψuδ−u) + (λδ−λ) Z
Ω
(uδ−u)ψuδ−u
+h(E′′(u)−λ)(uδ−u), ψuδ−u−ψδiX′,X
+kuδ−uk2L2
Z
Ω
f′(u2)u3ψuδ−u+1
4kuδ−uk4L2,
where we have used that Z
Ω
ψuδ−uu= 0.
Letψ0δ ∈Xδ be such that
kψuδ−u−ψ0δkH1 = min
ψδ∈Xδkψuδ−u−ψδkH1.
Noticing that kψ0δkH1 ≤ kψuδ−ukH1 ≤β−1Mkuδ−ukL2, we obtain from (9), (19) and (33) that there exists C∈R+such that for all 0< δ≤δ1,
kuδ−uk2L2 ≤ C
kuδ−uk2H1kuδ−ukL2+kuδ−ukH1kψuδ−u−ψδ0kH1
+kuδ−uk3L2+kuδ−uk2L2kuδ−uk2H1
. Therefore, there exists 0< δ0≤δ2 andC∈R+ such that for all 0< δ≤δ0,
kuδ−uk2L2≤C kuδ−uk2H1kuδ−ukL2+kuδ−ukH1kψuδ−u−ψδ0kH1
, (38)
which completes the proof of Theorem 1.
Remark 1 In the proof of Theorem 1, we have obtained bounds on|λδ−λ| from (32), using L2 estimates on wu,uδ and (uδ −u) to control the second term of the right hand side. Remarking that
∇wu,uδ = −uf(u2)u−f(u2δ)u−2f′(u2δ)u2δ(u−uδ)
(uδ−u)2 ∇uδ
−uδ f(u2δ)uδ−f(u2)uδ−2f′(u2)u2(uδ−u)
(uδ−u)2 ∇u
+2uuδ f′(u2δ)∇uδ+f′(u2)∇u
+ 2uδf(u2δ)−f(u2) uδ−u ∇uδ, we deduce from (8)-(9) that if uδ is uniformly bounded in L∞(Ω), then wu,uδ is uniformly bounded in X. It then follows from (32) that
|λδ−λ| ≤C kuδ−uk2H1+kuδ−ukX′ ,
an estimate which is an improvement of (29). In the next two sections, we will see that this approach (or analogous strategies making use of negative Sobolev norms of higher orders), can be used in certain cases to obtain optimal estimates on|λδ−λ| of the form
|λδ−λ| ≤Ckuδ−uk2H1.
3 Fourier expansion
In this section, we consider the problem inf
E(v), v∈X, Z
Ω
v2= 1
, (39)
where
Ω = (0,2π)d, withd= 1, 2 or 3, X =H#1(Ω),
E(v) =1 2
Z
Ω|∇v|2+1 2
Z
Ω
V v2+1 2 Z
Ω
F(v2).
We assume that V ∈ H#σ(Ω) for some σ > d/2 and that the function F satisfies (7)-(9) and is such that the functiont7→F(t2) is inCσ+1(R+,R) ifσ∈N, and in C[σ]+2(R+,R) ifσ /∈N. Actually, as we know that the unique positive solutionu to (39) stays away from 0 in Ω, the assumptions onF can be slightly relaxed, but we will not elaborate further on this technical detail.
The positive solutionuto (39), which satisfies the elliptic equation
−∆u+V u+f(u2)u=λu, then is inH#σ+2(Ω).
A natural discretization of this problem consists in using a Fourier basis. Denoting byek(x) = (2π)−d/2eik·x, we have for allv∈L2(Ω),
v(x) = X
k∈Zd
bvkek(x),
wherebvk is thekth Fourier coefficient ofv:
b vk=
Z
Ω
v(x)ek(x)dx= (2π)−d/2 Z
Ω
v(x)e−ik·xdx.
The approximation of the solution to (39) by the spectral Fourier approximation is based on the choice
Xδ =XeN = Span{ek, |k|∗≤N},
where |k|∗ denotes either the l2-norm or the l∞-norm of k (i.e. either |k| = (Pd
i=1|ki|2)1/2 or |k|∞ = max1≤i≤d|ki|). For convenience, the discretization pa- rameter for this approximation will be denoted asN.
EndowingH#r(Ω) with the norm defined by
kvkHr =
X
k∈Zd
1 +|k|2∗
r
|bvk|2
1/2
,
we obtain that for all s ∈ R, and all v ∈ H#s(Ω), the best approximation of v in H#r(Ω) for anyr≤sis
ΠNv= X
k∈Zd,|k|∗≤N
b
vkek. (40)
The more regularv (the regularity being measured in terms of the Sobolev norms Hr), the faster the convergence of this truncated series tov: for all real numbersr andswithr≤s, we have
∀v∈H#s(Ω), kv−ΠNvkHr ≤ 1
Ns−rkvkHs. (41) LetuN be a solution to the variational problem
inf
E(vN), vN ∈XeN, Z
Ω
v2N = 1
(42) such that (uN, u)L2 ≥0. Using (41), we obtain
ku−ΠNukH1 ≤ 1
Nσ+1kukHσ+2,
Besides, the unique solution to (24) solves the elliptic equation
−∆ψw+ V +f(u2) + 2f′(u2)u2−λ ψw= 2
Z
Ω
f′(u2)u3ψw
u+w−(w, u)L2u, from which we infer that ifw∈H#r(Ω) for some 0≤r≤σ, thenψwis in H#r+2(Ω) and satisfies
kψwkHr+2≤CkwkHr, (43) for some constantC independent ofw. In particular,
kψuN−u−ΠNψuN−ukH1 ≤ 1
NkψuN−ukH2 ≤ C
NkuN −ukL2.
A direct application of Theorem 1 then yields that there existsδ0>0 and C∈R+ such that for all 0< δ≤δ0,
kuN −ukH1 ≤ C
Nσ+1 (44)
kuN −ukL2 ≤ C
Nσ+2 (45)
|λN −λ| ≤ C
Nσ+2. (46)
The last estimate is slightly deceptive since, in the case of a linear eigenvalue problem (i.e. for −∆u+V u=λu) the convergence of the eigenvalues goes twice as fast as the convergence of the eigenvector in theH1-norm. We are going to prove that this is also the case for the nonlinear eigenvalue problem under study in this section, at least under some additional assumptions on the functionf.
Let us first come back to (32), which we rewrite as, λN −λ=h(Au−λ)(uN −u),(uN −u)iX′,X+
Z
Ω
wu,uN(uN−u) (47) with
wu,uN =u2Nf(u2N)−f(u2) uN−u . We then observe that uN is solution to the elliptic equation
−∆uN + ΠN
V uN +f(u2N)uN
=λNuN. (48) This implies that the sequence (uN)N∈N, which is uniformly bounded inH#1(Ω), is in fact also uniformly bounded inH#σ+2(Ω). Together with (9), this implies in turn that (wu,uN)N∈N is bounded in H#1(Ω)∩L∞(Ω). We therefore obtain from (47) that
|λN −λ| ≤C kuN −uk2H1+kuN −ukH−1
. (49) Let us now compute the H−1-norm of the error. Let w∈H#1(Ω). Proceeding as in Section 2, we obtain
Z
Ω
w(uN −u)
= −
Z
Ω
f(u2N)uN −f(u2)uN−2f′(u2)u2(uN −u) ΠNψw
+(λN −λ) Z
Ω
uN(ΠNψw−ψw) + (λN −λ) Z
Ω
(uN −u)ψw
+h(E′′(u)−λ)(uN −u), ψw−ΠNψwiX′,X +kuN−uk2L2
Z
Ω
f′(u2)u3ψw−1
2kuN−uk2L2
Z
Ω
uw.
Combining (19), (43), (44)-(46), (47) and the above equality, we obtain that there exists a constantC∈R+ such that for allN ∈Nand allw∈H#1(Ω),
Z
Ω
w(uN −u) ≤ C′ kuN −uk2H1+N−2kuN −ukH1
kwkH1
≤ C
Nmin(2(σ+1),σ+3)kwkH1. Therefore
kuN−ukH−1 = sup
w∈H#1(Ω)\{0}
Z
Ω
w(uN −u)
kwkH1 ≤ C
Nmin(2(σ+1),σ+3), (50) for some constantC∈R+ independent ofN. We end up with
|λN−λ| ≤ C
Nmin(2(σ+1),σ+3).
To proceed further, we need to make additional assumptions on the function f. Indeed, if we had a uniform bound onwu,uN in H#r(Ω) for somer >1, the above argument would lead to
|λN −λ| ≤ C kuN −uk2H1+kuN −ukH−r
≤ C
Nmin(2(σ+1),σ+r+2).
A simple case when this is true is whenF(t) = ct2 with c >0; in this particular case indeed, wu,uN = 2cu2N(uN +u) is uniformly bounded in H#σ+2(Ω). We can summarize the results obtained in this section in the following theorem.
Theorem 2 Assume that V ∈H#σ(Ω) for some σ > d/2 and that the functionF satisfies (7)-(9) and is such that the functiont7→F(t2)is inCσ+1(R+,R)ifσ∈N, and inC[σ]+2(R+,R)if σ /∈N. Then there existsC∈R+ such that for all N ∈N,
kuN −ukH1 ≤ C
Nσ+1 (51)
kuN−ukL2 ≤ C
Nσ+2 (52)
|λN−λ| ≤ C
Nmin(2(σ+1),σ+3). If in addition
wu,uN :=u2Nf(u2N)−f(u2) uN−u is uniformly bounded inH#σ+2(Ω), then
|λN −λ| ≤ C
N2(σ+1). (53)
In order to evaluate the quality of the error bounds obtained in Theorem 2, we have performed numerical tests with Ω = (0,2π),V(x) = sin(|x−π|/2) andF(t2) =t2/2.
The Fourier coefficients of the potentialV are given by Vbk =− 1
√2π 1
|k|2−14
, (54)
from which we deduce thatV ∈H#σ(0,2π) for allσ <3/2. It can be see on Figure 1 that kuN −ukH1, kuN −ukL2, kuN −ukH−1, and |λN −λ| decay respectively as N−2.67,N−3.67,N−4.67andN−5(the reference values foruandλare those obtained forN = 65). These results are in very good agreement with the upper bounds (51), (52), (50) and (53), which respectively decay as N−2.5+ǫ, N−3.5+ǫ, N−4.5+ǫ and N−5+ǫ, forǫ >0 arbitrarily small.
y = -2.67617 x + 0.205405 y = -3.67532 x + 0.434802 y = -4.6724 x + 0.677294 y = -5.00941 x - 0.275398 101
9. 100 2. 101 3. 101 4. 101 5. 101 6. 1017. 101
10-9 10-8 10-7 10-6 10-5 10-4 10-3
Figure 1: Numerical errorskuN−ukH1(+),kuN−ukL2(×),kuN−ukH−1 (∗), and
|λN −λ|(◦), as functions of 2N+ 1 (the dimension ofXeN) in log scales.
4 Finite element discretization
In this section, we consider the problem inf
E(v), v∈X, Z
Ω
v2= 1
, (55)
where
Ω is a rectangular brick ofRd, withd= 1, 2 or 3, X =H01(Ω),
E(v) = 1 2
Z
Ω|∇v|2+1 2
Z
Ω
V v2+1 2 Z
Ω
F(v2).
We assume thatV satisfies (6) and that the functionF satisfies (7)-(9). Througout this section, we denote by u the unique positive solution of (55) and by λ the corresponding Lagrange multiplier.
In the non periodic case considered here, a classical variational approximation of (1) is provided by the finite element method. We consider a family of regular triangu- lations (Th)h of Ω. This means, in the case when d= 3 for instance, that for each h >0,This a collection of tetrahedra such that
• Ω is the union of all the elements ofTh;
• the intersection of two different elements of Th is either empty, a vertex, a whole edge, or a whole face of both of them;
• the ratio of the diameter hK of any element K of Th to the diameter of its inscribed sphere is smaller than a constant independent ofh.
As usual,hdenotes the maximum of the diametershK,K∈ Th. The parameter of the discretization then isδ=h >0. For eachKinThand each nonnegative integer k, we denote byPk(K) the space of the restrictions toK of the polynomials withd variables and total degree lower or equal tok.
The finite element spaceXh,kconstructed fromThandPk(K) is the space of all con- tinuous functions on Ω vanishing on∂Ω such that their restrictions to any element K ofTh belong toPk(K). Recall thatXh,k⊂H01(Ω) as soon ask≥1.
We denote by πh,k0 and πh,k1 the orthogonal projectors on Xh,k for the L2 and H1 inner products respectively. The following estimates are classical: there exists C∈R+ such that for allr∈Nsuch that 1≤r≤k+ 1,
∀φ∈Hr(Ω)∩H01(Ω), kφ−πh,k0 φkL2 ≤ChrkφkHr, (56)
∀φ∈Hr(Ω)∩H01(Ω), kφ−πh,k1 φkH1 ≤Chr−1kφkHr. (57) Letuh,k be a solution to the variational problem
inf
E(vh,k), vh,k∈Xh,k, Z
Ω
vh,k2 = 1
(58) such that (uh,k, u)L2 ≥ 0. In this setting, we obtain the following a priori error estimates.
Theorem 3 Assume thatV satisfies (6) and that the functionF satisfies (7)-(9).
Then there existsh0>0 andC∈R+ such that for all 0< h≤h0,
kuh,1−ukH1 ≤ C h (59) kuh,1−ukL2 ≤ C h2 (60)
|λh,1−λ| ≤ C h2. (61) If in addition, V ∈H1(Ω), then there existsh0 >0 andC ∈R+ such that for all 0< h≤h0,
kuh,2−ukH1 ≤ C h2 (62) kuh,2−ukL2 ≤ C h3 (63)
|λh,2−λ| ≤ C h4. (64) Proof As Ω is a convex polygon, and asV andF satisfie (6) and (7)-(9) respec- tively, we haveu∈H2(Ω). We then use the fact thatψuh,k−u is solution to
−∆ψuh,k−u+ (V +f(u2) + 2f′(u2)u2−λ)ψuh,k−u
= 2 Z
Ω
f′(u2)u3ψuh,k−u
u+ (uh,k−u)−(uh,k−u, u)L2u, (65) to establish thatψuh,k−u∈H2(Ω)∩H01(Ω) and that
kψuh,k−ukH2 ≤Ckuh,k−ukL2 (66) for some constant C independent of h and k. The estimates (59), (60) and (61) then are directly consequences of Theorem 1, (57) and (66).
Under the additional assumption that V ∈H1(Ω), we obtain by elliptic regularity arguments that u∈ H3(Ω). TheH1 and L2 estimates (62) and (63) immediately follows from Theorem 1, (57) and (66). We also have
|λ2,h−λ| ≤Ch3 (67) for a constantCindependent ofh. In order to prove (64), we proceed as in Section 3.
We start from the equality
λ2,h−λ=h(Au−λ)(u2,h−u),(u2,h−u)iX′,X+ Z
Ωweh(u2,h−u) (68) where
weh=u22,hf(u22,h)−f(u2) u2,h−u .
We now claim thatuh,2converges touinL∞(Ω) whenhgoes to zero. To establish this result, we first remark that
kuh,2−ukL∞ ≤ kuh,2− Ih,2ukL∞+kIh,2u−ukL∞,
whereIh,2is the interpolation projector onXh,2. Asu∈H3(Ω)֒→C1(Ω), we have
h→0lim+kIh,2u−ukL∞= 0.
On the other hand, using the inverse inequality
∃C∈R+ s.t. ∀0< h≤h0, ∀vh∈Xh,2, kvh,2kL∞ ≤Cρ(h)kvh,2kH1, with ρ(h) = 1 ifd= 1,ρ(h) = 1 + lnhifd= 2 and ρ(h) =h−1/2 ifd= 3 (see [8]
for instance), we obtain
kuh,2− Ih,2ukL∞ ≤ Cρ(h)kuh,2− Ih,2ukH1
≤ Cρ(h) (kuh,2−ukH1+ku− Ih,2ukH1)
≤ C′ρ(h)h2 −→
h→0+ 0.
Hence the announced result. This implies in particular thatwehis bounded inH1(Ω), uniformly inh. Consequently, there existsC∈R+ such that for all 0< h≤h0,
|λh,2−λ| ≤C kuh,2−uk2H1+kuh,2−ukH−1
. (69) To estimate the H−1-norm ofuh,2−u, we write that for allw∈H01(Ω),
Z
Ω
w(uh,2−u)
= −
Z
Ω
f(u2h,2)uh,2−f(u2)uh,2−2f′(u2)u2(uh,2−u) πh,21 ψw
+(λN −λ) Z
Ω
uN(πh,21 ψw−ψw) + (λh,2−λ) Z
Ω
(uh,2−u)ψw
+h(E′′(u)−λ)(uh,2−u), ψw−πh,21 ψwiX′,X +kuh,2−uk2L2
Z
Ω
f′(u2)u3ψw−1
2kuh,2−uk2L2
Z
Ω
uw, (70)
where ψw is solution to
−∆ψw+ (V +f(u2) + 2f′(u2)u2−λ)ψw
= 2 Z
Ω
f′(u2)u3ψw
u+w−(w, u)L2u. (71)
