Convergence to equilibrium for discrete gradient-like flows and An accurate method for the motion of suspended particles in a Stokes fluid

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flows and An accurate method for the motion of

suspended particles in a Stokes fluid

Thanh Nhan Nguyen

To cite this version:

Thanh Nhan Nguyen. Convergence to equilibrium for discrete gradient-like flows and An accurate

method for the motion of suspended particles in a Stokes fluid. Numerical Analysis [math.NA]. Ecole

Polytechnique X, 2013. English. �pastel-00871089�

Présentée pour obtenir legrade de

DOCTEUR DE L'ÉCOLE POLYTECHNIQUE

Spé ialité : Mathématiques Appliquées

par

Thanh-Nhan Nguyen

Convergen e to equilibrium for

dis rete gradient-like ows

and

An a urate method for the motion of

suspended parti les in a vis ous uid

Soutenue le 01O tobre 2013 devant le jury omposé de :

M. FrançoisALOUGES Dire teur de Thèse

M. Jean-PaulCHÉHAB Rapporteur

Mme. Pauline LAFITTE-GODILLON Examinateur

M. Frédéri LAGOUTIÉRE Rapporteur

M. BenoîtMERLET Co-dire teurde Thèse

Thethesishastwo independent parts.

Therstpart on erns the onvergen etoward equilibriumofdis retegradientows or,with

more generality, of some dis retizations of autonomous systems whi h admit a Lyapunov

fun tion. The study is performed assuming su ient onditions for the solutions of the

ontinuousproblemto onvergetoward astationarystateastimegoestoinnity. Itisshown

that under mildhypotheses, the dis rete system has thesame property. This leads to new

resultson the large timeasymptoti behaviorof someknown non-linear s hemes.

These ondpart on erns thenumeri alsimulation of themotion ofparti lessuspendedina

vis ousuid. Itisshownthatthemostwidelyusedmethodsfor omputingthehydrodynami

intera tions between parti leslose theira ura y inthepresen eof largenon-hydrodynami

for es and when at least two parti les are lose from ea h other. This ase arises in the

ontext of medi al engineering for the design of nano-robots that an swim. This loss of

a ura y is due to the singular hara ter of the Stokes ow in areas of almost onta t. A

new method is introdu ed here. Numeri al experiments are realized to illustrate its better

a ura y.

Résumé

Lathèse omporte deuxparties.

La première traite de la onvergen e vers l'équilibre de ots de gradients dis rets ou plus

généralementdedis rétisationsd'unsystèmeautonomeadmettantunefon tiondeLyapunov.

En se plaçant dans une adre pour lequel les solutions du problème ontinu onverge vers

un état stationnaire en temps grand, il est démontré sous des hypothèses générale que le

systèmedis ret a la même propriété. Ce résultat onduit à des on lusionsnouvelles sur le

omportement en temps grandde s hémas numériques an iens.

Lase ondepartie on ernelasimulationnumériquedeparti ulesensuspensiondansunuide

visqueux. Il est montré que les méthodes utilisées a tuellement pour simuler l'intera tion

hydrodynamique entre parti ules perdent de leur pré ision quand de grandes for es

non-hydrodynamiquessontenjeuetqueaumoinsdeuxparti ulessontpro hesl'unedel'autre. Ce

assurvient,dansle ontextedel'ingénieriebiomédi ale,lorsdela on eptiondenano-robots

apables de nager. Cette perte de pré ision est due au ara tère singulier de l'é oulement

de Stokes dansles zones de presque onta t. Une nouvelle méthode est introduite i i. Des

Abstra t i

1 General introdu tion 1

1.1 Introdu tion en Français . . . 1

1.1.1 Objetde lapremière partie . . . 1

1.1.2 Objetde lase onde partie . . . 1

1.2 Introdu tion inEnglish. . . 2

I Convergen e to equilibrium for dis rete gradient-like ows 5 1.3 Introdu tion to thepaper[19℄ . . . 7

1.4 Convergen e to equilibrium for dis retizations of gradient-like ows on Rie-mannian manifolds . . . 10

1.4.1 Introdu tion. . . 11

1.4.2 The ontinuous ase . . . 14

1.4.3 Abstra t onvergen e results. . . 18

1.4.4 The

θ

-s hemeand aproje ted

θ

-s heme . . . 22

1.4.4.1 The

θ

-s heme in

R

d

. . . 22

1.4.4.2 A proje ted

θ

-s heme . . . 24

1.4.5 TheBa kward Euler S heme in

R

d

. . . 26

1.4.6 Harmoni maps andharmoni mapow . . . 29

1.4.7 Appli ation to the Landau-Lifshitzequations . . . 33

1.4.7.1 Spa edis retization . . . 34

1.4.7.2 Time-spa e dis retization oftheLandau-Lifshitz equations . 36 Bibliography 39 II An a urate method for the motion of suspended parti les in a Stokes uid 41 2 Introdu tion - Motivation 43 3 The Stokes problem in an exterior domain 47 3.1 Origin of theequations . . . 47

3.2 Fun tionspa es . . . 49

3.2.1 Spa esasso iatedto a boundeddomain. . . 49

3.2.2 Pressureelds. Homogeneous Sobolev spa es inan exteriordomain . . 50

3.2.3 HomogeneousSobolev spa es ofvelo ityelds inexteriordomains . . 51

3.3 Well-posednessand regularityresults . . . 52

3.3.1 Thefundamental solution andtheStokesequations in

R

3

. . . 53

3.3.2 Lo alregularity . . . 53

3.3.3 Asymptoti sas

|x| → ∞

. . . 54

3.4 Diri hlet-to-Neumann and Neumann-to-Diri hlet operators. . . 54

3.4.1 TheDiri hletto Neumann operatorin aboundeddomain . . . 55

3.4.2 TheDiri hletto Neumann operatorin anexteriordomain . . . 57

3.4.3 Jump of for es through an interfa e. A new Neumann to Diri hlet operator . . . 59

4 Spe tral dis retization of the hydrodynami intera tions 63 4.1 Thehydrodynami intera tions . . . 63

4.1.1 Setting ofthe problem . . . 63

4.1.2 Theboundary integral method . . . 65

4.1.3 Spe tral approximation . . . 67

4.2 De ompositioninve torial spheri al harmoni s . . . 68

4.2.1 Thebasisof ve torial spheri al harmoni s . . . 68

4.2.1.1 Spheri alharmoni s . . . 68

4.2.1.2 Legendrepolynomials . . . 71

4.2.1.3 Ve torial spheri al harmoni s . . . 71

4.2.2 TheStokesproblemina ballor inthe omplement of aball . . . 73

4.2.2.1 De omposition of velo ityand pressureeld. . . 73

4.2.2.2 De omposition of Neumannto Diri hletoperator . . . 76

4.2.3 Pra ti alimplementation oftheboundaryintegralmethodinthebasis ofve torialspheri al harmonni s . . . 78

4.2.3.1 Trun ationorder

L

max

. . . 78

4.2.3.2 Computationof dis reteNeumann to Diri letmatrix

N D

L

max

79 4.2.3.3 Thedis rete fri tionand mobilitymatri es . . . 80

5 The Stokesian dynami method for lose parti les 81 5.1 First numeri al tests. Di ulties for lose parti les . . . 81

5.2 Asymptoti sfor two lose parti les . . . 86

5.2.1 De ompositionof themotion . . . 86

5.2.2 Inner andouter region ofexpansion . . . 87

5.2.3 Asymptoti formulas ofthetotal for e andtorque. . . 90

5.2.3.1 Translation motionof spheres. . . 91

5.2.3.2 Tangential and rollingmotion ofspheres. . . 93

5.2.3.3 Rotationalmotion of spheres . . . 95

5.3 Stokesian Dynami method . . . 95

5.3.1 Mainidea . . . 96

5.3.2 Limitation oftheStokesian Dynami s . . . 96

6 The orre tion method 101 6.1 Singular-regular splitting ofthehydrodynami intera tions . . . 101

6.2 Dis retization . . . 104

6.3 Numeri al results . . . 107

6.3.1 Three parti les . . . 107

6.3.2 Four parti les . . . 109

6.4 Numeri al determinationof thetrun ation orders . . . 111

6.4.1 Corre tiontrun ation order . . . 112

6.4.2 Trun ationorder forsolving theproblem. . . 114

6.5 Con lusions and perpe tives . . . 116

General introdu tion

1.1 Introdu tion en Français

Cemanus ritde thèse omportedeuxparties indépendantes.

1.1.1 Objet de la première partie

Nous traiterons tout d'abord de ots de gradient dis rets et plus généralement de

systèmes d'EDO autonomes qui admettent une fon tion de Lyapunov stri te. Nous nous

intéressons au omportement en temps long de solutions dis rètes de ertains s hémas

asso iés à de tels systèmes. Nous nousplaçons dans des onditions où leproblème ontinue

admet une solution globale

u(t)

qui admet une limite

ϕ

quand letemps

t

tend vers l'inni.

Nousmontrerons que pour ertains s hémas la solutiondis rète admet elle aussi une limite

ϕ

∆t

etque dansle asoù

ϕ

estun point de minimumlo aldelafon tion de Lyapunov,

ϕ

∆t

onverge vers

ϕ

quandlepasde temps

∆t

onvergevers

0

.

Plan de la première partie

Letravailprésentéest onstituéd'uneintrodu tionetde lareprodu tiond'unarti le [19℄

publié onjointement ave Benoît Merlet.

1.1.2 Objet de la se onde partie

La se onde partie traite de l'approximation numérique des intéra tions hydrostatiques

entre parti ules en suspension dans un uide visqueux. La motivation de e travail vient

d'une part de l'étude de suspensions denses et d'autre part de la né essité de produire des

simulations numériques pré ises de mi ro ou nano-nageurs arti iels. Dans le as de es

derniers, omme dans le as de nano-nageurs vivants (spermatozoïdes, ba téries ou algues

mon- ellulaires) l'énergie mobilisable pour la nage est relativement grande, e qui permet

à deux parties du nageur qui sont à une distan e très faible l'une de l'autre d'avoir des

vitessesrelativessensiblementdiérentesbienquelesfor eshydrostatiquess'yopposent. Ces

situations où deux objets à une distan e

d

faible l'un de l'autre ont des vitesses diérentes

réent des densités de for e hydrostatique d'une part élevées et d'autre part singulières au

sens où elles sont lo alisées dans une région de diamètre de l'ordre de

√

d

. Le ara tère lo alisé de es densités de for e fait qu'elles sont mal appro hées par les méthodes basées

dansun uidevisqueux.

Nous proposons une nouvelle méthode, plus oûteuse, mais qui permet une approximation

pré ise. Nous omparonslesperforman esde etteméthode ave laDynamiqueStokesienne.

Plan de la se onde partie

Dans la Chapitre 3, nous présentons tous d'abord le système de Stokes ainsi que les espa esfon tionnels etlesrésultatsthéoriques standards( ara tère bienposédeséquations,

régularité dessolutions). Nousnous plaçons ensuitedans le asde

N

parti ulessphériques

plongées dansunuide visqueux ave des onditions de non-glissement sur lebord des

par-ti ules. Nous présentons laméthode d'approximation spe traledans e adre, en parti ulier

la dé omposition desvitesses etdensités de for e dans une base d'Harmoniques Sphériques

Ve torielles (Chapitre 4). Nousexpliquons ensuiteàpartir d'uneétudenumérique pourquoi ette dé ompositon spe trale n'est pas e a e dans le as où deux parti ules pro hes ont

desvitessessensiblement diérentes(Se tion5.2). La méthodedelaDynamiqueStokesienne qui palie en partie à ette faiblesse est présentée dans laSe tion 5.3. Nous montrons aussi leslimitesde etteméthodesdansle asoùune troisièmeparti ulesesitue danslevoisinage

d'une paire de parti ules pro hes. Nous présentons notre méthode et sa dis rétisation dans

les Se tions 6.1 et 6.2. Les résultats numériques sont exposés dans la Se tion 6.3. Enn, dansune dernièrese tion nousdis utonsles hoix de paramètres de dis rétisation.

1.2 Introdu tion in English

This manus riptis madeof two independent parts whi h have their ownintrodu tions.

We rst study dis rete gradient ows and more generallydis rete s hemes asso iated to

quasi-gradient ows, that is autonomous systems of ODEs whi h admit a stri t Lyapunov

fun tion. Under some general hypotheses, the ontinuous problemadmits a global solution

u(t)

whi h onverges toward somelimit

ϕ

asthetime

t

goesto innity. We showthatunder some mildhypotheses, thedis retesolution asso iated to somestandard s hemes also admit

a limit

ϕ

∆t

. We alsoprove thatifmoreover

ϕ

is alo alminimizer of theLyapunovfun tion

then

ϕ

∆t

onverges to

ϕ

asthetimestep

∆t

goesto 0.

The se ond part on erns the numeri al simulation of the hydrodynami intera tions

between small parti les in a vis ous uid. It is motivated by thestudy of dense suspension

of parti les and also by the study of the motion of living mi roorganisms (sperm ells,

ba teria, uni ellular algae) or of arti ial mi ro or nanorobots. In these ases the energy

whi h is available for the motion is (relatively) large and the swimmers may move lose

parts of their bodies with dierent velo ities even if the hydrodynami for es strongly

oppose su h movement. Su h situations where two obje ts at a small distan e

d

from one

another have dierent velo ities reate hydrostati for e densities whi h on the one hand

have large magnitudes in a small region with a diameter of the order of

√

d

. Due to their lo alized nature, these for e densities are poorly approximated by methods based on a

whi hisawidelyusedmethodtosimulatethemotionofparti lessuspendedinavis ousuid.

Weproposeanewmethod,whi h ismoreexpensive,butallowsa urate approximations.

Convergen e to equilibrium for

1.3 Introdu tion to the paper [19℄

Inthis part,we onsider autonomoussystems ofODE su has

˙u = G(u),

t

_{≥ 0,}

(1.3.1) with

u(t)

∈ R

d

and

G : U

⊂ R

d

_{→ R}

d

.

We assumethatthe systemadmitsa stri t Lyapunov fun tion

F

,that is

d

dt

[F (u)] (t)

≤ 0

foreverysolution and every

t

≥ 0,

andmoreover,

d

dt

[F (u)] (t

0

) = 0

=

⇒

u(t) = u(t

0

)

for

t

≥ t

0

.

Morepre isely,we studysomedis retizationsofthesystem(1.3.1)andstudytheasymptoti behavioras

t

n

→ ∞

ofthe dis rete solutions.

Themost simple situation isthe ase ofa gradient ow

G =

_−∇F.

Inthis ase, if

F

isof lass

C

1

and bounded from below,we have

F (u(t))

_−→

t↑∞

F

∞

∈ R.

Moreover, the

ω

-limit set

ω[u] :=

n

v

_{∈ R}

d

:

_{∃ (t}

n

)

↑ ∞

su hthat

u(t

n

)

→ v

o

isa onne ted subset of the riti alpointsof

F

.

We restri tourstudy to situationsfor whi h

F

satisesaŠojasiewi z inequalityat some

point

ϕ

∈ ω[u]

,thatis: thereexists

ν

∈ [0, 1/2)

,

γ > 0

and

σ > 0

su h that

|v − ϕ| < σ

=

_⇒

_{|F (v) − F (ϕ)|}

1−ν

_{≤ γ|∇F (v)|.}

(1.3.2)

Thisinequalityimplies that

u

hasalimit at innity:

u(t)

−→

t↑∞

ϕ,

(1.3.3)

andwe evenhave thestrongerresult

Z

R

₊

| ˙u| < ∞.

The importan e of the Šojasiewi z inequality (1.3.2) omes from a famous result by Šojasiewi z [17℄whi h statesthatif

F : U

⊂ R

d

_{→ R}

is(real)analyti thensu hinequality

holdsin theneighborhood of any point

ϕ

∈ U

. (Thisis non-trivial only when

ϕ

isa riti al

The onvergen e(1.3.3)isnottrueingeneral. Asa ounterexample,letusbuildafun tion

F : R

2

_{→ R}

asfollows. Givena urve

Γ

parameterized by

γ : R

+

→ R

2

,

γ(s) =



1 +

1

1 + s



(cos s, sin s) .

We dene the tangent ve tor on this urve as

τ (t) = ˙γ/

| ˙γ|

and the normal unit ve tor as

n(t) = (

_−τ

2

, τ

1

)

. Letus x

η > 0

small enough su h thatthemapping at 0,themapping

Φ : (t, s)

_{∈ R}

+

× (−1, 1)

7−→

γ(t) +

η

1 + t

2

n(t)

isone to one. Wethen set

F (x, y) =















g(t)ψ(s)

if

(x, y) = Φ(t, s)

for some

(t, s)

∈ R

+

× (−1, 1),

0

if

(x, y)

6∈ Φ(R

+

× (−1, 1)).

where

g

∈ C

∞

_{([0, +}

_{∞], [0, 1], [0, 1])}

issupported in

[1/2, +

∞)

and

g(t) = e

1−t

for

t

≥ 1

,and

ψ

_{∈ C}

_c

∞

(

_{−1, 1)}

admitsa lo alminima at

s = 0

,with

ψ(0) = 1

and

ψ

′

_{(0) = ψ}

′′

_{(0) = 0}

.

We easily he k that

F

∈ C

∞

c

(R

2

)

andthat the urve

{γ(s) : s ≥ 1}

is thetraje toryof a solution

u

of thegradient ow

˙u =

−∇F (u)

,inparti ular

ω[u] = S

1

,see Figure1.1.

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

−1.5

−1

−0.5

0

0.5

1

1.5

2

0

0.1

0.2

0.3

0.4

0.5

y

x

z

Figure 1.1: Counterexample. Graph of an energy fun tion

F

with a traje tory satisfying

When (1.3.1) isnot a gradient ow, we still have thefollowing result.

Proposition 1.3.1. Assumethat there exists

α > 0

su h that

(

_{−∇F, G) ≥ αk∇F k kGk}

in

U,

(1.3.4)

andthat

u : R

+

→ U

solves(1.3.1) . Ifthereexists

ϕ

∈ ω[u]

su hthat

F

satisesaŠojasiewi z

inequality in the neighborhood of

ϕ

, then

u(t)

→ ϕ

as

t

↑ ∞

.

The ondition (1.3.4)is alledtheangle ondition.

The dis rete ase

Inthe paperreprodu ed after this introdu tion, we onsider theissuewhether a solution of

a numeri al s heme approximating (1.3.1) also satises these onvergen e properties. The pra ti al interest of this work is to establish under mild hypotheses that if

ϕ = lim

∞

u(t)

is a lo al minimizer of

F

and if

(u

∆t

n

)

is a numeri al approximation of

u(t)

obtained by a standardnumeri al s heme withtimestep

∆t > 0

,thenfor

∆t

smallenough

(u

n

)

onverges and

lim

∆t↓0



lim

n↑∞

u

∆t

n



= ϕ.

When

ϕ

is not an isolated lo al minimizer of

F

, the on lusion is not trivial and may be

wrong for some numeri al s hemes. Even for a onverging and energy de reasing numeri al

s heme, the numeri al solution ouldslip towards another lo al minimizer

ϕ

∆t

on the same energy level

F (ϕ

∆t

_{) = F (ϕ)}

and with

ϕ

∆t

_{6−→ ϕ}

as

∆t

↓ 0

.

Ina pre edingwork [18℄,Merlet and Pierre haveestablished thatif

(u

n

)

⊂ R

d

solves

u

n+1

∈

Argmin



F (v) +

|v − u

n

|

2

2∆t

: v

∈ R

d



,

for

n

≥ 0,

(that is

(u

n

)

solvesthe Euler s heme withtimestep

∆t > 0

asso iated to thegradient ow

˙u =

−∇F (u)

) and if

F

∈ C

1

_(R

d

₎

satises a Šojasiewi z inequality in the neighborhood of

somepoint

ϕ

,with

ϕ

_{∈ ω[(u}

n

)] =

\

p≥0

{u

k

: p

≥ k},

then

u

n

→ ϕ

.

Theseauthors alsoestablishsimilar resultsfor the

θ

-s heme for

1/2

≤ θ ≤ 1

.

Herewe ontinue thisanalysisinthe aseofanautonomoussystemwhi hadmitsastri t

Lyapunovfun tion

F

. Weobtainpositiveresultsassuming

(a)

that

F

satisestheŠojasiewi z

inequality,

(b)

that

F

satisesaone-sidedLips hitz ondition(i.e.

u

7→ F (u)+λ|u|

2

is onvex

for

λ

largeenough)and

(c)

that(1.3.4)isstrengthentoanangleand omparability ondition: thereexists

α > 0

su h that

(

−∇F, G) ≥

α

2

k∇F k

2

₊

_kGk

2

in

R

Assuming moreover that

G

is Lips hitz ontinuous, we also obtain positive results for the

θ

-s heme.

In all these ases, we provide expli it onvergen e rates whi h are similar to the ontinuous

ase.

Wealso onsiderthe ase of anautonomous systemonan embedded manifold

M ⊂ R

d

,

i.e. we onsiderautonomous systemssu h that the ondition

u(0)

∈ M

imply

u(t)

∈ M

for

t

_{≥ 0}

. Ingeneralusing astandard numeri al s heme, thenumeri alsolution doesnot satisfy

u

n

∈ M

for

n

≥ 1

. For this reason, we onsider a family of proje ted

θ

-s hemes dened as

u

0

∈ M

andfor

n

≥ 0

,















v

n+1

− u

n

∆t

− θG(v

n+1

)

− (1 − θ)G(u

n

) = 0,

u

n+1

:= Π

M

v

n+1

,

where

Π

M

denotestheorthogonal proje tion on themanifold

M

.

We establish onvergen e to equilibrium results for su h s hemes. As an illustration, we

showthattheyapplytosomespa e-timedis retizationsoftheharmoni mapowandofthe

Landau-Lifshitz equations of mi romagnetism. For these ows, thenumeri al solutions are

subje ted to the onstraint

u

n

∈ (S

2

)

N

,

where

N

is thenumberof degrees offreedom asso iatedto thespa e dis retization.

1.4 Convergen e to equilibrium for dis retizations of

DierentialandIntegralEquations Volume26,Numbers5-6(2013),Pages571-602

Convergen e to equilibrium for

dis retizations of gradient-like ows

on Riemannian manifolds

Benoît Merlet Thanh Nhan Nguyen

Abstra t

Inthispaper,we onsiderdis retizationsofsystemsofdierential equationsonmanifolds

that admit a stri t Lyapunov fun tion. We study the long time behavior of the dis rete

solutions. In the ontinuous ase, if a solution admits an a umulation point for whi h a

Šojasiewi zinequalityholdsthenitstraje tory onverges. Herewe ontinuetheworkstarted

in [18℄ by showing that dis rete solutions have the same behavior under mild hypotheses.

In parti ular, we onsider the

θ

-s heme for systems with solutions in

R

d

and a proje ted

θ

-s heme for systems dened on an embedded manifold. As illustrations,we show thatour resultsapplytoexistingalgorithms: 1)Alouges'algorithm for omputingminimizingdis rete

harmoni mapswithvaluesinthesphere;2)adis retizationoftheLandau-Lifshitzequations

ofmi romagnetism.

1.4.1 Introdu tion

Inthis paper,we onsider timedis retizations ofthenon-linear dierential system,

˙u = G(u),

t

_{≥ 0, u(t) ∈ M,}

(1.4.1)

where

M

⊂ R

d

isa

C

2

-embedded manifoldwithout boundaryand

G

isa ontinuoustangent

ve tor eldon

M

. More pre isely, we are interested inthelong-time behavior and stability

propertiesofthe global solutions of (1.4.1) . Ifthe ontinuous system (1.4.1) admitsa stri t Lyapunov fun tion

F

∈ C

1

_{(M, R)}

andiftheset ofa umulationpoints

ω(u) :=

_{{ϕ ∈ M : ∃(t}

n

)

↑ ∞

su hthat

u(t

n

)

→ ϕ}

is non-empty, then

t

7→ F (u(t))

in a non-in reasing fun tion onverging to

F (ϕ)

where

ϕ

_{∈ ω(u)}

.

Under additional assumptions, namelyif

F

satisesa Šojasiewi z inequality in a

neigh-borhood of

ϕ

and if

G(u)

and

−∇F (u)

satisfy an angle ondition, then one an prove that

u(t)

does indeed onverge to

ϕ

(see the papers by Lageman [16℄, by Chill et al. [10℄ and

the more re ent paper by Bárta et al. [7℄). Under an additional omparability ondition

between

kG(u)k

and

k∇F (u)k

,weevenhave onvergen erates dependingontheŠojasiewi z

exponent.

Noti ethatif

ϕ

isanisolated lo alminimizer of

F

,thentheabove onvergen e property

is almost obvious and the Šojasiewi z inequality is not required. On the other hand these

results are not trivial (and wrong in general) when a onne ted omponent of the riti al

set of

F

does not redu e to a single point. A typi al example is given by the fun tion

F : R

2

_{7→ R, x 7→ (kxk}

2

_{− 1)}

2

for whi h the setof minimizersis

S

1

.

If we are on erned with numeri al simulations, it is of interest to know whether the

above asymptoti properties also holdfor numeri al solutions. Consider,for some time-step

∆t > 0

, a sequen e

(u

n

)

⊂ M

su h that

u

n

approximates the exa t solution

u

at time

t

n

= n∆t

. Su hasequen e ouldbebuilt bymeansof anystandardor reasonablenumeri al

s heme. Mimi king the ontinuous ase, ourrst goalis to establishthefollowing property:

Result 1. If

ϕ

is ana umulationpoint ofthesequen e

(u

n

)

,then

u

n

→ ϕ

. When

ϕ

isa lo alminimizer of

F

,weexpe ta morepre ise stabilityresult:

Result 2. Let

ϕ

∗

_{∈ M}

be a lo al minimizer of

F

. For every

η > 0

there exists

0 < ε < η

su h that

ku

N

− ϕ

∗

k < ε

=

⇒

ku

n

− ϕ

∗

k < η, ∀n ≥ N.

Moreover, inthis ase, the sequen e

(u

n

)

onverges to some

ϕ

∈ M

.

Itturnsout thatthislastpropertyleadsto auniform onvergen eresult. Indeed,assume

that the exa t solution

u

onverges to a lo al minimizer

ϕ

∗

of

F

, then for

T

large enough,

we have

ku(t) − ϕ

∗

k < ε/2,

∀t ≥ T.

Ifthes hemeisuniformly onvergent onniteintervals(areasonablequery)thenfor

∆t > 0

small enough,we have,

ku

n

− u(t

n

)

k < ε/2,

for

0

≤ t

n

≤ T + 1.

Inparti ular,

ku

N

− ϕ

∗

_{k < ε}

where

T

≤ N∆t < T + 1

. Applying Result2,we on ludethat

for

∆t > 0

smallenough, we have

ku

n

− u(t

n

)

k ≤ η,

∀n ≥ N.

We theninfer,

lim

∆t↓0

sup

_n≥0

ku

n

− u(t

n

)

k = 0.

As a onsequen e, denoting

ϕ(∆t) := lim

n→∞

u

n

, we also have

ϕ(∆t)

→ ϕ

∗

as

∆t

→ 0

.

Thus, the numeri al s heme provides a method to approximate the limit

ϕ

∗

. Thisproperty

motivates our interestfor Result2.

On e the onvergen e of the sequen e is known, we will try to pre ise the onvergen e

rateand establish:

Thefun tion

t

7→ f(t)

shouldde reaseto

0

as

t

goesto

+

∞

. Typi ally

f

isanexponential

ora rational fun tion(see (1.4.12) below).

As in the ontinuous ase, the ba kground assumptions on (1.4.1) to obtain Results of type 1, 2 and 3 are a Šojasiewi z inequality, the angle ondition and (for the onvergen e rates)the omparability onditionthat wewill des ribe inthenext se tion. Forthes heme,

on top of usual onsisten y property, the basi additional required assumption is that

F

shouldbea stri t Lyapunov fun tion forthes hemewithan estimateof theform

F (u

n+1

) + µ

ku

n+1

− u

n

k

2

∆t

≤ F (u

n

),

(1.4.2)

for some

µ > 0

. In the ase of a gradient ow

G =

−∇F

, and if

∇F

satises a one-sided

Lips hitz ondition,thisstabilitypropertyisnaturallysatisedbytheba kwardEulers heme

and,under a regularity assumption on

G

, by the

θ

-s heme for

0

≤ θ < 1

. In this paper,we

fo uson these s hemes andwe assumethat

∇F

satisesaone sided Lips hitz ondition.

In a previous paper, Merlet and Pierre [18℄ (see also [8, Theorem 24℄) have studied the

longtimebehaviorof some time-dis retizations ofthegradient system,

˙u =

_{−∇F (u),}

t

_{≥ 0, u ∈ R}

d

.

(1.4.3)

Their results have been generalized to some se ond order perturbations in [13℄. A losely

relatedquestion on erns the onvergen e of theproximal algorithm asso iated to the

mini-mization of

F

inniteor innite dimension (see [1,6,8℄and referen es therein). Asinthe present work,the key assumption for onvergen e results in these papersis the Šojasiewi z

inequality. Hereweextendtheresultsof[18℄ongradientowsin

R

d

by onsidering

gradient-likesystemsonamanifold: weestablishResultsoftype1,2and3forthe

θ

-s hemesasso iated tosu hsystems.

Thesequel isorganizedasfollows. In thenext se tion,weset thenotation and themain

hypotheses. We also prove the onvergen e result in the ontinuous ase. Elements of this

proofareusedinSe tion 1.4.5.

Our results on erning the

θ

-s heme and the proje ted

θ

-s heme will be obtained as a

onsequen e of general abstra t results of type 1, 2 and 3 that we rst establish in Se -tion 1.4.3. We will highlight there the essential hypotheses required for these onvergen e toequilibrium results. We believethat thisgeneral setting enablesto qui kly he kwhether

onvergen e to equilibrium properties inthe ontinuous ase transpose to the solutions of a

numeri al s heme inspe i situations.

InSe tion 1.4.4.1,weapply theabstra t situationto the

θ

-s hemes asso iatedto (1.4.1)

inthe ase

M = R

d

.

In Se tion 1.4.4.2, we onsiderthe ase of an embedded manifold by paying attentionto the onstraint

u(t)

∈ M

. Of ourse, underusual hypotheses ensuringthe unique solvability

of (1.4.1) for

u(0)

∈ M

(e.g. assuming that

G

is lo ally Lips hitz), the traje tory of the solution will remain on

M

. This is no longer true for general time dis retizations. For this

reason,we introdu eandstudyalinearized

θ

-s hemesupplementedbyaproje tionstepthat

enfor esthe onstraint

u

n

∈ M

.

We onsider the ba kward Euler s heme in the ase

M = R

d

(Se -problem (even in the ase of thegradient-like system(1.4.1) ) allows us to weaken the regu-larityhypotheseson

G

.

Eventually,we apply our methods to some on rete problems. First, we onsiderin

Se -tion 1.4.6 a s heme byAlouges [3℄ designed for the approximation of minimizing harmoni maps withvaluesin the sphere

S

l−1

. We establishthatthesequen e built by thealgorithm

does onverge to a dis rete harmoni map. The original result was onvergen e up to

ex-tra tion. Then, in Se tion 1.4.7, we apply our results on erning the proje ted

θ

-s heme to a dis retization of the Landau-Lifshitz equations of mi romagnetism (again proposed by

Alouges[4℄). Theseexamples illustrate our general results for inboth ases thetraje tories

lie ona non-atmanifold (

(S

l−1

₎

N

). Moreover, inthelastexample, theunderlying

ontinu-oussystem isnot agradient ow but merely a systemon the form(1.4.1) admiting a stri t Lyapunovfun tion.

1.4.2 The ontinuous ase

From now on,

M

is a

C

2

-Riemannian manifold without boundary. Without loss of

gen-erality,we assumethat

M

is embedded in

R

d

and that the inner produ ton every tangent

spa e

T

u

M

is the restri tionof theeu lidian innerprodu t on

R

d

.

We onsider a tangent ve tor eld

G

∈ C(M, T M)

and a fun tion

F

∈ C

1

_{(M, R)}

. We

assumethat

−∇F

and

G

satisfytheangle ondition dened below.

Denition 1.4.1. We say that

G

and

∇F

satisfy the angle ondition ifthere exists a real

number

α > 0

su h that

hG(u), −∇F (u)i ≥ αkG(u)kk∇F (u)k,

∀u ∈ M.

(1.4.4)

Remark 1.4.1. In this denition and below, when we onsider a dierentiable fun tion

F : M

_{→ R}

, we write

∇

M

F

or simply

∇F

to denote the gradient of

F

with respe t to the

tangent spa e of

M

. In parti ular,

∇F (u) ∈ T

u

M

. If the fun tion

F

is also dened on a neighborhood

Ω

of

M

in

R

d

, the notation

∇F

is ambiguous. In this ase

∇F (u)

denotesthe

gradient of

F

in

R

d

, that is

∇F (u) = (∂

x

1

F (u),

· · · , ∂

x

d

F (u))

and

∇

M

F (u) = Π

T

u

M

∇F (u)

where

Π

T

u

M

denotesthe orthogonal proje tion on the tangent spa e

T

u

M

⊂ R

d

.

Weassume moreover that

F

isa stri t Lyapunov fun tion for (1.4.1) :

Denition 1.4.2. We saythat

F

is aLyapunov fun tion for (1.4.1) iffor every

u

∈ M

,we

have

hG(u), −∇F (u)i ≥ 0

. If moreover,

∇F (u) = 0

implies

G(u) = 0

thenwe saythat

F

is

a stri tLyapunovfun tion.

As we already noti ed, the key tool of the onvergen e results presented here is a

Šo-jasiewi zinequality.

Denition 1.4.3. Let

ϕ

∈ M

.

1)Wesaythatthefun tion

F

satisesaŠojasiewi zinequalityat

ϕ

ifthereexists

β, σ > 0

and

ν

∈ (0, 1/2]

su h that,

The oe ient

ν

is alleda Šojasiewi zexponent.

2) The fun tion

F

satises a Kurdyka-Šojasiewi z inequality at

ϕ

if there exists

σ > 0

anda non-de reasingfun tion

Θ

∈ C(R

+

, R

+

)

su hthat

Θ(0) = 0,

Θ > 0

on

(0, +

∞),

1/Θ

∈ L

1

loc

(R

+

)

(1.4.6)

and,

Θ (

_{|F (u) − F (ϕ)|) ≤ k∇F (u)k,}

_{∀u ∈ B(ϕ, σ) ∩ M.}

(1.4.7)

Noti e that the rst denition is a parti ular ase of the se ond one with

Θ(f ) =

(1/β)f

1−ν

. The interestof therstdenition relieson thefollowing fundamental result:

Theorem 1 (Šojasiewi z [17℄, see also [15℄). If

F : Ω

⊂ R

d

_{→ R}

is real analyti in some

neighborhood of a point

ϕ

, then

F

satises the Šojasiewi zinequalityat

ϕ

.

Remark1.4.2. TheŠojasiewi zinequalityonlyprovidesinformationat riti al pointsof

F

.

Indeed, if

ϕ

is nota riti al point of

F

, then by ontinuity of

∇F

, the Šojasiewi zinequality

issatised in someneighborhood of

ϕ

.

It is well known that the Kurdyka-Šojasiewi z inequality implies the onvergen e of the

bounded traje tories of the gradient ow (1.4.3) as

t

goes to innity. Here we state the onvergen e resultinthemoregeneral aseof agradient-like system:

Theorem2. Assumethat

F

isastri tLyapunovfun tionfor (1.4.1)andthat

G,

∇F

satisfy the angle ondition (1.4.4). Let

u

be a global solution of (1.4.1) and assume that there exists

ϕ

∈ ω(u)

su h that

F

satises the Kurdyka-Šojasiewi z inequality (1.4.7) at

ϕ

. Then

u(t)

_{→ ϕ}

as

t

→ +∞

.

Remark 1.4.3. In many appli ations, the Kurdyka-Šojasiewi z hypothesis holds at every

point. Moreover, for nite dimensional systems the fa t that

ω(u)

is not empty is often the

onsequen e of a oer ivity onditionon

F

,

F (u)

−→ +∞,

as

kuk → ∞.

(1.4.8)

Proof. The proofstated herefollows [10℄. First we write

d

dt

[F (u(t))]

(1.4.1)

=

hG(u(t)), ∇F (u(t))i

(1.4.4)

≤

−αkG(u(t))kk∇F (u(t))k ≤ 0,

and the fun tion

F (u)

is non-in reasing. By ontinuity of

F

and sin e

ϕ

∈ ω(u)

,

F (u(t))

onverges to

F (ϕ)

as

t

goes to

+

∞

. Changing

F

by an additive onstant if ne essary, we

mayassume

F (ϕ) = 0

,sothat

F (u(t))

≥ 0

for every

t

≥ 0

.

If

F (u(t

0

)) = 0

for some

t

0

≥ 0

then

F (u(t)) = 0

for every

t

≥ t

0

andtherefore, (sin e

F

is a stri t Lyapunov fun tion),

u

is onstant for

t

≥ t

0

. Inthis ase, there remains nothing toprove.

Hen e we may assume

F (u(t)) > 0

for every

t

≥ 0

. Sin e

F

satises a

Kurdyka-Šojasiewi z inequality at

ϕ

, there exist

σ > 0

and a fun tion

Θ

∈ C(R

+

, R

+

)

satisfy-ing (1.4.6) and(1.4.7) . Letus dene

Φ(f ) =

Z

f

0

1

Letus take

ε

∈ (0, σ)

. There exists

t

0

large enough su h that

ku(t

0

)

− ϕk + α

−1

Φ(F (u(t

0

))) < ε.

Let us set

t

1

:= inf

{t ≥ t

0

:

ku(t) − ϕk ≥ ε}

. By ontinuity of

u

we have

t

1

> t

0

. Then for

every

t

∈ [t

1

, t

0

)

, using the angle ondition (1.4.4) and the Kurdyka-Šojasiewi z inequality, we have

−

_dt

d

Φ(F (u(t))) =

hG(u), −∇F (u)i

Θ(F (u(t)))

≥ αkG(u(t))k = αku

′

_(t)

_k.

(1.4.10)

Integrating on

[t

0

, t)

for any

t

∈ [t

0

, t

1

)

,we get

ku(t) − ϕk ≤ ku(t) − u(t

0

)

k + ku(t

0

)

− ϕk ≤

Z

t

1

t

0

u

′

_(s)

_{ds + ku(t}

0

)

− ϕk

≤ α

−1

Φ(F (u(t))) +

ku(t

0

)

− ϕk < ε.

This inequality implies

t

1

= +

∞

. Eventually, the estimate (1.4.10) yields

˙u

∈ L

1

_(R

+

)

and we on lude that

u(t)

onverges to

ϕ

as

t

goes to innity.

In the ase of a gradient ow and if the Šojasiewi z inequality is satised, we have an

expli it onvergen e rate thatdependson the Šojasiewi z exponent. In orderto extend this

resultto gradient-like systems,theangle ondition isnot su ient:

Denition 1.4.4. We saythat

G

and

∇F

satisfy the angle and omparability ondition if

there existsareal number

γ > 0

su h that

hG(u), −∇F (u)i ≥

γ

2



kG(u)k

2

+

_{k∇F (u)k}

2



,

_{∀u ∈ M.}

(1.4.11)

Remark1.4.4. Noti ethatthis onditionimpliestheangle ondition(1.4.4) . Infa t (1.4.11) isequivalent to the fa tthat there exists

α > 0

su h that for every

u

∈ M

,

hG(u), −∇F (u)i ≥ αkG(u)kk∇F (u)k

and

α

−1

kG(u)k ≥ k∇F (u)k ≥ αkG(u)k.

Theorem 3. Under the hypotheses of Theorem

2

,

assume moreover that

∇F

and

G

satisfy the angle and omparability ondition (1.4.11) and that

F

satises a Šojasiewi z inequality withexponent

0 < ν

≤ 1/2

, then there exist

c, µ > 0

su hthat,

ku (t) − ϕk ≤

(

c e

−µt

if

ν = 1/2,

c t

−ν/(1−2ν)

if

0 < ν < 1/2,

∀ t ≥ 0.

(1.4.12)

Proof. By Theorem 2, we know that

u(t)

onverges to

ϕ

as

t

goes to innty. As in the pre eding proof,we mayassume

F (ϕ) = 0

and

F (u(t)) > 0

forevery

t

≥ 0

. Let

Φ

bedened

by (1.4.9) with

Θ(f ) := (1/β)f

1−ν

the expli it formula

Φ(f ) = (β/ν)f

ν

. Next let

t

1

≥ 0

su h that

ku(t) − ϕk ≤ σ

for every

t

_{≥ t}

1

. By (1.4.10) ,for every

t

≥ t

1

,we have

ku(t) − ϕk ≤

Z

+∞

t

u

′

_(s)

_{ds ≤}

Z

+∞

t

γ

−1

H

′

(s)ds = γ

−1

H(t).

(1.4.13)

Usingtheangleand omparability ondition(1.4.11) andtheŠojasiewi zinequality (1.4.5), we ompute

−H

′

(t) = β [F (u(t))]

ν−1



−

d

dt

[F (u(t))]



(1.4.11)

≥

β [F (u(t))]

ν−1

γ

2

k∇F (u(t))k

2

(1.4.5)

≥

γβ

₂

[F (u(t))]

ν−1

1

β

2

[F (u(t))]

2−2ν

₌

γ

2β

[F (u(t))]

1−ν

_{= λ [H(t)]}

1−ν

_ν

_.

where

λ = C(γ, β, ν) > 0

. Summing up,we have

H

′

(t) + λ [H(t)]

1−ν

ν

≤ 0,

∀t ≥ t

₁

.

Inthe ase

ν = 1/2

,we get

H

′

_{(t) + λH(t)}

_{≤ 0}

. Writing

H(t) = e

−λt

_g(t)

we on lude that

g

isnon-in reasing, so

H(t)

≤ c e

−λt

for every

t

≥ t

1

.

In the ase

0 < ν < 1/2

, we set

K(t) := [H(t)]

−(1−2ν)/ν

. This fun tion satises

K

′

_(t)

_≥

λν/(1

_{− 2ν)}

, whi h implies

K(t)

≥ at

for some

a > 0

and for

t

large enough. Hen e

H(t)

_{≤ (at)}

−ν/(1−2ν)

_{= ct}

−ν/(1−2ν)

.

Combining this estimatewith(1.4.13) ompletes theproof.

Inthesequel,we usesome toolsrelatedto Riemannian metri s on

R

d

thatwe introdu e

here.

Denition1.4.5. Let

g

beaRiemannianmetri on

R

d

. Were allthatthegradient

∇

g

F (u)

of

F

withrespe tto themetri

g

ata point

u

is dened by

h∇F (u), Xi = h∇

g

F (u), X

i

_g

,

∀X ∈ R

d

We write

h., .i

g

(to be pre ise, we should write

h., .i

g(u)

) for the inner produ t on the tangent spa eat the point

u

. We alsowrite

k.k

g

for theindu ed norm.

In [7℄, Bárta et al. establish the remarkable result that when (1.4.1) admits a stri t Lyapunovfun tionthen,uptoa hangeofmetri , (1.4.1)isagradientsystem. Thefollowing theoremisa dire t orollaryof Theorem 1and Theorem2 in[7℄.

Theorem4. Assume that

F

isastri t Lyapunovfun tion. Thenthere existsa Riemannian

metri

g

on

M :=

˜



u

∈ R

d

_{: G(u)}

_{6= 0}

su hthat

G =

−∇

g

F

.

Moreover,if

∇F

and

G

satisfytheangleand omparability ondition (1.4.11)then

g

is equiv-alent tothe Eu lidean metri . Namely, there exist

c

1

, c

2

> 0

su h that

c

1

kXk ≤ kXk

g(u)

≤ c

2

kXk,

∀X ∈ R

d

,

∀u ∈ ˜

M .

(1.4.14)

Denition 1.4.6. We saythat

∇F

satisesthe one-sidedLips hitz onditionifthere exists

c

_{≥ 0}

su hthat:

h∇F (u) − ∇F (v), u − vi ≥ −cku − vk

2

,

∀u, v ∈ M.

(1.4.15)

Let usdene the set ofa umulationpointsof thesequen e

(u

n

)

⊂ M

:

ω((u

n

)) :=



ϕ

∈ M :

thereexistsa subsequen e

(u

n

k

)

s.t.

u

n

k

−−−→

k→∞

ϕ



.

To end thisse tion, we re all the mainhypothesesintrodu ed above:

•

ω(u

n

)

isnot empty. (H0)

(In appli ations, thishypothesis ismainly a onsequen e of (1.4.2)and (1.4.8) .)

•

The Kurdyka-Šojasiewi z inequality holdsat some

ϕ

∈ ω(u

n

)

. (H1)

•

∇F

and

G

satisfytheangleand omparability ondition (1.4.11). (H2)

•

∇F

satisesthe one-sidedLips hitz ondition (1.4.15) . (H3)

For onvergen e rateresults, (H1) is repla ed by

•

TheŠojasiewi z inequalityholds at some

ϕ

∈ ω(u

n

)

. (H1')

Otherassumptionsontheregularityof

G

and

F

willbemadeinthestatementoftheresults.

1.4.3 Abstra t onvergen e results

In this se tion we prove Results of type 1, 2 and 3 for abstra t sequen es

(u

n

)

⊂ M

satisfying the two additional onditions introdu ed below. In the ase

M = R

d

, Results of

type 1 and 2 are known (see Absil et al. [1℄). The proofs are identi al in the ase of an

embedded manifold but we provide them for ompleteness and be ause we use them in the

proofof Result3.

Let usintrodu e our rst ondition:

∃C ≥ 0, ∀n ≥ 0, F (u

n

)

− F (u

n+1

)

≥ Ck∇

M

F (u

n

)

kku

n

− u

n+1

k.

(H4)

We need moreovera dis reteversion ofthe stri t Lyapunovhypothesis:

∀n ≥ 0, F (u

n+1

) = F (u

n

)

=

⇒

u

n+1

= u

n

.

(H5)

Werst state aResultof type1.

Theorem 5 ([1℄ Theorem 3.2). Let

(u

n

)

⊂ M

. Assume that hypotheses

(H0)

,

(H1)

,

(H4)

and

(H5)

hold. Thenthe sequen e

(u

n

)

onverges to

ϕ

as

n

goes toinnity.

Proof. By(H4)thesequen e

(F (u

n

))

isnon-in reasing,soby ontinuityof

F

andhypothesis

(H0)

, we know that

F (u

n

)

onverges to

F (ϕ)

whi h an be assumedto be 0. By (H5), we mayassume that

F (u

n

)

is de reasing, sin e in the other ase, the sequen e is onstant and thus onverge to

ϕ

. Sin e

F

satises the Kurdyka-Šojasiewi z inequality at

ϕ

, there exist

σ > 0

and anon-de reasing fun tion

Θ

satisfying (1.4.6) and (1.4.7) . Let

Φ

bethe fun tion

dened by(1.4.9) . Wehave for

n

≥ 0

,

Φ(F (u

n

))

− Φ(F (u

n+1

)) =

Z

F (u

n

)

F (u

n+1

)

ds

Θ(s)

≥

1

Θ(F (u

n

))

(F (u

n

)

− F (u

n+1

)) .

Using (H4) , weobtain

Φ(F (u

n

))

− Φ(F (u

n+1

))

≥ C

k∇

M

F (u

n

)

k

Θ(F (u

n

))

ku

n+1

− u

n

k.

(1.4.16)

By

(H0)

and the onvergen e of

(F (u

n

))

to

0

,there exists

¯

n

su hthat

ku

n

¯

− ϕk +

1

C

Φ(F (u

¯

n

)) < σ.

Letus dene

N := sup

_{{n ≥ ¯n : ku}

k

− ϕk < σ, ∀¯n ≤ k ≤ n} ,

andassume by ontradi tion that

N

is nite. For every

n

¯

≤ n ≤ N

, we have

ku

n

− ϕk < σ

, sowe anapply (1.4.7) with

u = u

n

and dedu efrom (1.4.16)

ku

n+1

− u

n

k ≤ (1/C) {Φ(F (u

n

))

− Φ(F (u

n+1

))

} ,

∀¯n ≤ n < N + 1.

(1.4.17)

Summingthese inequalities, we get

N

X

n=¯

n

ku

n+1

− u

n

k ≤

1

C

Φ(F (u

¯

n

)).

(1.4.18) Inparti ular,

ku

N +1

− ϕk ≤

1

C

Φ(F (u

n

¯

)) +

ku

n

¯

− ϕk < σ,

whi h ontradi ts the denition of

N

. So

N = +

∞

, and the onvergen e of the sequen e

follows from(1.4.18).

We nowestablisha resultoftype 2.

Theorem 6 ([1℄ Proposition 3.3). Let

ϕ

be a lo al minimizer of

F

su h that

F

satises a Kurdyka-Šojasiewi z inequality in a neighborhood of

ϕ

. Consider a sequen e

(u

n

)

⊂ M

and assume that (H4) holds. Then,for every

η > 0

there exists

ε

∈ (0, η)

onlydepending on

F

,

η

and the onstant

C

in (H4) su h that

ku

n

¯

− ϕk < ε

=

⇒

ku

n

− ϕk < η, ∀n ≥ ¯n.

Proof. We assumewithoutlossofgeneralitythat

F (ϕ) = 0

. Sin e

ϕ

is alo alminimizer of

F

,thereexists

ρ > 0

su h that

∀u ∈ R

d

,

_{ku − ϕk < ρ ⇒ F (u) ≥ 0.}

(1.4.19)

Moreover, sin e

F

satisestheKurdyka-Šojasiewi z inequalityat

ϕ

,thereexist

σ > 0

anda

fun tion

Θ

satisfying (1.4.6) and (1.4.7) .

Let

η > 0

andlet us set

¯

η := min(ρ, σ, η),

We x

ε

∈ (0, η)

su h thatfor every

u

∈ M

ku − ϕk < ε

=

_⇒

_{ku − ϕk + (1/C)Φ(F (u)) < ¯η.}

Thenwe onsiderasequen e

(u

n

)

satisfying(H4)andweassumethatthereexists

n

¯

≥ 0

su h

that

ku

¯

n

− ϕk < ε

. Then, asinthe proof oftheprevious result, we dene

N := sup

{n ≥ ¯n : ku

k

− ϕk < ¯η, ∀¯n ≤ k ≤ n} ,

and assume by ontradi tion that

N

is nite. As in the proof of the pre eding result, we

establishand sumthe Kurdyka-Šojasiewi z inequalities(1.4.17) for

n

¯

≤ n ≤ N

to get:

N

X

n=¯

n

ku

n+1

− u

n

k ≤

1

C

{Φ(F (u

n

¯

))

− Φ(F (u

N

))

} .

By denition of

N

and (1.4.19), we have

F (u

N

)

≥ 0

so that

Φ(F (u

N

))

≥ 0

and (1.4.18) holds. We dedu e

ku

N +1

− ϕk ≤

1

C

Φ(F (u

n

¯

)) +

ku

¯

n

− ϕk < ¯η,

whi h ontradi tsthedenitionof

N

. The onvergen eof

(u

n

)

thenfollowsfrom (1.4.18). Remark 1.4.5. The result does not hold if we only assume that

ϕ

is a riti al point of

F

.

In this ase even if

u

n

¯

isvery lose to

ϕ

, the sequen e may es ape the neighborhood of

ϕ

by taking values

F (u

n

) < F (ϕ)

. In this ase the proof is no more valid. Indeed, we still have the key estimate

N

X

n=¯

n

ku

n+1

− u

n

k ≤ c{Φ (F (u

¯

n

))

− Φ(F (u

N

))

} ,

but we an not bound the right handside by

cΦ(F (u

n

¯

))

.

In order to prove a onvergen e rate resultof type 3,we need to supplement (H4) with thefollowing hypothesis: there exists

C

2

> 0

su h thatfor every

n

≥ 0

,

ku

n+1

− u

n

k ≥ C

2

k∇F (u

n

)

k.

(H6)

In the ase ofnumeri al dis retizations of thegradient ow (1.4.3) (ormore generallyof a gradient-like system (1.4.1)), the quantity

ku

n+1

− u

n

k

behaves like

∆t

k∇F (u

n

)

k

, where

∆t

is the time step. For these appli ations, the onstant

C

2

intheabove hypothesis should s ale as

∆t

: we expe t

C

2

= C

′

2

∆t

. In this ontext, the fa tors

C

2

n

in the onvergen e rates (1.4.20) below, have the form

C

′

2

t

n

. So, these rates are uniform with respe t to the time step

∆t

. Infa twe re over the onvergen e ratesof the ontinuous ase (withpossibly

Theorem 7. Let

(u

n

)

⊂ M

. Assume that hypotheses

(H0)

,

(H1

′

₎

and

(H5)

hold and that

there exist

C, C

2

> 0

su h that (H4) and (H6) hold for

n

≥ 0

. Thenthere exists

¯

n

≥ 0

su h that for all

n

≥ ¯n

ku

n

− ϕk ≤

(

λ

1

e

−λ

2

C

2

n

if

ν = 1/2,

λ

2

(C

2

n)

−ν/(1−2ν)

if

0 < ν < 1/2.

(1.4.20)

where

ν

istheŠojasiewi zexponentof

F

atpoint

ϕ

and

λ

1

, λ

2

arepositive onstantsdepending on

C, β

and

ν

.

Proof. First let us re all some fa ts from the proof of Theorem 5. We know that

(u

n

)

onverges to

ϕ

and thatthe sequen e

(F (u

n

))

isnon-in reasing and onverges to

F (ϕ)

that weassumeagaintobezero. Let

˜

n

≥ 0

su hthat

ku

n

−ϕk < σ

for

n

≥ ˜n

,we anapply(1.4.18)

with

n = n

¯

and

N = +

∞

for every

n

≥ ˜n

. Here, the fun tion

Φ

dened by (1.4.9) hasthe

expli itform

Φ(f ) = (β/ν)f

ν

andestimate (1.4.18) yields

ku

n

− ϕk ≤

β

Cν

[F (u

n

)]

ν

,

∀n ≥ ¯n.

(1.4.21)

Next,let us dene thefun tion

K : (0, +

∞) → (0, +∞)

by

K(x) =







− ln x

if

ν = 1/2

1

(1

_{− 2ν)x}

1−2ν

if

0 < ν < 1/2

Thesequen e

(K(F (u

n

)))

isnon-de reasingand tendsto innity. Using (H4) and(H6) ,we have

K(F (u

n+1

))

− K(F (u

n

)) =

Z

F (u

n

)

F (u

n+1

)

dx

x

2−2ν

≥

F (u

n

)

− F (u

n+1

)

[2F (u

n+1

)]

2−2ν

(H4) (H6)

≥

CC

2

k∇F (u

n

)

k

2

/ [F (u

n+1

)]

2−2ν

.

Applying(1.4.7) in the right hand sideof thelast inequality, weget

K(F (u

n+1

))

− K(F (u

n

))

≥ CC

2

/β

2

,

∀n ≥ ¯n.

Summingfrom

n

¯

to

n

− 1

,weget thatthereexists

c

1

∈ R

su h that

K(F (u

n

))

≥ (CC

2

/β

2

)n + c

1

,

∀n ≥ ¯n.

(1.4.22)

Nowlet us onsiderthe ase

ν = 1/2

,we have

K(F (u

n

)) =

− ln(F (u

n

))

,sowe get

F (u

n

)

≤ λe

−(CC

2

/β

2

_)n

,

∀n ≥ ¯n,

with

λ = e

−c

1

. Re alling (1.4.21),theTheorem isproved inthis ase. Eventually, if

0 < ν < 1/2

,(1.4.22) reads

F (u

n

)

≤



1

− 2ν

(CC

2

/β

2

)n + c

1



1/(1−2ν)

,

∀n ≥ ¯n.

1.4.4 The

θ

-s heme and a proje ted

θ

-s heme

In this se tion, we show that the onvergen e results of Se tion 1.4.3 apply to some numeri al s hemes asso iated to system (1.4.1) under the set of hypotheses

(H0)

,

(H1)

,

(H2)

,

(H3)

(

(H0)

,

(H1

′

₎

,

(H2)

,

(H3)

for the onvergen e rate). We also need some

regularityassumptions on

G

and

F

.

1.4.4.1 The

θ

-s heme in

R

d

Werst onsiderthe

θ

-s hemeinthe ase

M = R

d

. Re allthat fora xed

θ

∈ [0, 1]

,the

θ

-s hemeasso iated to equation(1.4.1) reads:

u

n+1

− u

n

∆t

= θG(u

n+1

) + (1

− θ)G(u

n

).

(1.4.23)

Lemma 1.4.7. Let

θ

∈ [0, 1]

and let

(u

n

)

be a sequen e that omplies to the

θ

-s heme (1.4.23) . Assume that

G

is Lips hitz ontinuous and that hypotheses (H2) and (H3)hold. Thenthere exist

µ

1

, µ

2

, ∆t

′

_{> 0}

su h thatfor

∆t

∈ (0, ∆t

′

₎

,

F (u

n+1

) + µ

1

ku

n+1

− u

n

k

2

∆t

≤ F (u

n

),

∀n ≥ 0,

(1.4.24) and

ku

n+1

− u

n

k

∆t

≥ µ

2

k∇F (u

n

)

k,

∀n ≥ 0.

(1.4.25)

Proof. Firstwe establish(1.4.25). Werewrite the

θ

-s heme intheform

u

n+1

− u

n

∆t

= G(u

n

) + θ [G(u

n+1

)

− G(u

n

)] .

Denoting

K

≥ 0

the Lips hitz onstant of

G

on

R

d

and using the omparability

ondi-tion(1.4.11) , we dedu e

(1 + K∆t)

ku

n+1

− u

n

k

∆t

≥ kG(u

n

)

k

(1.4.11)

≥

γ

2

k∇F (u

n

)

k.

So (1.4.25) holds with

µ

2

= γ/4

assoonas

∆t

∈ (0, 1/K)

.

Next we prove(1.4.24) . Byassumption

(H2)

we an applyTheorem4 andthereexistsa metri

g

on

˜

M := R

d

_{\ {v : G(v) = 0}}

satisfying (1.4.14) and su h that

h−∇F (u), wi = hG(u), wi

g(u)

,

∀u ∈ ˜

M , w

∈ R

d

.

(1.4.26)

Letus set

δ

n

:= u

n

− u

n+1

andwrite theTaylorexpansion,

F (u

n

) = F (u

n+1

) +

Z

1

0

∇F (u

n+1

+ tδ

n

) dt, δ

n



= F (u

n+1

) +

h∇F (u

n+1

), δ

n

i +

Z

1

0

∇F (u

n+1

+ tδ

n

)

− ∇F (u

n+1

), δ

n



dt.

Applyingassumption (H3) with

u = u

n+1

+ tδ

n

and

v = u

n+1

,we get

F (u

n

)

− F (u

n+1

)

− h∇F (u

n+1

), δ

n

i ≥ −c

Z

1

0

t

_kδ

n

k

2

dt,

thatis,

F (u

n+1

)

− F (u

n

)

≤ h∇F (u

n+1

), u

n+1

− u

n

i + (c/2)ku

n+1

− u

n

k

2

.

If

u

n+1

∈ ˜

M

,thenwe mayapply (1.4.26)with

u = u

n+1

and get

F (u

n+1

)

− F (u

n

)

≤ hG(u

n+1

), u

n

− u

n+1

i

g(u

n+1

)

+ (c/2)

ku

n+1

− u

n

k

2

_.

(1.4.27)

If

u

n+1

6∈ ˜

M

, then by the omparability ondition we have

G(u

n+1

) =

∇F (u

n+1

) = 0

and thisestimatestillholds ifwe set

h·, ·i

g(u

n+1

)

tobetheusuals alarprodu t. Addingtheterm

0 =



u

n+1

− u

n

∆t

− θG(u

n+1

)

− (1 − θ)G(u

n

) , u

n

− u

n+1



g(u

n+1

)

totheright hand sideof (1.4.27),we get

F (u

n+1

)

− F (u

n

)

≤ −(1/∆t)ku

n+1

− u

n

k

2

g(u

n+1

)

+ (c/2)

ku

n+1

− u

n

k

2

+ (1

− θ) hG(u

n+1

)

− G(u

n

), u

n

− u

n+1

i

_g(u

_n+1

₎

.

Combining this estimatewith(1.4.14) ,we obtain

F (u

n+1

)

− F (u

n

)

≤ −µ

∆t

ku

n+1

− u

n

k

2

∆t

,

with

µ

∆t

:= c

2

1

−∆t(c/2+(1−θ)Kc

2

2

)

where

K

istheLips hitz onstantof

G

. Theparameter

µ

∆t

being larger than the positive onstant

µ

1

:= c

2

1

/2

for

∆t

small enough (1.4.24) is proved.

Corollary1.4.1. Let

θ

∈ [0, 1]

andlet

(u

n

)

bethesequen edenedbythe

θ

-s heme (1.4.23). Assumehypotheses

(H2)

,

(H3)

hold,that

G

is Lips hitz andthat

∆t

∈ (0, ∆t

′

₎

. Then:

•

If

(H0)

,

(H1)

hold,the sequen e

(u

n

)

onverges to

ϕ

.

•

If

F

satisesa Kurdyka-Šojasiewi zinequality intheneighborhoodof somelo al min-imizer

ϕ

then forevery

η > 0

there exists

ε

∈ (0, η)

su h that

ku

n

¯

− ϕk < ε

=

⇒

ku

n

− ϕk < η, ∀n ≥ ¯n.

•

If

(H0)

,

(H1

′

₎

hold, the sequen e

(u

n

)

onverges to

ϕ

with onvergen e rates given by(1.4.20) with

C

2

= µ

1

µ

2

2

∆t

.

Proof. From (1.4.24) and (1.4.25) of Lemma 1.4.7, we easily obtain (H4), (H5) and (H6)

with

C = µ

1

µ

2

and

C

2

= µ

1

µ

2