Variational integrators of fractional Lagrangian systems in the framework of discrete embeddings

0.11 MB·

by  Loïc Bourdin

MonografíasMatemáticasGarcíadeGaldeanovv,1–10(yyyy)
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VARIATIONAL INTEGRATORS OF
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J FRACTIONAL LAGRANGIAN SYSTEMS IN

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THE FRAMEWORK OF DISCRETE

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A EMBEDDINGS
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. Loïc Bourdin
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a Abstract.Thispaperisasummaryofthetheoryofdiscreteembeddingsintroducedin[5].
m
Adiscreteembeddingisanalgebraicprocedureassociatinganumericalschemetoagiven
[ ordinarydifferentialequation. Lagrangiansystemspossessavariationalstructurecalled

Lagrangianstructure. Wearespeciallyinterestedintheconservationatthediscretelevel
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ofthisLagrangianstructurebydiscreteembeddings. Wethenreplaceinthisframework
v
thevariationalintegratorsdevelopedin[10,ChapterVI.6]andin[12].Finally,weextend
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8 the notion of discrete embeddings and variational integrators to fractional Lagrangian
8 systems.
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Keywords:Lagrangiansystems,Variationalintegrator,Fractionalcalculus.
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AMSclassification:70H03,37K05,26A33.
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6 Introduction
1
: ThetheoreticalframeworkofembeddingsofdynamicalsystemsisinitiatedbyCressonand
v
i Darsesin[7]. Areviewofthesubjectisgivenin[6]. Anembeddingofanordinaryorpartial
X
differentialequationis a way to give a sense to this equation overa largerset of solutions.
r As an example, the stochastic embedding developed in [7] allows to give a meaning of a
a
differentialequationoverthesetofstochasticprocesses.
WearespeciallyinterestedinLagrangiansystemscoveringalargesetofdynamicalbe-
haviors and widely used in classical mechanics, [2]. These systems possess a variational
structurecalledLagrangianstructure,i.e. theirsolutionscorrespondtocriticalpointsofLa-
grangian functionals, [2, p.57]. The Lagrangian structure is intrinsic and induces strong
constraintsonthequalitativebehaviorofthesolutions. Theconservationofthisstructureby
embeddingseemsthenimportant. In[7],theauthorsconstructstochasticembeddingswhich
preserve the variational structure of Lagrangian systems, i.e. the generalized solutions are
alsocharacterizedascriticalpointsofgeneralizedLagrangianfunctionals.
Thispaper is a summaryof the theoryof discrete embeddingsintroducedin [5] where,
as in [7], we are interested in the conservation of the Lagrangian structure of Lagrangian
systems. Wethenreferto[5]formoredetailsandfortheproofofsomeresults.
Adiscreteembeddingisanalgebraicprocedureassociatinganumericalschemetoagiven
differentialequation,inparticulartoagivenLagrangiansystem. Ontheotherhand,defining
adiscreteembeddinginducesadiscretizationoftheLagrangianfunctionalassociatedandwe
2 LoïcBourdin
can developa discrete calculus of variations on this one: this leads to a numericalscheme
called variational integrator. The variational integrators, developed in [10, Chapter VI.6]
and [12], are then numerical schemes for Lagrangian systems preserving their variational
structures.
Thus, we proposethe followingdefinition: a discrete embeddingis said to be coherent
if the two discrete versionsobtained(the directone and the variationalintegrator)of a La-
grangian system coincide. Hence, a coherentdiscrete embeddingconservesat the discrete
leveltheLagrangianstructureofaLagrangiansystem.
Recently, many studies have been devoted to fractional Lagrangian systems, [1], [7].
They arise for examplein fractionaloptimalcontroltheory, [9]. They are difficultto solve
explicitly,itistheninterestingtodevelopefficientnumericalschemestosuchsystems.
Somepreliminaryresultsonfractionaldiscreteoperatorsandonthediscretizationoffrac-
tionalEuler-Lagrangeequationshavebeendiscussedbyseveralauthors,[3],[4],[8]. Inthis
paper,weextendthediscreteembeddingpointofview,thecorrespondingproblemofcoher-
enceandtheassociatednotionofvariationalintegratortothefractionalcase.
The paper is organized as follows. In Section 1, we define the notion of discrete em-
beddings of differential equations. Section 2 recalls definitions and results concerningLa-
grangian systems and we apply the previous theory of discrete embeddings to Lagrangian
systems. Then, we recall the strategy of variational integrators of Lagrangian systems in
the framework of discrete embeddingsand we finally present the problem of coherence of
a discrete embedding. Section 3 is devoted to the extension of discrete embeddingsto the
fractionalcase.
§1.Notionofdiscrete embeddings
Inthispaper,weconsiderclassicalandfractionaldifferentialsystemsinRd whered ∈N∗ is
the dimension. Thetrajectoriesofthesesystemsarecurvesqin C0([a,b],Rd)wherea < b
aretworeals. Forsmoothenoughfunctionsq,wedenoteq˙ =dq/dtandq¨ =d2q/dt2.
1.1. Discreteembeddings
Definition1. Definingadiscreteembeddingmeansgivingadiscreteversionofthefollowing
elements: the curves q ∈ C0([a,b],Rd), the derivative operator d/dt and the functionals
a:C0([a,b],Rd)−→R. Moreprecisely,itmeansgiving:
• anapplicationq7−→qhwhereqh ∈(Rd)m1,
• adiscreteoperator∆:(Rd)m1 −→(Rd)m2 discretizingthedifferentialoperatord/dt,
• anapplicationa7−→ahwhereah :(Rd)m1 −→R,
wherem ,m ∈N∗.
1 2
InordertoillustrateDefinition1,wedefinebackwardandforwardfinitedifferencesembed-
dings. Foralltherestofthepaper,wefixσ = ±andN ∈ N∗. Wedenotebyh = (b−a)/N
VariationalintegratorsoffractionalLagrangiansystemsintheframeworkofdiscreteembeddings 3
thestepsizeofthediscretizationandτ=(tk)k=0,...,N thefollowingpartitionof[a,b]:
∀k=0,...,N, t =a+kh.
k
Definition2(caseσ=−). WecallbackwardfinitedifferencesembeddingdenotedbyFDE−
thedefinitionofthefollowingelements:theapplication
disc: C0([a,b],Rd) −→ Rd N+1 ,
q 7−→ ((cid:16)q(tk(cid:17)))k=0,...,N
andthediscreteoperator
∆ : Rd N+1 −→ Rd N
−
Q=(Qk(cid:16))k=0(cid:17),...,N 7−→ (cid:16)Qk−(cid:17)hQk−1 k=1,..,N.
(cid:16) (cid:17)
Definition3(caseσ = +). WecallforwardfinitedifferencesembeddingdenotedbyFDE+
thedefinitionofthefollowingelements:theapplicationdiscandthediscreteoperator
∆+ : Rd N+1 −→ Rd N
Q=(Qk(cid:16))k=0(cid:17),...,N 7−→ (cid:16)Qk−(cid:17)hQk+1 k=0,..,N−1.
(cid:16) (cid:17)
Let us notice that the discrete analogous of d/dt in FDEσ is then −σ∆ . We use these
σ
notationsinordertobeuniformwiththefractionalnotations(seeSection3).
1.2. Directdiscrete embeddings
Definingadiscreteembeddingallowsustodefineadirectdiscreteversionofagivendiffer-
entialequation:
Definition4. LetbefixedadiscreteembeddingasdefinedinDefinition1andlet(E)bean
ordinarydifferentialequationofunknownq∈C0([a,b],Rd)givenby:
O(q)=0, (E)
whereOisadifferentialoperatorshapedasO= f(·)(d/dt)i◦g(·)where f,g arefunctions.
i i i i i
Then,thedirectdiscreteembeddingof (E)is(EPh)thesystemofequationsofunknownqh ∈
(Rd)m1 givenby:
Oh(qh)=0, (E )
h
whereOhisthediscretizedoperatorofOgivenbyOh = f(·)∆i◦g(·).
i i i
Asanexample,weconsidertheNewton’sequationwithfrPictionofunknownq∈C0([a,b],Rd)
givenby:
∀t∈[a,b], q¨(t)+q˙(t)+q(t)=0. (NE)
Then, thedirectdiscreteembeddingof (NE) withrespectto FDE−is(NE )the systemof
h
equationsofunknownQ∈(Rd)N+1givenby:
Q −2Q +Q Q −Q
∀k =2,...,N, k k−1 k−2 + k k−1 +Q =0. (NE )
h2 h k h
4 LoïcBourdin
Thedirectdiscreteembeddingofanordinarydifferentialequationisstronglydependenton
theformofthedifferentialoperatorO(andnotonitsequivalenceclass). TheprocessO −→
Oh is not an application. For example, the discretized operator Oh of O = d/dt◦ sin(·) =
d/dt(·)cos(·)isdifferentdependingonthewritingofO.
1.3. Directdiscrete embeddings ofLagrangiansystems
We recallnowclassicaldefinitionsandtheoremsconcerningLagrangiansystems. We refer
to[2]foradetailedstudyandforadetailedproofofTheorem1.
Definition5. ALagrangianfunctionalisanapplicationdefinedby:
L: C2([a,b],Rd) −→ R
b
q 7−→ L(q(t),q˙(t),t)dt
Z
a
whereLisaLagrangiani.e. aC2applicationdefinedby:
L: Rd×Rd×[a,b] −→ R
(x,v,t) 7−→ L(x,v,t).
An extremal (or critical point) of a Lagrangian functional L is a trajectory q such that
DL(q)(w) = 0 for any variations w (i.e. w ∈ C2([a,b],Rd), w(a) = w(b) = 0), where
DL(q)(w) is the differential of L in q along the direction w. Extremals of a Lagrangian
functionalcanbecharacterizedassolutionofadifferentialequationoforder2:
Theorem 1 (Variational principle). Let L be a Lagrangian functionalassociated to the La-
grangianLandletq ∈ C2([a,b],Rd). Then,qisanextremalofLifandonlyifqissolution
oftheEuler-Lagrangeequationgivenby:
∂L d ∂L
∀t∈]a,b[, (q(t),q˙(t),t)− (q(t),q˙(t),t) =0. (EL)
∂x dt ∂v !
WenowapplydefinitionsofSection1onLagrangiansystems.
Proposition1. LetLbeaLagrangianandlet(EL)beitsassociatedEuler-Lagrangeequation.
Thedirectdiscreteembeddingof(EL)withrespecttoFDEσisgivenby:
∂L ∂L
(Q,−σ∆ Q,τ)+σ∆ (Q,−σ∆ Q,τ) =0, Q∈(Rd)N+1. (1)
σ σ σ
∂x ∂v !
Wereferto[5]foraconcreteexampleillustratingTheorem1andProposition1.
§2. Discreteembeddings andvariationalintegrators ofLagrangian
systems
A directdiscrete embeddingis only based on the formof the differentialoperatorwhich is
dependentofthecoordinatessystemandconsequentlyisnotintrinsic. Then,anaturalques-
tion arises: whatcan besaid aboutthe conservationofintrinsic propertiesofadifferential
VariationalintegratorsoffractionalLagrangiansystemsintheframeworkofdiscreteembeddings 5
equation by a discrete embedding? This paper is devoted to the conservation by discrete
embeddings of the Lagrangian structure of Lagrangian systems. More precisely, Theorem
1 shows that (EL) possesses a variational structure: the direct discrete embedding being a
procedure mainly algebraic, does (1) possess a variationalstructure too? It is not always
true.
However,a variationalintegrator,developedin [10, ChapterVI.6] andin [12], is a dis-
cretization of a Lagrangian system preserving its variational structure. Indeed, it is based
onthediscreteanalogousofthevariationalprincipleonadiscreteversionoftheassociated
Lagrangianfunctional.
Inourframework,thediscretizationoftheLagrangianfunctionalisinducedbygivinga
discreteembedding.
2.1. DiscreteLagrangian functionals anddiscrete calculus ofvariations
In this subsection, as an example, we are going to work exclusively in the framework of
FDEσ. Giving FDEσ induces the discretization of a Lagrangian functional as long as
a quadrature formula is fixed in order to approximate integrals. We choose the usual σ-
b
quadratureformulaof Gauss: for a continuousfunction f on [a,b],we discretize f(t)dt
a
byh k∈Iσ f(tk)whereI+ ={0,...,N−1}andI− ={1,...,N}. R
P
ThisprocessdefinestheGaussfinitedifferencesembeddingdenotedbyGauss-FDEσ. Such
achoiceallowstokeepatthediscretelevelthefollowingfundamentalresult:
b
q˙(t)dt=q(b)−q(a) Gauss-FDEσ // h (−σ∆ Q) = Q −Q .
σ k N 0
Z
a Xk∈Iσ
Proposition2. LetLbeaLagrangianfunctionalassociatedtoaLagrangianL. Thediscrete
LagrangianfunctionalassociatedtoLwithrespecttoGauss-FDEσisgivenby:
Lσ : Rd N+1 −→ R
h
Q=(Qk(cid:16))k=0(cid:17),...,N 7−→ h L(Qk,(−σ∆σQ)k,tk).
Xk∈Iσ
Once the discrete version of the Lagrangian functional is formulated, we can develop a
discrete calculus of variations on it: this leads to a variational integrator. Let L be a La-
grangian functional and Lσ the discrete Lagrangian functional associated with respect to
h
Gauss-FDEσ. A discrete extremal (or discrete critical point) of Lσ is an element Q in
h
(Rd)N+1 such that DLσ(Q)(W) = 0 for any discrete variations W (i.e. W ∈ (Rd)N+1,
h
W = W = 0). Discrete extremals of Lσ can be characterized as solution of a system
0 N h
ofequations:
Theorem 2 (Discrete variational principle). Let Lσ be the discrete Lagrangian functional
h
associatedtotheLagrangianLwithrespecttoGauss-FDEσ.Then,Qin(Rd)N+1isadiscrete
extremal of Lσ if and only if Q is solution of the following system of equations (called
h
discreteEuler-Lagrangeequation)givenby:
∂L ∂L
(Q,−σ∆ Q,τ)−σ∆ (Q,−σ∆ Q,τ) =0, Q∈(Rd)N+1. (ELσ)
∂x σ −σ ∂v σ ! h
6 LoïcBourdin
(ELσ)isobtainedfrom(EL)byvariationalintegrator. Itsvariationaloriginallowsustosay
h
thatitpossessesaLagrangianstructure. Then,wehaveconservationatthediscretelevelof
theLagrangianstructurebyvariationalintegrator.
Let us note that an asymmetry appears in (ELσ): indeed, we have a composition between
h
thetwodiscreteoperators∆+ and∆−. Wenoticethatthisasymmetrydoesnotappearinthe
continuousspacein(EL).
2.2. Problem ofcoherence ofa discrete embedding
Hence, defining a discrete embedding leads to two discrete versions of an Euler-Lagrange
equation: thefirstone(1) obtainedbydirectdiscreteembeddingandthesecondone(ELσ)
h
correspondingtoavariationalintegrator.Thedirectdiscreteembeddingisanalgebraicproce-
dure(respectingforexamplethelawofsemi-groupofthedifferentialoperatord/dt). Onthe
contrary,avariationalintegratorismainlybasedonadynamicalapproachviatheextremals
ofafunctional.
However, we are interested in the conservation at the discrete level of the Lagrangian
structure of Lagrangiansystems. We then proposethe followingdefinition: a discrete em-
beddingis said to be coherentif the two numericalschemescoincide. Precisely, a discrete
embeddingiscoherentifitmakesthefollowingdiagramcommutative:
Lagrangianfunctional Functionaldiscretization //DiscreteLag.functional
Variationalintegrator
Variationalprinciple Discr.var.principle
(cid:15)(cid:15) (cid:30)(cid:30) (cid:15)(cid:15)
//
Euler-Lagrangeequation Numericalscheme
Directdiscreteembdedding
Thus,acoherentdiscreteembeddingprovidesadirectdiscreteversionofaLagrangiansys-
tempreservingitsLagrangianstructure.
The previous study leads to a default of coherence of Gauss-FDEσ. Indeed, algorithms
obtainedbydirectdiscreteembedding(1)andobtainedbydiscretevariationalprinciple(ELσ)
h
donotcoincide. Theproblemistounderstandwhythereisnotasymmetryappearinginthe
directdiscreteembedding?Itseemsthatwemissdynamicalinformationsintheformulation
ofLagrangiansystemsatthecontinuouslevelwhicharepointedupinthediscretespacewith
theasymmetricdiscreteoperators(−σ∆σ)σ=±. Nevertheless,thisdefaultofcoherencecanbe
correctedusingadifferentwritingoftheinitialEuler-Lagrangeequation.
2.3. Rewriting oftheEuler-Lagrange equation and discrete embeddings
TheusualwaytoderivedifferentialequationsinPhysicsistobuiltacontinuousmodelusing
discretedata. However,thisprocessgivesonlyaninformationinonedirectionoftime. As
VariationalintegratorsoffractionalLagrangiansystemsintheframeworkofdiscreteembeddings 7
aconsequence,adiscreteevaluationofthevelocitycorrespondsingeneralatthecontinuous
level to the evaluation of the right or left derivative. In general, we replace the right (or
left) derivative by the classical derivative d/dt. However, this procedure assumes that the
underlying solution is differentiable. This assumption is not only related to the regularity
ofthesolutionsbutalsotothereversibilityofthesystems(therightandleftderivativesare
equal). In this section, we introduce asymmetric Lagrangian systems which are obtained
with functionalsdependingonly on left or only on rightderivatives. We provein this case
thatGauss-FDEσiscoherent.
Definition6. For f :[a,b]−→Rd smoothenoughfunction,wedenote:
f(t)− f(t−h)
∀t∈]a,b], d f(t)= lim
−
h→0+ h
and
f(t)− f(t+h)
∀t∈[a,b[, d+f(t)= lim .
h→0+ h
Although we have d−f = −d+f = f˙for a differentiable function f, it is interesting to use
thesenotationsinordertokeepdynamicalinformations.
Definition7. AnasymmetricLagrangianfunctionalisanapplication:
Lσ : C2([a,b],Rd) −→ R
b
q 7−→ L(q(t),−σd q(t),t)dt
σ
Z
a
whereLisaLagrangian.
Then,bycalculusofvariations,weobtainthefollowingcharacterizationoftheextremalsof
anasymmetricLagrangianfunctional:
Theorem 3(Variationalprinciple). LetLσ be anasymmetricLagrangianfunctionalassoci-
atedtotheLagrangianLandletq ∈ C2([a,b],Rd). Then,qisanextremalofLσ ifandonly
ifqissolutionoftheasymmetricEuler-Lagrangeequation:
∂L ∂L
∀t∈]a,b[, (q(t),−σd q(t),t)−σd (q(t),−σd q(t),t) =0. (ELσ)
σ −σ σ
∂x ∂v !
Hence, (ELσ)possessesa variationalstructure. Isitconservedbydiscrete embeddings? In
ordertoembed(ELσ),wehavetodiscretizetwodifferentialoperatorsatthesametime. We
thendefinethefollowingasymmetricversionofGauss-FDEσ:
Definition8. WecalltheasymmetricversionofGauss-FDEσthedefinitionofthefollowing
elements: theapplicationdisc,theσ-quadratureformulaofGaussandthediscreteoperators
∆− and∆+ discretizingrespectivelytheoperatorsd− andd+.
Proposition3. TheasymmetricversionofGauss-FDEσisa coherentdiscreteembedding.
Indeed, the direct discrete embeddingand the variationalintegrator of (ELσ) in the frame-
workoftheasymmetricGauss-FDEσleadtothesamenumericalscheme:(ELσ).
h
Wenoticethattherewriting(ELσ)of(EL)providesadditionaldynamicalinformationswhich
allowstheasymmetricGauss-FDEσtounifythealgebraicandthedynamicalapproachesin
the discretization of a Lagrangiansystem. Moreover, this rewriting can be justified by the
fractionalcalculusaswewillseeinSection3.
8 LoïcBourdin
§3.Discrete embeddings and variationalintegrators offractional
Lagrangiansystems
3.1. Fractionalderivatives and fractional Lagrangiansystems
Fractionalcalculusis the generalizationof the derivativenotionto realorders. We refer to
[11], [14] for many different ways generalizing this notion. For the whole paper, we fix
0 < α < 1andforanyr ∈ N∗,wedenotebyα = (−α)(1−α)...(r−1−α)/r!andα = 1.
r 0
WearegoingtousetheclassicalnotionsofGrünwald-Letnikov. Thefollowingdefinitionis
extractedfrom[13].
Definition 9. Let f be an elementof C1([a,b],Rd). The Grünwald-Letnikovfractionalleft
derivativeoforderαwithinferiorlimitaof f is:
n
1
∀t∈]a,b], Dαf(t)= lim α f(t−rh)
− nhh=→t−0ahα Xr=0 r
andtheGrünwald-Letnikovfractionalrightderivativeoforderαwithsuperiorlimitbof f is:
n
1
∀t∈[a,b[, Dαf(t)= lim α f(t+rh).
+ nhh=→b0−thα Xr=0 r
Recently, an important activity has been devoted to fractional Lagrangian systems for the
purpose of optimal control, mechanics, engineering and Physics, [1], [3], [9]. We recall
definitions and results concerning these fractional systems, we refer to [1] for a detailed
studyandforadetailedproofofTheorem4.
Definition10. AfractionalLagrangianfunctionaloforderαisanapplicationdefinedby:
Lσ,α : C2([a,b],Rd) −→ R
b
q 7−→ L(q(t),−σDαq(t),t)dt
σ
Z
a
whereLisaLagrangian.
WecangiveacharacterizationofextremalsofafractionalLagrangianfunctionalassolutions
ofafractionaldifferentialequation:
Theorem 4 (Variationalprinciple). Let Lσ,α be a fractionalLagrangian functionalof order
α associated to the Lagrangian L and let q be an element of C2([a,b],Rd). Then, q is an
extremalofLσ,α ifandonlyifqissolutionofthefractionalEuler-Lagrangeequation:
∂L ∂L
∀t∈]a,b[, (q(t),−σDαq(t),t)−σDα (q(t),−σDαq(t),t) =0. (ELσ,α)
∂x σ −σ ∂v σ !
Wereferto[1]foradetailedproof.Hence,inthefractionalcase,wefindanasymmetryagain
makingalinkwiththeasymmetricrewritingof(EL)into(ELσ).
As in the classicalcase, we concludethat(ELσ,α) possesses a Lagrangianstructureand we
areiterestedbyitsconservationatthediscretelevelbydiscreteembeddings.
VariationalintegratorsoffractionalLagrangiansystemsintheframeworkofdiscreteembeddings 9
3.2. Discreteembeddings offractional Lagrangian systems
Thereexistmanystudiesconcerningthediscretizationoffractionaldifferentialequationsbut
withoutthe pointof view of discrete embeddings. We referto [3], [4]. By referringto the
notionofGrünwald-Letnikov[8],wegivethefollowingdefinition:
Definition 11. The Gauss Grünwald-Letnikov embedding denoted by Gauss-GLEσ is the
definitionofthefollowingelements:theapplicationdisc,theσ-quadratureformulaofGauss
andthediscreteoperators
∆α : (Rd)N+1 −→ (Rd)N
−
k
1
Q=(Qk)k=0,...,N 7−→ hα αrQk−r
 Xr=0 k=1,..,N
and
∆α : (Rd)N+1 −→ (Rd)N
+
N−k
1
Q=(Qk)k=0,...,N 7−→ hα αrQk+r .
ThesediscreteoperatorsarerespectivelythediscretevXre=r0sionsofDk=α0,.a.,nNd−1Dα.
− +
We are first interested in the variational integrator of (ELσ,α) in the framework of Gauss-
GLEσ. GivingGauss-GLEσallowsustoformulatethediscreteversionofafractionalLa-
grangianfunctional:
Proposition4. LetLσ,αbethefractionalLagrangianfunctionalassociatedtotheLagrangian
L. The discrete fractionalLagrangianfunctionalassociated to Lσ,α with respect toGauss-
GLEσisgivenby:
Lσ,α : Rd N+1 −→ R
h
Q=(Qk(cid:16))k=0(cid:17),...,N 7−→ h L(Qk,(−σ∆ασQ)k,tk).
Xk∈Iσ
Then,discreteextremalsofthediscretefractionalLagrangianfunctionalcanbecharacterized
assolutionsofasystemofequations:
Theorem5(Discretevariationalprinciple). LetLσ,αbeadiscretefractionalLagrangianfunc-
h
tionalassociated to the Lagrangian L with respecttoGauss-GLEσ. Then, Q in (Rd)N+1 is
adiscreteextremalofLσ,α ifandonlyifQissolutionofthefollowingsystemofequations,
h
calledthediscretefractionalEuler-Lagrangeequation:
∂L ∂L
(Q,−σ∆αQ,τ)−σ∆α ( (Q,−σ∆αQ,τ))=0, Q∈(Rd)N+1. (ELσ,α)
∂x σ −σ ∂v σ h
Weconcludewiththefollowingproposition:
Proposition 5. Gauss-GLEσ is a coherentdiscrete embedding. Indeed, the direct discrete
embeddingandthevariationalintegratorof (ELσ,α)intheframeworkofGauss-GLEσlead
tothesamenumericalscheme:(ELσ,α).
h
10 LoïcBourdin
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LMA,UniversityofPau-PostaladdressIPRABP1155PauCedex(France)
[email protected]