Rule of the mesh in the Finite Element Method

Rule of the mesh in the Finite Element Method

M. Duprez

V. Lleras

A. Lozinski

1CEntre de REcherche en MAthématiques de la DÉsision, Paris

2IMAG, Montpellier

3Laboratoire de Mathématiques de Besançon

Séminaire d’Analyse et Probabilités

29/10/19

Outline

1

Framework : Finite Element Method (FEM)

2

Geometrical quality of a mesh

3

Mesh which is not conforming to the boundaries

Outline

1

Finite Element Method (FEM)

2

Geometrical quality of a mesh

3

Mesh which is not conforming to the boundaries

Framework

Considered equation

−∆u=f, inΩ u= 0, on∂Ω i.e.the problem

Findu∈H₀¹(Ω)s.t. :

⇔(∇u,∇v)L²(Ω)= (f, v)L²(Ω)∀v∈H0¹(Ω).

General framework

Findu∈V s.t. : a(u, v) =l(v)∀v∈V where

• l(·)linear and continuous:|l(v)|6C1kvk ∀v∈V

• a(·,·)bilinear continuous:|a(u, v)|6C2kukkvk ∀u, v∈V andcoercive:|a(v, v)|>Ckvk²∀v∈V

1

Framework : Finite element method

LetVh=hφk∈V :k∈ {1, ..., N}i ⊂V.

System (1) will be approximate by Finduh∈Vhs.t. :

a(uh, vh) =l(vh)∀vh∈Vh

⇔

FindUh∈R^Ns.t. : AhUh=Fh

where

Ah=a(φk, φj)kj

Fh=l(φk)k

Thus

uh=X Uhkφk.

Michel Duprez Rule of the mesh in the Finite Element Method 2

Framework : Lagrange continuous finite element

Piecewise linear Lagrange FE order 1 (P¹) Lagrange interpolation

Order 1 (P1), dimension 2 Order 2 (P2), dimension 1

3

Outline

1

Framework : Finite Element Method (FEM)

2

Geometrical quality of a mesh

3

Mesh which is not conforming to the boundaries

Challenge

Questions

â Geometrical criterion on the mesh to ensure the convergence ? ku

− uk −→

h→0

0 where h := max

in mesh

diam ( Cell )

â Optimal order in the a priori estimates for P

(

|u − u

|

6 Ch|u|

,

|u − u

|

6 Ch

|u|

.

4

Framework : geometrical condition

• Minimum angle condition âdim 2 : Zlámal 68’

min{angle}> α >0

• Ciarlet condition âdim n : Ciarlet 78’

hK

ρK

< γ

hK

ρK

• Maximum angle condition

âdim 2 : Babuska-Aziz 76’, Barnhill-Gregory 76’, Jamet 76’

âdim 3 : Krizek 92’

max{angle}< β < π

• Local damage âdim 2 : Kucera 2016

âdim n : Duprez-Lleras-Lozinski 2019

Michel Duprez Rule of the mesh in the Finite Element Method 5

Idea of the proof

Findu∈V s.t. :

a(u, v) =l(v)∀v∈V (1)

Finduh∈Vhs.t. :

a(uh, vh) =l(vh)∀vh∈Vh (2) Lemma (Local interpolation)

For each cellK∈ Thandu∈H²(K)

|u− Ihu|1,K6Ch|u|K,2, whereIhis theLagrange interpolation operator.

Lemma (Céa, see Ern-Guermond’s book) Letuanduhsolution to (1) and (2). Then

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω.

Thus

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω6C|u− Ihu|1,Ω6Ch|u|Ω,2.

6

Counter-example [Babuska-Aziz 1976]

Let A

= (−1, 0), A

= (0, 1) and A

= (0, ε).

Consider u(x

, x

) = x

. It holds

|u − I

u|

> 1 ε .

→ large interpolation error

Michel Duprez Rule of the mesh in the Finite Element Method 7

Inclusion of meshes [Hannukaimen-Korotov-Krizek 2012]

N²

N

TriangulationT_h¹

N²

N

TriangulationT_h² SinceV_h¹ ⊂V_h²,

|u−u²h|1,Ω6C inf

v_h²∈V² h

|u−vh²|1,Ω6C inf

v_h¹∈V¹ h

|u−vh¹|1,Ω6C|u−Ih¹u|1,Ω6Ch|u|Ω,2.

TriangulationT_h¹ TriangulationT_h²

8

Local damage on the mesh : example

Assume the degenerated cells are isolated :

Michel Duprez Rule of the mesh in the Finite Element Method 9

Local damage on the mesh : exact assumption

Assumption

LetK₁^deg, ..., K_I^degbe the degen. cells

h_Kdeg i

/ρ_Kdeg i

> c0.

Assume that hP_i/ρP_i 6c1. hKi,j/ρKi,j 6c1. h_Knd

i /ρ_Knd i 6c1. Pei∩Pej=∅

#Pei6M

K_i^nd K_i^d

K_i,1 Ki,2

Ki,3

Ki,4

Ki,5

x₀

Pi:=K_i^d∪K_i^nd Pei:=Pi∪jKi,j

Theorem (D.-Lleras-Lozinski 2019)

Letu∈V anduh∈Vhbe the solutions to continuous and discrete Systems.

Then, under Assumption 1,

|u−uh|1,Ω≤Ch|u|2,Ω and |u−uh|0,Ω≤Ch²|u|2,Ω.

10

Sketch of proof

Modified Lagrange interpolation operator

Definition

Forv∈H²(Ω), definedeIh(v)∈Vhs.t.

Ieh=IhonK6∈P˜i

Ieh(x0) =Ext(I_h|Knd i )(x0) whereIhis the standard Lagrange interpolation operator.

K_i^nd K_i^d

K_i,1 Ki,2

Ki,3

Ki,4

Ki,5

x₀

Proposition (D.-Lleras-Lozinski 2019)

Under Assumption 1, we have for allv∈H²(Ω)∪H₀¹(Ω)

|v−Ieh(v)|1,Ω≤Ch|v|2,Ω. âUse the local interpolation estimate ofIhonPi

Michel Duprez Rule of the mesh in the Finite Element Method 11

Sketch of proof

Proposition (D.-Lleras-Lozinski 2019)

Under Assumption 1, we have for allv∈H²(Ω)∪H₀¹(Ω)

|v−Ieh(v)|1,Ω≤Ch|v|2,Ω.

Lemma (Céa, see Ern-Guermond’s book) Letuanduhsolution to (1) and (2). Then

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω.

Thus

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω6C|u−Iehu|1,Ω6Ch|u|Ω,2.

12

Poor conditioning of the system matrix

Proposition (D.-Lleras-Lozinski 2019)

Suppose that the meshThcontains a degenerate cellK^deg

ρ_Kdeg=ε, h_Kdeg>C1h.

Then theconditioning numberassociated to the bilinear formainVhsatisfies

κ(A) :=kAk2kA⁻¹k2> C hε

Idea :

φk φj

ε

1 a(φk,φj) = 1/ε

Michel Duprez Rule of the mesh in the Finite Element Method 13

An alternative scheme

We approximate

Findu∈V s.t. :

a(u, v) =l(v)∀v∈V (1) by the finite element formulation

Finduh∈Vhs.t. :

ah(uh, vh) =l(vh)∀vh∈Vh (2) where the bilinear formahis defined for alluh, vh∈Vhby

ah(uh, vh) :=a_Ωnd h

(uh, vh)+X

i

aP_i(Iehu_h,Iehv_h)

+X

i

1 h²_P

i

((Id−Ieh)uh,(Id−Ieh)vh)P_i,

withΩ^nd_h := (∪Pi)^c.

14

A priori estimates and conditioning

Theorem (A prioriestimate, D.-Lleras-Lozinski 2019)

LetΩconvex andu∈V, uh∈Vhbe the solutions to Systems (1) and (2).

Then, under Assumption 1, we have (ε= 0ifn= 3) _|u−

uh|1,Ω≤Cεh^1−ε|u|2,Ω, ku−uhk0,Ω≤Cεh^2−ε|u|2,Ω.

Proposition (Conditioning, D.-Lleras-Lozinski 2019)

Suppose that Assumption 1 holds and the union of cellsωxattached to each nodex ofThsatisfies

c1hⁿ≤ |ωx| ≤c2hⁿ. Then, the conditioning number of the matrixAsatisfies

κ(A) :=kAk2kA⁻¹k26Ch⁻².

Michel Duprez Rule of the mesh in the Finite Element Method 15

Sketch of proof

Lemma (Coercivity ofah)

ah(vh, vh)>Ckvhk²0,Ω ∀vh∈Vh.

Lemma (Continuity ofah)

ah(uh, vh)6(C/h²)kuhk0,Ωkvhk0,Ω ∀uh, vh∈Vh.

kAk2 = sup

v∈R^N

(Av,v)

|v|²₂ = sup

v∈R^N

ah(vh, vh)

|v|²₂ 6Chⁿ sup

v_h∈V_h

ah(vh, vh)

kvhk²₀ 6Chⁿ⁻² kA⁻¹k2= sup

v∈R^N

|v|²2

(Av,v) = sup

v∈R^N

|v|²2

ah(vh, vh) 6Ch⁻ⁿ sup

v_h∈V_h

kvhk²0

ah(vh, vh) 6Ch⁻ⁿ Thus

κ(A) :=kAk2kA⁻¹k26C/h².

16

Equivalent formulation

Ki^nd

K_i^deg Fi

Example of patchPi=K_i^nd∪K_i^deg.

Proposition (D.-Lleras-Lozinski 2019) For alluh, vh∈Vh, it holds

ah(uh, vh) =a_Ωnd

h (uh, vh) +P

i

|P_i|

|K^nd i |a_Knd

i (uh, vh) +κnP

i

|K^deg_i |³ h²_P

i

|F_i|²[∇uh]F_i·[∇vh]F_i

withκn:=_(n+1)(n+2)²ⁿ² .

Michel Duprez Rule of the mesh in the Finite Element Method 17

Simulation

LetΩ := (0,1)×(0,1)andf(x, y) = 2π²sin(πx) sin(πy).

Consider the system

_Findu∈H1 0(Ω)s.t. :

(∇u,∇v)_L2(Ω)= (f, v)_L2(Ω)∀v∈H₀¹(Ω).

Exact solution :u(x, y) = sin(πx) sin(πy)∀(x, y)∈Ω.

Cartesian meshesThs.t. :

h_Kdeg =handρ_Kdeg ∼h².

18

Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

10⁻² 10⁻¹

10⁻¹ 10⁰ 10¹

1 1

h

Conditioning×_h¹2

2 degenerated cells, standard scheme

Michel Duprez Rule of the mesh in the Finite Element Method 19

Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

|u−I^ehuh|1,Ω

10⁻² 10⁻¹

10⁻¹ 10⁰ 10¹

h

Conditioning×_h¹2

2 degenerated cells, alternative scheme

20

Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

10⁻³ 10⁻² 10⁻¹

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

h

ku−uhk0,Ω

|u−uh|1,Ω

|u−I^ehuh|1,Ω

5.5 % of degenerated cells standard scheme (left), alternative scheme (right)

Michel Duprez Rule of the mesh in the Finite Element Method 21

Work in progress

Application in biomechanics : generation of patient specific meshes

Collaborations A. Lozinski

âLMB, Besançon C. Lobos

â^Chile M. Bucki

âTexisence, Grenoble V. Lleras

âIMAG, Montpellier

[Bijar-Rohan-Perrier-Payan 2016]

22

Conclusion

• New sufficient geometrical conditions âLocal damages : optimal convergence

• New operator ofinterpolation

âConvergence of a operator of interpol.6=convergence to the solution

• Alternative formulationfor a good conditioning of the systemmatrix âQuasi optimal convergence

• Necessary and sufficientgeometrical conditions remainopen âInclusion of meshes ?

Michel Duprez Rule of the mesh in the Finite Element Method 23

Outline

1

Framework : Finite Element Method (FEM)

2

Geometrical quality of a mesh

3

Mesh which is not conforming to the boundaries

Framework : previous results

Fictitious domain methods : non-matching meshes

Previous results

• First work âSaul’ev 63’

• XFEM

âMoes-Bechet-Tourbier 2006 âHaslinger-Renard 2009

• CutFEM

âBurman-Hansbo 2010-2014 âLozinski 2019

Methods

• Extended finite element space

• Standard penalizations

• Nitsche penalizations

• Ghost penalty

• Lagrange multipliers

Michel Duprez Rule of the mesh in the Finite Element Method 24

Framework : previous results

Lagrange multipliers approximated byP0-FE on the cut trianglesT_h^γ:

Finduh∈Vh, λh∈Wh={µh∈L²(Ω^γ_h) :µh|T∈P0(T)∀T ∈ T_h^Γ}:

Z

Ω

∇uh· ∇vh+ Z

Γ

λhvh= Z

Ω

f vh ∀vh∈Vh

Z

Γ

µhuh=σh X

E∈T^γ h

[λh][µh] ∀µh∈Wh

âBurman-Hansbo 2010

Difficulty :

The actual formulation contain aintegral on the real boundary.

25

φ-FEM : formal approach

Initial problem

Consider a domainΩand

( _{Findus.t. :}

−∆u=f in Ω u= 0 on ∂Ω

φ-FEM (formal) approach

Assume thatΩandΓare given by alevel-setfunctionφ: Ω :={φ <0}andΓ :={φ= 0}.

Consider the problem

Findvs.t. :

−∆(φv) =f in Ω Thenu:=φvis solution to the initial problem.

Michel Duprez Rule of the mesh in the Finite Element Method 26

φ-FEM : weak formulation

Assume thatΩandΓare given by a level-set functionφ: Ω :={φ <0}andΓ :={φ= 0}.

Suppose thatφis nearΓas the signed distance toΓ.

Consider the finite element approximation Z

Ω_h

∇(φvh)· ∇(φwh)− Z

∂Ω_h

∂

∂n(φvh)φwh+ Stab.

Term = Z

Ω_h

f φwh ∀wh∈Wh,

Thenuh:=vhφas approximation of the initial problem.

27

φ-FEM : a priori error estimates

Consider the finite element approximation :ah(vh, wh) =lh(wh) ∀wh∈Wh, where











ah(vh, wh) = Z

Ω_h

∇(φhvh)· ∇(φhwh)− Z

∂Ω_h

∂

∂n(φhvh)φhvh+G¹h(˜uh, vh) lh(wh) =

Z

Ω_h

f φhwh+G²h(wh),

withφh=Ih(φh),G¹_handG²_hstands for the theghost penaltygiven by



















G¹_h(vh, wh)=σh X

E∈F_Γ

Z

E

h _∂

∂n(φhvh) i h _∂

∂n(φhwh) i

+σh²P

T∈T^Γ h

R

T∆(φhvh)∆(φhwh) G²_h(wh)=−σh² X

T∈T^Γ h

Z

T

f∆(φhwh)

and

T_h^Γ={T ∈ Th:T∩Γh6=∅} (Γh={φh= 0});

FΓ={E(an internal edge ofTh)such that∃T ∈ Th:T∩Γh6=∅andE∈∂T}.

Michel Duprez Rule of the mesh in the Finite Element Method 28

φ-FEM : a priori error estimate

Continuous problem a(u, w) =l(w)∀w∈V

Finite element formulation

ah(vh, wh) =lh(wh)∀wh∈V_h^(k)⊂V

Theorem (D.-Lozinski 2019)

Suppose that the meshThis uniform (with some weak assumptions), and

f∈H^k(Ωh∪Ω). Letu∈H^k+2(Ω)be the continuous solution andwh∈V_h^(k)be the discret solution. Denotinguh:=φhwh, it holds

|u−uh|1,Ω∩Ω_h≤Ch^kkfkk,Ω∪Ω_h

withC=C(φ). Moreover, supposingΩ⊂Ωh

ku−uhk0,Ω≤Ch^k+1/2kfkk,Ω_h.

29

φ-FEM : key lemmas

Lemma (Hardy inequality, D.-Lozinski 2019)

We assume that the domainΩis given by the level-setφregular enough. Then, for anyu∈H^k+1(O)vanishing onΓ,

u φ k,O

≤Ckukk+1,O.

Lemma (Local interpolation, Ern-Guermond’s book) Fork∈N^∗, each cellK∈ Thandu∈H²(K)

|u− Ihu|1,K6Ch^k|u|K,k+1, whereIhis the Lagrange interpolation operator.

Michel Duprez Rule of the mesh in the Finite Element Method 30

φ-FEM : conditioning

Theorem (Conditioning)

Assume thatThis uniform (and satisfies the assumptions). Then the condition numberκ(A) :=kAk2kA⁻¹k2of the matrixAassociated to the bilinear formah

onV_h^(k)satisfies

κ(A)≤Ch⁻².

Here,k · k2stands for the matrix norm associated to the vector 2-norm| · |2.

31

φ-FEM : Simulations

LetΩbe the circle of radius√

2/4centered at the point(0.5,0.5)and the surrounding domainO= (0,1)². The level-set functionφgiving this domainΩis taken as

φ(x, y) = (x−1/2)²−(y−1/2)²−1/8.

We useφ-FEM to solve numerically Poisson-Dirichlet problem with the exact solution given by

u(x, y) =φ(x, y)×exp(x)×sin(2πy).

φ <0 Ω

0 1

1

Michel Duprez Rule of the mesh in the Finite Element Method 32

φ-FEM : Simulations

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

ku−uhk0,Ωh/kuk0,Ωh

|u−u_h|1,Ωh/|u|1,Ωh

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

1 1

1 2

h

ku−uhk0,Ωh/kuk0,Ωh

|u−u_h|1,Ωh/|u|1,Ωh

FIGURE–Relative errors ofφ-FEM fork= 1. Left :φ-FEM with ghost penaltyσ= 20; Right :φ-FEM without ghost penalty (σ= 0).

33

φ-FEM : Simulations

10⁻² 10⁻¹

10¹ 10² 10³ 10⁴ 10⁵

1 2

h

Conditioning

10⁻² 10⁻¹

10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

1 4

h

Conditioning

FIGURE–Condition numbers forφ-FEMk= 1. Left :φ-FEM with ghost penaltyσ= 20; Right :φ-FEM without ghost penalty (σ= 0).

Michel Duprez Rule of the mesh in the Finite Element Method 34

φ-FEM : Simulations

10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹ 10⁰ 10¹ 10² 10³ 10⁻⁵

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

σ

h= 1.41×10⁻² h= 7.07×10⁻³ h= 3.54×10⁻³ h= 1.77×10⁻³

10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹ 10⁰ 10¹ 10² 10³ 10⁻²

10⁻¹ 10⁰ 10¹

σ

h= 1.41×10⁻² h= 7.07×10⁻³ h= 3.54×10⁻³ h= 1.77×10⁻³

FIGURE–Influence of the ghost penalty parameterσon the relative errors forφ-FEMk= 1. Left :ku−uhk0,Ω_h/kuk0,Ω_h; Right :

|u−uh|1,Ω_h/|u|1,Ω_h.

35

φ-FEM : Simulations

10⁻³ 10⁻² 10⁻¹

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

1 2

1 3

h

ku−uhk0,Ωh/kuk0,Ωh

|u−uh|1,Ωh/|u|1,Ωh

10⁻² 10⁻¹

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

1 3

1 4

h

ku−uhk0,Ωh/kuk0,Ωh

|u−uh|1,Ωh/|u|1,Ωh

FIGURE–Relative errors ofφ-FEM. Left :k= 2; Right :k= 3.

Michel Duprez Rule of the mesh in the Finite Element Method 36

φ-FEM : Simulations

10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹ 10⁰ 10¹ 10² 10³ 10⁻⁸

10⁻⁶ 10⁻⁴ 10⁻² 10⁰ 10²

σ

h= 1.41×10⁻² h= 7.07×10⁻³ h= 3.54×10⁻³ h= 1.77×10⁻³

10⁻⁵10⁻⁴10⁻³10⁻²10⁻¹ 10⁰ 10¹ 10² 10³ 10⁻⁶

10⁻⁴ 10⁻² 10⁰ 10² 10⁴

σ

h= 1.41×10⁻² h= 7.07×10⁻³ h= 3.54×10⁻³ h= 1.77×10⁻³

FIGURE–Influence of the ghost penalty parameterσon the relative errors forφ-FEM andk= 2. Left :ku−uhk0,Ω/kuk0,Ω; Right :

|u−uh|1,Ω/|u|1,Ω.

37

φ-FEM : Simulations

We now choose domainΩgiven by the level-set

φ(x, y) =−(y−πx−π)×(y+x/pi−π)×(y−πx+π)×(y+x/pi+π).

It is thus the rectangle with corners 2π²

π²+1,^π_π³₂^−π₊₁

,(0, π),

−_π^2π2+1² ,−^π_π³2^−π+1

, (0,−π). We useφ-FEM to solve numerically Poisson-Dirichlet problem inΩwith the right-hand side given by

f(x, y) = 1.

Ω

0 4

4

Michel Duprez Rule of the mesh in the Finite Element Method 38

φ-FEM : Simulations

10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

kuref−uhk0,Ω/kurefk0,Ω

|uref−uh|1,Ω/|uref|1,Ω

10⁻² 10⁻¹

10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

1 2

1 3

h

kuref−uhk0,Ω/kurefk0,Ω

|uref−uh|1,Ω/|uref|1,Ω

FIGURE–Relative errors ofφ-FEM. Left :k= 2; Right :k= 3. The reference solutionu_ref is computed by a standardF EM on a sufficiently fine fitted mesh onΩ.

39

Conclusion

Results

Optimal convergence of φ-FEM in the H

semi-norm Quasi-optimal convergence of φ-FEM in the L

norm

â

Discrete problem well conditioned

Perspectives

Neumann or Robin boundary conditions â

Dynamic equation : heat equation

Michel Duprez Rule of the mesh in the Finite Element Method 40

Thanks for your attention !

φ-FEM : mesh assumptions

Assumption

The approximate boundaryΓhcan be covered by element patches{Πi}i=1,...,N_Π

having the following properties :

Each patchΠiis a connected set composed of a mesh elementTi∈ Th\ T_h^Γ and some mesh elements cut byΓh. More precisely,Πi=Ti∪Π^Γ_i with Π^Γi ⊂ T_h^Γcontaining at mostMmesh elements ;

T_h^Γ=∪^N_i=1^ΠΠ^Γ_i ;

ΠiandΠjare disjoint ifi6=j.

Michel Duprez Rule of the mesh in the Finite Element Method 41

φ-FEM : mesh assumptions

Assumption

The boundaryΓcan be covered by open setsOi,i= 1, . . . , Iand one can introduce on everyOilocal coordinatesξ1, . . . , ξdwithξd=φsuch that all the partial derivatives∂^αξ/∂x^αand∂^αx/∂ξ^αup to orderk+ 1are bounded by someC0>0.

Morover,|φ| ≥monO \ ∪i=1,...,IOiwith somem >0.

42