Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
Etude de l’ordre de convergence de laméthode des volumes finis
Frédéric PASCAL
CMLA, ENS de Cachan
Joint work with
Daniel BOUCHE and Jean-Michel GHIDAGLIA
CMLA, ENS de Cachan
Groupe de travail "Méthodes numériques" LJLL, 08/10/07
Etude de l’ordre de convergence de la méthode des volumes finis – p. 1/27
Supraconvergence
Definition :Supraconvergence is observed when the global error of a Finite Differences scheme appliedto an ODE or PDE has a better behavior than that indicated by the local error.
Examples :
Order 2 scheme with order 1 local error
Order 1 (therefore convergent) scheme with a non-zero convergent local error.
Consequences :Lax Theorem is useless
Applications :This loss of accuracy real for the local error and apparent for the global error is observedwith non uniform grids
Etude de l’ordre de convergence de la méthode des volumes finis – p. 2/27
Lax TheoremFinite differences scheme : (h −→ 0 : parameter of discretization, Lh assumed linear)
Lh(uh) = F
Stability : (c cst ind. on h) Truncation error : (u smooth exact sol.)
‖L−1h ‖ ≤ c Lh(u) = ǫh + F
Global error :
‖eh‖ := ‖uh − u‖ = ‖L−1h (Lh(eh))‖ ≤ c‖Lh(eh)‖ = c‖ǫh‖
⇓Lax Theorem :
If the scheme is consistant (i.e. if ‖ǫh‖ h→0−→ 0) then uhh→0−→ u with a rate at least equal to the
local error rate.
Etude de l’ordre de convergence de la méthode des volumes finis – p. 3/27
Diffusion in dimension 1
−u′′ = f with Dirichlet B.C.xxx xx
∆∆
i−1 i i+1
i i+1
0 N
FD Scheme : − 2ui−1
∆i(∆i + ∆i+1)+
2ui
∆i∆i+1− 2ui+1
∆i+1(∆i + ∆i+1)= f(xi) et C.B.
Local Error : ǫih =∆i − ∆i+1
3u′′′(xi) + O(h2) = O(h) for non uniform grid
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
u
Solution Exacte (1.−x).*(atan(10.*(x−0.5))+atan(10*0.5))
10−4
10−3
10−2
10−1
100
101
102
103
104
h
Max
h /
Min
hMax h / Min h
10−4
10−3
10−2
10−1
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
h
Err
Inf
5≤ max h / Min h ≤ 4191
Erreur Inf2.198 log(h) + 0.819
Error in O(h2)
Keller, 78 : applied a FD scheme to a 1st order equiv. system =⇒ O(h2)
Manteuffel et White, 86 : discretization of the 1st order system =⇒ 2nd order eqwith Lh = D0D1, then ǫh = D1γ + ǫ =⇒ O(h2)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 4/27
Mathematical Analysis
Supraconvergence :ǫh = O(hp−1) and numerically ‖eh‖ = O(hp)
Wendroff and White, Comp. Math. Appl., 89 :
=⇒ correction of the error for the mathematical analysis
if ǫh = O(hp−1)
but ǫh = Lh(γ) + ǫ
with γ = O(hp) and ǫ = O(hp)
then Lh(eh + γ) = −ǫ=⇒ ‖eh + γ‖ = O(hp)
=⇒ ‖eh‖ = O(hp)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 5/27
Equation of Transport in dimension 1 (1/2)
−u′ + f = 0 and u(0) given
xxx xx
∆∆
i−1 i i+1
i i+1
0 N
xi−1/2 xi+1/21
FD Scheme : u0 given and − ui − ui−112(∆i + ∆i+1)
+ f(xi) = 0
Local error : ǫih =−∆i+1 + ∆i
∆i+1 + ∆iu′(xi) + O(h)= O(1) for non uniform grid
10−4
10−3
10−2
10−1
100
101
102
103
h
Max
i ∆i /
Min
i ∆i
Maxi ∆
i / Min
i ∆
i
10−4
10−3
10−2
10−1
10−5
10−4
10−3
10−2
10−1
h
||u−
u h|| ∞
||u−uh||∞
0.989 log(h) − 0.543
Numerical O(h) error
Etude de l’ordre de convergence de la méthode des volumes finis – p. 6/27
Transport in dimension 1 (2/2)
Correction :
ǫih =γi − γi−1
12(∆i + ∆i+1)
+ ǫi
with γi = −1
2∆i+1u
′(xi+1/2) = O(h)
and ǫi =∆i+1(u
′(xi+1/2) − u′(xi)) + ∆i(u′(xi) − u′(xi−1/2))
∆i+1 + ∆i= O(h)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 7/27
Linear Convection Problem
a constant vector
Stationnary problem : 0 < b0 ≤ b(x) ≤ b1 :
8
<
:
bu+ (a · ∇)u = f in Ω
u(x) = ψ(x) on ∂Ω−
Unsteady problem :
8
>
>
>
<
>
>
>
:
∂u
∂t+ (a · ∇)u = 0 in Ω
u(x, 0) = φ(x) in Ω
u(x, t) = ψ(x, t) on ∂Ω− × [0,∞[
Etude de l’ordre de convergence de la méthode des volumes finis – p. 8/27
Finite Volume Method
Explicit, Finite volume scheme : unj ≈ 1
|Kj |
Z
Kj
u(x, tn)dx ≡ Unj such that 1 ≤ j ≤ Nv
un+1j − un
j
∆tn+
1
|Kj |
0
B
@
X
k∈N+(j)
a ·Nj,kunj +
X
k∈N−
0(j)
a ·Nj,kunk +
X
k∈N−
b(j)
a ·Nj,kφ(gj,k)
1
C
A= 0
nj,k1nj,k4
2j,kn
gj,k
gj,k
2
gj,k4
gj,k3
1gj
nj,k3
k1
k3
a2
k4
j
k
k
Nj,k = |Kj ∩Kk|nj,k
k1, k3 ∈ N+(j) and k2, k4 ∈ N−0 (j)
Hyp :hnd
j
|Kj |≤ κ1 and ♯N (j) ≤ κ2 , ∀Kj
Etude de l’ordre de convergence de la méthode des volumes finis – p. 9/27
Local and Global error
FV scheme :
(1) un+1j −Ln
j
`
unkk
´
+ Lnbord = 0 where
Lnj
`
ξkk
´
= ξj − ∆tn
|Kj |
0
B
@
X
k∈N+(j)
a ·Nj,kξj +X
k∈N−
0(j)
a ·Nj,kξk
1
C
A
Local error :
(2) u(gj , tn+1) −Ln
j
`
u(gk, tn)k
´
+ Lnbord = ∆tn ǫ
nj
Global error : enj = un
j − u(gj , tn)
⇒ (3) en+1j − Ln
j
`
enkk
´
= −∆tn ǫnj
Etude de l’ordre de convergence de la méthode des volumes finis – p. 10/27
Stability
Theorem 1 :
Under CFL∆tn
minj τj≤ 1 with τj =
|Kj |P
k∈N+(j) a ·Nj,k
i) ‖Ln(ξ)‖p ≤ ‖ξ‖p where ||ξ||p =
0
@
NvX
j=1
|Kj ||ξj |p1
A
1/p
and ||ξ||∞ = max1≤j≤Nv
|ξi|
ii) ‖en‖p ≤ ||e0||p +
n−1X
i=0
∆ti||ǫi||p
Lax Theorem :Local error estimate transfers to global error estimate
Etude de l’ordre de convergence de la méthode des volumes finis – p. 11/27
Consistency
Local error : ǫnj = Gnj + In
j + B. C. discretization error
• Error on B.C. discretization assumed small
• Centered part :
Gnj =
u(gj , tn+1) − u(gj , tn)
∆tn+
1
|Kj |X
k∈N (j)
a ·Nj,ku(gj,k, tn)= O(h)
• Upwind part :
Inj =
1
|Kj |
0
@
X
k∈N+(j)
a ·Nj,k
“
u(gj , tn) − u(gj,k, tn)”
+X
k∈N−
0(j)
a ·Nj,k
“
u(gk, tn) − u(gj,k, tn)”
1
C
A= O(1)
FVM is not consistent in the FD sense
Etude de l’ordre de convergence de la méthode des volumes finis – p. 12/27
Correction
Can we write ǫnj =1
∆tn
“
γn+1 −Lnj (γn)
”
+ ǫnj with γn ≡ γnj j = O(h) and ǫnj = O(h) ?
Geometric corrector : γnj = −Γj · ∇u(gj , t
n) where
Γ ≡ Γjj depends only on the mesh and vector a and is solution of
X
k∈N+(j)
a ·Nj,kΓj +X
k∈N−
0(j)
a ·Nj,kΓk =X
k∈N+(j)
a ·Nj,k(gj,k − gj) +X
k∈N−
0(j)
a ·Nj,k(gj,k − gk)
⇐⇒ (I −B)Γ = ∆ with (Bψ)j =
X
k∈N−
0(j)
a ·Nj,kψk
X
k∈N+(j)
a ·Nj,k
Etude de l’ordre de convergence de la méthode des volumes finis – p. 13/27
Existence of the geometric corrector
Theorem :σ(B) ⊂ z ∈ C , |z| < 1
et (I −B)−1 =∞X
i=0
Bi
Dependence cone : let J a volume of control
C(J) = K /∃ J1, · · · , Jn , a ·NJ1,K < 0, a ·NJ2,J1< 0, · · · , a ·NJ,Jn
< 0
a J
K
J1
J2 J3
Etude de l’ordre de convergence de la méthode des volumes finis – p. 14/27
Existence of the geometric corrector
Theorem :σ(B) ⊂ z ∈ C , |z| < 1
et (I −B)−1 =∞X
i=0
Bi
Proof :
a) ‖Bx‖∞ ≤ ‖x‖∞ sinceX
k∈N (j)
a ·Nj,k = 0
b) ∀J ∈ T , there is at least K ∈ C(J) sharing a face with ∂Ω−
Input Boundary plays an important role
c) By contradictionLet λ / |λ| = 1 et x / Bx = λx.
For Kj / |xj | = maxk
|xk|, we get N−b (Kj) = ∅ and ∀k ∈ N−
0 (j), |xk| = |xj |.
By induction, ∀K ∈ C(Kj), N−b (K) = ∅ contradiction with b)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 14/27
Main result
Theorem : let γnj = −Γj · ∇u(gj , t
n) and let assume u smooth enough
i) local quasi-uniformity of the mesh1
κ3|Kk| ≤ |Kj | ≤ κ3|Kk| , ∀h < h0, ∀Kj ∈ T h, ∀k ∈ N (j)
ii) I.C.. : ||(u0j − ϕ(gj))j ||p ≤ κ4h ,
iii) B.C. : |unk − ψ(gj,k, tn)| ≤ κ5h2
then under CFL ,∀p ∈ [1,+∞], (α ∈]0, 1])
∃cp > 0 / ‖Γ‖p ≤ cphα =⇒ ‖u(·, tn) − (unj )j‖p ≤ cphα ∀tn ≤ T
Extension :
Implicit scheme
Mesh where interfaces are not hyperplane
The center of gravity gj can be replaced by any point at a distance less than h fromgj .
Etude de l’ordre de convergence de la méthode des volumes finis – p. 15/27
Study of the Geometric corrector (1)
Dimension 1 :
gj,j−1 gjxx jxj−1/2
gj,j+1j+1/2
e_jej∆ jx
a > 0
un+1j − un
j
∆tn+ a
unj − un
j−1
∆xj= 0
8
<
:
aΓ1 = a (x 32
− x1) , j = 1
a (Γj − Γj−1) = a (xj+ 12
− xj) − a (xj− 12
− xj−1) , j ≥ 2=⇒ Γj =
∆xj
2
Remark : γnj = −∆j
2
∂u(xj , tn)
∂x=⇒ en
j = unj − u(xj+1/2, tn) + O(h2)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 16/27
Study of the Geometric corrector (2)
Theorem :
−Dimension 2
−T0 unstructured coarse mesh of triangles or quadrangles
−Th obtained by uniformly refinement
then ‖Γ‖p ≤ ch
Proof :a) Start with one triangle : ♯T0 = 1
−−
−−
12
−+
−+
+−
−
+
−
2n
1
− +
m
T2,2−
2,2T+
T+1,2
+
2,1T+
+1
+1
a
+−+
++
+
+
+
−
−−
−
−+
12
12
++
m
n
T
T2,2
2,2
+
−
+1
+1
a
=⇒ |Γǫm,n| ≤
C(a, T )
ℓ+ 1
Etude de l’ordre de convergence de la méthode des volumes finis – p. 17/27
Study of the Geometric corrector (2)
Theorem :
−Dimension 2
−T0 unstructured coarse mesh of triangles or quadrangles
−Th obtained by uniformly refinement
then ‖Γ‖p ≤ ch
Proof :b) By induction on the number of volumes
Lemma : convex volumes in 2d
∃ a broken line of interfaces that are “en-lighted” on the same side which devidethe domain in 2
A
A
A
A
A
S
S
S
S
S
F
44
4
3
3
32
2
2
1
I1
aP2
P3
n
nn
n
Etude de l’ordre de convergence de la méthode des volumes finis – p. 17/27
Study of the Geometric corrector (3)
Numerical test with independent meshes :
a=0
a=pi/4
10−2
10−1
100
10−3
10−2
10−1
h
||Γ|| 1
θ = π/4
1.052 log(h) − 0.550
10−2
10−1
100
10−3
10−2
10−1
h
||Γ|| ∞
θ = π/4
0.905 log(h) − 0.469
10−2
10−1
100
10−3
10−2
10−1
100
h
Err
Err L ∞ 0.874 log(h) + 0.054 Err L1 0.988 log(h) − 0.593
10−2
10−1
100
10−3
10−2
10−1
h
||Γ|| 1
θ = 0
1.033 log(h) − 0.502
10−2
10−1
100
10−3
10−2
10−1
h
||Γ|| ∞
θ = 0
0.435 log(h) − 0.713
10−2
10−1
100
10−3
10−2
10−1
100
h
Err
Err L ∞ 0.441 log(h) − 0.427 Err L1 0.992 log(h) − 0.850
‖.‖1 ‖.‖∞
O(h)
‖.‖1 ‖.‖∞
O(h1/2)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 18/27
Counter Example Peterson (Sinum 91)
T
TT
T
+
+
−
−
+
T−
T −T+
T+
T+
0,1
1,0 3,02,1
0,1 2,1
1,2
1,2
0,3
0,3 = 2 = 4
T
a
= 8
j
i
Numbering of triangles and definition of gj :
+ ++
m,n m,n
(T ,G )m,n m,n
(T ,G )
0,n
0,n
(T ,G )0,n
(T ,G )0,n− − −−
+
+
m+1,n−1
+ +
m−1,n−1 m−1,n−1(T ,G ) (T ,G )m+1,n−1
+
Etude de l’ordre de convergence de la méthode des volumes finis – p. 19/27
Counter Example Peterson (Sinum 91)
T
TT
T
+
+
−
−
+
T−
T −T+
T+
T+
0,1
1,0 3,02,1
0,1 2,1
1,2
1,2
0,3
0,3 = 2 = 4
T
a
= 8
j
i
Geometric Corrector :
Γ+m,n = 1
2(Γ+
m−1,n−1 + Γ+m+1,n−1) , m ≥ 1 , n ≥ 1
Γ+0,n = Γ
+1,n−1 + h
2~i , n = 1, 3, . . . , 2ℓ− 1
Γ+m,0 = 0 , m = 1, 3, . . . , 2ℓ− 1 .
Etude de l’ordre de convergence de la méthode des volumes finis – p. 19/27
Counter Example Peterson (Sinum 91)
T
TT
T
+
+
−
−
+
T−
T −T+
T+
T+
0,1
1,0 3,02,1
0,1 2,1
1,2
1,2
0,3
0,3 = 2 = 4
T
a
= 8
j
i
Explicit estimation equal to the analytical error estimation :
Γ+2p,2q+1 =
8
>
>
<
>
>
:
0 , 0 ≤ q ≤ ℓ2− 1 , q + 1 ≤ p ≤ ℓ− q − 1
qX
k=p
1
22k+1
“ 2k
k − p
”
h~i , 0 ≤ q ≤ ℓ− 1 , 0 ≤ p ≤ min(q, ℓ− q − 1)
||Γ(ℓ)||∞ =1√πh1/2 + O(h) et ||Γ(ℓ)||1 = O(h)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 19/27
Counter Example Peterson:oblique incidence
T
TT
T
+
+
−
−
+
T−
T −T+
T+
T+
0,1
1,0 3,02,1
0,1 2,11,2
1,2
0,3
0,3 = 2 = 4
T
= 8
j
i
a
θ
Geometric corrector :
Γ+m,n = pΓ+
m−1,n−1 + qΓ+m+1,n−1 , m 6= 0 , n ≥ 1
Γ+0,n = q
pΓ
+1,n−1 + h~β , n = 1, 3, . . . , 2ℓ− 1
Γ+m,0 = 0 , m = 1, 3, . . . , 2ℓ− 1 ,
p =cosα
cosα+ sinα, p+ q = 1 , q < 1/2 < p , α =
π
4− θ , θ ∈]0, π/4] , ~β constant
Etude de l’ordre de convergence de la méthode des volumes finis – p. 20/27
Counter Example Peterson:oblique incidence
T
TT
T
+
+
−
−
+
T−
T −T+
T+
T+
0,1
1,0 3,02,1
0,1 2,11,2
1,2
0,3
0,3 = 2 = 4
T
= 8
j
i
a
θ
Results :
||Γ(ℓ)||∞ = O(h) et ||Γ(ℓ)||1 = O(h)
A matrix formulation (square of a matrix of transition of a Markov process)
A probabilist approach : a sum of random walk
Etude de l’ordre de convergence de la méthode des volumes finis – p. 20/27
Second order scheme (1d)
1st order :duj
dt+ a
uj − uj−1
∆xj= 0
Γj − Γj−1 =∆xj
2− ∆xj−1
2
2nd order :duj
dt+ a
fj+ 12
− fj− 12
∆xj= 0 with fj+ 1
2
= a
0
@uj +∆xj
2
uj+1 − uj−1
∆xj+ 12
+ ∆xj− 12
1
A
γj = −Γj∂2u(xj , t)
∂x2
8
>
>
>
>
<
>
>
>
>
:
Γ′j − Γ′
j−1 = ρj − ρj−1 with ρj =∆xj(∆xj−1 + ∆xj − ∆xj+1)
8
Γ′j = Γj + αj(Γj+1 − Γj−1) with αj =
∆xj
∆xj−1 + 2∆xj + ∆xj+1
Γ′j = O(h2) and ‖Γ‖p ≤ c‖Γ′‖p
Etude de l’ordre de convergence de la méthode des volumes finis – p. 21/27
Non constant vectora (1d)
A Riemann finite volume approach :
un+1j − un
j
∆tn+
1
∆xj(Φj+ 1
2
(un) − Φj− 12
(un)) = 0
with Φj+ 12
(un) =aj+ 1
2
(unj + un
j+1)
2− σj+ 1
2
aj+ 12
(unj+1 − un
j )
2
=σj+ 1
2
+ 1
2aj+ 1
2
unj −
σj+ 12
− 1
2aj+ 1
2
unj+1
= a+
j+ 12
unj − a−
j+ 12
unj+1
un+1j = un
j − ∆tn
∆xj
“
a+j+1/2
unj − a−
j+1/2un
j+1 − a+j−1/2
unj−1 + a+
j−1/2un
j
”
with aj+1/2 ≡ a(xj+1/2) = a+j+1/2
− a−j+1/2
and σj+ 12
= sign(aj+ 12
)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 22/27
Analysis
Stability : Under∆tn
∆xj(a+
j+ 12
+ a−j− 1
2
) ≤ 1 then ‖Ln(ξ)‖p ≤ (1 + ∆tn‖ax‖∞) ‖ξ‖p
Consistency : ǫnj = Gnj + In
j
Gnj =
u(xj , tn+1) − u(xj , tn)
∆tn+f(xj+ 1
2
, tn) − f(xj− 12
, tn)
∆xj= O(∆t) + O(h)
Inj =
Φj+ 12
(Un) − Φj− 12
(Un) + f(xj− 12
, tn) − f(xj+ 12
, tn)
∆xj.
Corrector : γnj = u(xj , tn) − u(xj + σj
∆xj
2, tn) = −σj
∆xj
2
∂u(xj , tn)
∂x+ O(h2)
σj = sign(a(xj))
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
cvtM
, u
a(x) = x with Riemann Scheme
10−3
10−2
10−1
10−4
10−3
10−2
10−1
h
Err
Inf
a(x) = x with Riemann Scheme
Erreur Inf0.967 log(h) − 0.459
For γnj = −Γj
∂u(xj ,tn)
∂x,
the equation satisfied by Γj
has the same form as constanta but is no longer 0 but O(h2)
near a change of sign of a
Etude de l’ordre de convergence de la méthode des volumes finis – p. 23/27
Non constant vectora(x) (1d)
A flux scheme :
un+1j − un
j
∆tn+
1
∆xj
“
Ψj+ 12
(un) − Ψj− 12
(un)”
= 0
with Ψj+ 12
(un) =aju
nj + aj+1u
nj+1
2− σj+ 1
2
aj+1unj+1 − aju
nj
2
=σj+ 1
2
+ 1
2aju
nj −
σj+ 12
− 1
2aj+1u
nj+1
un+1j = un
j − ∆tn
∆xj
„
1 − σj+1/2
2(aj+1u
nj+1 − aju
nj ) −
1 + σj−1/2
2(aj−1u
nj−1 − aju
nj )
«ff
×aj
with σj+1/2 = sign(a(xj+1/2))
In term of fluxes : ϕnj = aju
nj satisfy
ϕn+1j − ϕn
j
∆tn+
aj
∆xj
1 − σj+ 12
2(ϕn
j+1 − ϕnj ) +
1 + σj− 12
2(ϕn
j − ϕnj−1)
!
= 0
Etude de l’ordre de convergence de la méthode des volumes finis – p. 24/27
Analysis
Stability on the flux : under∆tn
∆xjaj
σj+ 12
+ σj− 12
2≤ 1 then
‖LnF (ξ)‖p ≤ (1 + c∆tn‖ax‖∞) ‖ξ‖p
Consistency : ǫnj = Gnj + ajI
nj
Gnj =
f(xj , tn+1) − f(xj , tn)
∆tn+ aj
f(xj+ 12
, tn) − f(xj− 12
, tn)
∆xj= O(∆t) + O(h)
Inj =
Ψj+ 12
(Fn) − Ψj− 12
(Fn) + f(xj− 12
, tn) − f(xj+ 12
, tn)
∆xj.
Correction on flux : γnj = f(xj , tn) − f(xj + σj
∆xj
2, tn) = −σj
∆xj
2
∂f(xj , tn)
∂x+ O(h2)
−1 −0.5 0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
cvtM
, u
a(x) = x with Flux Scheme
10−3
10−2
10−1
10−4
10−3
10−2
h
Err
Inf F
lux
a(x) = x with Flux Scheme
Erreur Inf Flux0.964 log(h) − 0.824
Error estimate on the flux
Etude de l’ordre de convergence de la méthode des volumes finis – p. 25/27
The nonlinear case
Nonlinear problem :∂u(x, t)
∂t+∂f(u(x, t))
∂x= 0
Scheme :un+1
j − unj
∆tn+
1
∆xj(Φn
j+ 12
(un) − Φnj− 1
2
(un)) = 0
Φnj+ 1
2
(un) =f(un
j+1) + f(unj )
2− σn
j+ 12
f(unj+1) − f(un
j )
2
with σnj+ 1
2
= sign(snj+ 1
2
) where snj+ 1
2
=
8
>
<
>
:
f(unj+1) − f(un
j )
unj+1 − un
j
if unj+1 6= un
j
f ′(unj ) if un
j+1 = unj
Error analysis : enj = un
j − u(xj + δj∆xj
2, tn) with δj = sign(f ′(u(xj , ·))
Under ∆tn
∆xj
σn
j+ 12
−1
2snj+ 1
2
+σn
j− 12
+1
2snj− 1
2
!
≤ 1
||(enj )N
j=1||∞ ≤ C′∞h
||(enj )N
j=1||1 ≤ C′1h (for global quasi-uniform meshes)
Etude de l’ordre de convergence de la méthode des volumes finis – p. 26/27
Conclusion
Geometric corrector helps in finding optimal error estimates
Bounded domains and B.C. are taken into account
Lost of order of convergence for non smooth solution is not due to the mesh
Open question and Perspective :
Probabilist approach to study the corrector
Extension in 2d for non constant vector and 2nd order scheme
Study of the norm of the corrector is open in 2d
Etude de l’ordre de convergence de la méthode des volumes finis – p. 27/27