![Page 1: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/1.jpg)
1P. Ackerer, IMFS, Barcelona 2006
About Discontinuous Galerkin Finite Elements
P. Ackerer, A. Younès
Institut de Mécanique des Fluides et des Solides, Strasbourg, France
![Page 2: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/2.jpg)
2P. Ackerer, IMFS, Barcelona 2006
OUTLINE
1. Introduction
2. Solving advective dominant transport 2.1. Eulerian methods: Finite Volumes, Finite Elements
3. Galerkin Discontinuous Finite Elements 3.1. 1D discretization3.2. General formulation3.3. Numerical integration3.4. Slope limiter
4. Numerical experiments 4.1. 2D – 3D benchmarks4.2. Comparisons with finite volumes
5. On going works
![Page 3: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/3.jpg)
3P. Ackerer, IMFS, Barcelona 2006
xjxj-1xj-2 xj+1
xj-1/2 xj+1/2Finite volumes
xjxj-1xj-2 xj+1
Finite elements
xjxj-1xj-2 xj+1
Discontinuous finite elements
x
n+1
n
n-1
t
j j+1j-1j-2
Space/time discretization
Introduction
![Page 4: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/4.jpg)
4P. Ackerer, IMFS, Barcelona 2006
C Cu 0
t x
Finite differences method (FD):
2 2
2
2 2
2
f x ff (x x) f (x) x ...
x 2 x
f x ff (x x) f (x) x ...
x 2 x
f f (x x) f (x)
x xf f (x x) f (x x)
x 2 x
n* n*n 1 nj j 1j j
C CC Cu 0
t x
n* n*n 1 nj 1 j 1j j
C CC Cu 0
t 2 x
Basic ideas:
1. Use Taylor’s (1685-1731) series 2. Replace the derivatives
Richardson (1922) was first to apply FD to weather forecasting. It required 3 months' worth of calculations to predict weather for next 24 hours.
Introduction
![Page 5: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/5.jpg)
5P. Ackerer, IMFS, Barcelona 2006
n 1 nj j n* n*
j 1/ 2 j 1/ 2
C Cx u C C
t
j 1/ 2 j j 1
1C C C
2 j 1/ 2 jC C
xjxj-1xj-2 xj+1
xj-1/2 xj+1/2
uFV have a very strong physical meaning
n* n*n 1 nj j 1j j
C CC Cu 0
t x
n* n*n 1 nj 1 j 1j j
C CC Cu 0
t 2 x
Finite Volumes methods
Introduction
![Page 6: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/6.jpg)
6P. Ackerer, IMFS, Barcelona 2006
n 1 nj j n n
j 1 j
C Cx u C C
t
Some key numbers (1D)
n 1 n nj j j 1
u tC C (1 ) C
x
2 2
2
2
2
C x CC(x x) C(x) x ...
x 2 x
C C(x x) C(x) x C
x x 2 x
C Cu 0
t x
2
2
C C(x x) C(x) x Cu u 0
t x 2 x
u xD or
2u x
Grid Peclet number 2D
To reduce numerical diffusion
u tCFL 1
x
To avoid oscillation for this scheme
(R. Courant, K. Friedrichs & H. Lewy ,1924)
Introduction
![Page 7: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/7.jpg)
7P. Ackerer, IMFS, Barcelona 2006
Galerkin Finite Elements method
Basic ideas:
1. Approximate the unknown function by a sum of ‘simple’ functionsne
j jj 1
C(x, t) (x)C (t)
i j
j ii j
1 if x x(x )
0 if x x
j jC(x , t) C (t)with so that
CL(C) uC D C 0
t
xjxj-1xj-2 xj+1
FE
2. The numerical solution should be as close as possible to the exact solution over the domain
L(C(x, t)) (x) 0
d(x)for any
iL(C(x, t)) (x) 0
dwith i=1 to ne,which leads to ne equations with ne unknowns
xjxj-1xj-2 xj+1
FV
u
Introduction
![Page 8: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/8.jpg)
8P. Ackerer, IMFS, Barcelona 2006
Basic ideas:
3. Choose i i(x) (x) which leads to
j j
jj j j j i
j j
C
u C D C d 0t
n 1 nj j j j
j j n 1 n 1j j j j i
j j
C C
u C D C d 0t
4. Standard Euler/implicit scheme for time discretization, for example
written for i=1 to ne.
The next steps are more or less easy mathematics ...
Introduction
![Page 9: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/9.jpg)
9P. Ackerer, IMFS, Barcelona 2006
Galerkin Discontinuous Finite elements method
Basic ideas:
1. Approximate the unknown function by a sum of ‘simple’ functions INSIDE an element E
xjxj-1xj-2 xj+1
FE
xjxj-1xj-2 xj+1
DFE
u
Discontinuous Finite Elements
2. Defining on node/edge/face A inside of E and on edge/face A outside of E
inAC
outAC
inj 1C
outj 1C
j jj 1
C(x, t) (x)Y (t)
Yj(t) : degree of freedom (nodal conc., ….)
![Page 10: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/10.jpg)
10P. Ackerer, IMFS, Barcelona 2006
Basic ideas:
3. Second order explicit Runge-Kutta scheme
2
tt,tt
n 1 2 nE An in nE E
E AA EE E A
QC Cw dE UC w dE C wds
t 2 A
/
, ,./
E,AQ
A
: the flux through A, positive if pointed outside
: norm of A (length, surface).
Step 1:
*,n 1 nE,An 1/ 2 in or out,n+1/2E E
E AA EE E A
QC Cw dE UC . w dE wC ds
t A
in,n+1/2A,Ein or out,n+1/2
A out,n+1/2A,E
C for outflowC
C for inflow
Step 2:
Discontinuous Finite Elements
![Page 11: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/11.jpg)
11P. Ackerer, IMFS, Barcelona 2006
Basic ideas:
4. Oscillations avoided by slope limitation
xjxj-1xj-2 xj+1
XCo
nc
55 60 65 70 75
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2DFG No Limit.
XCo
nc
55 60 65 70 75
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2DFG with Limit.
Discontinuous Finite Elements
![Page 12: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/12.jpg)
12P. Ackerer, IMFS, Barcelona 2006
C(uC)
t
C
tw dx w dx C w x C w x
E i iE
i i i i i i uC u( 1 1( ) ( ))
Hyperbolic 1D
C
tw dx uC w dx i i
EE
.( )
Variational form
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
i 1
i
x x(x)
x
i
i 1
x x(x)
x
i i i 1 i 1C(x, t) (x)C (t) (x)C (t)
Linear approximation
DGFE : 1D discretization
![Page 13: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/13.jpg)
13P. Ackerer, IMFS, Barcelona 2006
i iw (x) (x)
Galerkin formulation
*
i i iEE
*
i 1 i 1 i 1EE
Cw dx uC w dx + uC
tC
w dx uC w dx uCt
Discretization
i i 1
i i 1
n 1 n 1 n n *,n
i i 1 i
n 1 n 1 n n *,n
i i 1 i 1
u t u t 6 t2C C C (2 3 ) C (1 3 ) u C
x x xu t u t 6 t
C 2C C (1 3 ) C (2 3 ) u Cx x x
Explicit formulation leads to a local system:
xi+1xixi-1 xi+2
E
DGFE : 1D discretization
![Page 14: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/14.jpg)
14P. Ackerer, IMFS, Barcelona 2006
xi+1xixi-1 xi+2
E
DGFE : 1D discretization
t t, t t / 2 Step 1:
i i 1
i i 1
n 1/ 2 n 1/ 2 n n n
i i 1 i
n 1/ 2 n 1/ 2 n n n
i i 1 i 1
u t / 2 u t / 2 6 t / 22C C C (2 3 ) C (1 3 ) u C
x x xu t / 2 u t / 2 6 t / 2
C 2C C (1 3 ) C (2 3 ) u Cx x x
Step 2:
i i 1
i i 1
n 1 n 1 n 1/ 2 n 1/ 2 n 1/ 2,in or out
i i 1 i
n 1 n 1 n 1/ 2 n 1/ 2 n 1/ 2,in or out
i i 1 i 1
u t u t 6 t2C C C (2 3 ) C (1 3 ) u C
x x xu t u t 6 t
C 2C C (1 3 ) C (2 3 ) u Cx x x
t t, t t
![Page 15: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/15.jpg)
15P. Ackerer, IMFS, Barcelona 2006
XCo
nc
55 60 65 70 75
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2DFG No Limit.
XCo
nc
55 60 65 70 75
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2DFG with Limit.
Slope limitation
xi+1xixi-1 xi+2
En 1 n 1 n 1 n 1
i i 1 i i 1
n 1
E 1 E i E 1 E
n 1
E E 1 i 1 E E 1
C C C Cx x
2 2min(C ,C ) C max(C ,C )
min(C ,C ) C max(C ,C )
DGFE : 1D discretization
![Page 16: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/16.jpg)
16P. Ackerer, IMFS, Barcelona 2006
C
tuC .( )
General formulation
Variational formn 1 n *
E E A
E E,AA EE E A
C C C ww dE UC . w dE Q ds
t A
A : norm of A (length, surface).
A,EQ : the flux through A, positive if pointed outside
Polynomial approximation
E 1 2 3C (X, t) Y (t) xY (t) yY (t) Linear (2D):
E 1 2 3 4C (X, t) Y (t) xY (t) yY (t) xyY (t) Bi-Linear (2D):
DGFE : General formulation
![Page 17: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/17.jpg)
17P. Ackerer, IMFS, Barcelona 2006
Standard interpolation functions
(1,1)
(0,0)
1
4 3
2
Bilinear interpolation
1
2
3
4
(x,y)=(1-x)(1-y),
(x,y)=x(1-y),
(x,y)=xy,
(x,y)=y(1-x).
Linear interpolation
(1,1)
(0,0)
1
2
3
(x, y) 1,
(x, y) x x,
(x, y) y y.
E x yC (X, t) C(t) x x C (t) y y C (t)
DGFE : General formulation
![Page 18: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/18.jpg)
18P. Ackerer, IMFS, Barcelona 2006
Step 1 : t t, t t / 2
n 1/ 2 n in ,n
nE E A
E E,AA EE E A
C C CdE UC . dE Q ds
t / 2 A
A,EQ
A
: the flux through A, positive if pointed outside
: norm of A (length, surface).
Step 2 :
n 1 n in or out,n+1/2
n 1/ 2E E A
E E,AA EE K A
n
C C CdE UC . dE Q ds
t A
t t, t t
outA
inA C ,C A,EQ: depending on the sign of
DGFE : General formulation
![Page 19: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/19.jpg)
19P. Ackerer, IMFS, Barcelona 2006
x y
n
E 2E
2
C C C
1 E 0 0C w dE
(x x) 0 (x x) (x x)(y y)
(y y) 0 (x x)(y y) (y y)
n 1 n
E E
EE E
C Cw dE and UC . w dE
t
Numerical integration (1)
Exact integration in reference element for E
DGFE : Numerical integration
![Page 20: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/20.jpg)
20P. Ackerer, IMFS, Barcelona 2006
Exact numerical integration with Simpson’s rule (pol. Ordre 2)
EI f ( ) 4f ( ) f ( )
6
i jxx x j k
EI f ( ) f ( ) f ( )
3
ix xx
i
kjj
EI f ( ) 4f ( ) 16f ( )
36 i
x xx
Numerical integration (2)*
A
A
C wds
A
DGFE : Numerical integration
![Page 21: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/21.jpg)
21P. Ackerer, IMFS, Barcelona 2006
0 i 0,i 0 imin(C ,C ) C max(C ,C )
*
*
, , ,
, , .
W Ex x
N Sy y
C M C C C C C
C M C C C C C
sign( ) min( , , ), if sign( ) sign( ) sign( ),(a,b,c)=
0 otherwise.
a a b c a b cM
DGFE : Slope limiting
![Page 22: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/22.jpg)
22P. Ackerer, IMFS, Barcelona 2006
Step 3 : Multidimensional slope limiter (Bilinear function)
Ei
min(i)/max (i) : min/max ofover each element containing i
*,n 1
EC min(E)/max (E) : min/max value of
over each element which has a common node with E.
*,n 1
EC
E
nn 2n 1 n 1 n 1 *,n 1
E,1 E,nn E,i E ,ii 1
J(C ,...,C ) C C
Optimization :
Constraints :n 1 *,n 1
E EC C
n 1
E,imin(i) C max(i)
*,n 1 *,n 1
E E C max(E) or C min(E)
thenn 1 *,n 1
E,i EC C
Extrema :
DGFE : Slope limiting
![Page 23: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/23.jpg)
23P. Ackerer, IMFS, Barcelona 2006
1rd order Upwind
Centered 3rd order Upwind
ImplicitCFL=1
CFL=5
CFL=1
CFL=5
CFL=1
CFL=5
Crank-Nicholson
CFL=1
CFL=5
CFL=1
CFL=5
CFL=1
CFL=5
1rd order BDFCFL=1
CFL=5
CFL=1
CFL=5
CFL=1
CFL=5
Flux discretisation
Tim
e di
scre
tiza
tion
DGFE, CFL=1
FE, CFL=1
FE, CFL=5
DGFE : Numerical experiments
1D Benchmarks
![Page 24: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/24.jpg)
24P. Ackerer, IMFS, Barcelona 2006
Bilinear, CFL=0,6 Linear, CFL=0,6
Linear, CFL=0,6Bilinear, CFL=0,1
DGFE : Numerical experiments
2D Benchmarks
![Page 25: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/25.jpg)
25P. Ackerer, IMFS, Barcelona 2006
1 Ty x vVelocity field
DGFE : Numerical experiments
3D Benchmarks
![Page 26: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/26.jpg)
26P. Ackerer, IMFS, Barcelona 2006
Finite volume Bilin. DGFE
DGFE : Numerical experiments
![Page 27: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/27.jpg)
27P. Ackerer, IMFS, Barcelona 2006
Finite volume (CFL = 0.50) D-GFE (CFL = 0.50) CFL=0,1 CFL=0,5 CFL=1 CFL=2 CFL=5 CFL=10
E.F.D 0.363 0.684 0.887 1.109 1.383 1.549
V.F 1.276 1.343 1.406 1.491 1.622 1.713
V.F.2 0.987 1.054 1.123 1.225 1.398 1.554
EFD : 10000 cells, 30 000 unk.
VF : 10000 cells, 10 000 unk., VF 2: 40000 cells, 40 000 unk.
DGFE : Numerical experiments
Comparisons with Finite Volumes
![Page 28: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/28.jpg)
28P. Ackerer, IMFS, Barcelona 2006
Discontinuous Galerkin: well known algorithms
DGFE : Summary
Efficient in tracking fronts
Well adapted to change interpolation order from one element to the other
BUT
Explicit scheme ……
Summary
![Page 29: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/29.jpg)
29P. Ackerer, IMFS, Barcelona 2006
DGFE : On going work
Implicit upwind formulation n 1 n in or out ,*
*E E A
E E,AA EE E A
C C CdE UC . dE Q ds
t A
A,EQ
A
: the flux through A, positive if pointed outside
: norm of A (length, surface).
Time domain decomposition
![Page 30: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/30.jpg)
30P. Ackerer, IMFS, Barcelona 2006
X20 40 60 80 100
Xt
t+t
t+3t/4
t+t/2
t+t/4
Time domain decomposition
DGFE : On going work
DGFE, CFL=1
x 0.2;4.0
![Page 31: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/31.jpg)
31P. Ackerer, IMFS, Barcelona 2006
n 1 n in or out ,*
*E E A
E E,AA EE E A
C C CdE UC . dE Q ds
t A
Implicit upwind formulation
DGFE : On going work
* n n 1
E E EC (1 )C C
![Page 32: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/32.jpg)
32P. Ackerer, IMFS, Barcelona 2006
DGFE : On going work
![Page 33: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/33.jpg)
33P. Ackerer, IMFS, Barcelona 2006
DGFE : On going work
![Page 34: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/34.jpg)
34P. Ackerer, IMFS, Barcelona 2006
Next to come ….
DGFE : On going work
![Page 35: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/35.jpg)
35P. Ackerer, IMFS, Barcelona 2006
![Page 36: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/36.jpg)
36P. Ackerer, IMFS, Barcelona 2006
![Page 37: P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e175503460f94b02ad6/html5/thumbnails/37.jpg)
37P. Ackerer, IMFS, Barcelona 2006