P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,

1P. Ackerer, IMFS, Barcelona 2006

About Discontinuous Galerkin Finite Elements

P. Ackerer, A. Younès

Institut de Mécanique des Fluides et des Solides, Strasbourg, France

[email protected]


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


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


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


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


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


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


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


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., ….)


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:



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.



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


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



xi+1xixi-1 xi+2

E


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


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 )



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


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)



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



Step 2 :

n 1 n in or out,n+1/2

n 1/ 2E E A

E E,AA EE K A

n


t A

t t, t t

outA

inA C ,C A,EQ: depending on the sign of



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


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


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


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


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


Bilinear, CFL=0,6 Linear, CFL=0,6

Linear, CFL=0,6Bilinear, CFL=0,1


2D Benchmarks


1 Ty x vVelocity field


3D Benchmarks


Finite volume Bilin. DGFE



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.


Comparisons with Finite Volumes


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


DGFE : On going work

Implicit upwind formulation n 1 n in or out ,*

*E E A

E E,AA EE E A


t A

A,EQ

A



Time domain decomposition


X20 40 60 80 100

Xt

t+t

t+3t/4

t+t/2

t+t/4

Time domain decomposition


DGFE, CFL=1

x 0.2;4.0


n 1 n in or out ,*

*E E A

E E,AA EE E A


t A

Implicit upwind formulation


* n n 1

E E EC (1 )C C






Next to come ….





Documents

P. Ackerer, IMFS, Barcelona 2006 1 About Discontinuous Galerkin Finite Elements P. Ackerer, A. Younès Institut de Mécanique des Fluides et des Solides,