outline on reliability of higher order ﬁnite element...

On reliabilityof FEM in FSI

Svacek, P.

Outline

Introduction

MathematicalModel

NumericalApproximation

NumericalResults

Conclusions

On reliability of higher order finite elementmethod in fluid-structure interaction problems

Svacek, P.

Department of Math. Sciences The University of Texas at El Paso

CTU, Fac. of Mech. Eng., Dep. of Technical Math.

REC’06, February 22-24


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Conclusions

Thanks to

M. Horacek, CAS, IT, Czech Rep.

M. Feistauer, Charles University, Czech Rep.

M. Cecrdle, J. Malecek, Aeronautical Test and ResearchInstitute.


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Conclusions

Outline

1 Introduction

2 Mathematical ModelGeneral ModelSimplificationsInterface Conditions

3 Numerical ApproximationTime DiscretizationSpace discretization of Fluid Model

4 Numerical ResultsFluid Flow ApproximationFluid Approximation over Moving StructureAeroelastic Simulations

5 Conclusions


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Conclusions

Fluid-structure interaction problem

Air Flows Interacts with Elastic WingWind Tunnel in Aeronautical Test and Research Institute,

Prague


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Conclusions

Computational Aeroelasticity

numerical simulation of both fluid and structure motion

fluid-structure mutual interaction

structure motion → fluid characterizationaerodynamical forces → structural motion

Goals

determine the safe region (critical velocity)

simulate post-critical regimes (nonlinear aeroelasticity)


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Mathematical Model

Computational Aeroelasticity

fluid flow model

elastic structure

interface conditions


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Fluid Models

incompressible viscous flow (NS eq.)

RANS equations - turbulence models


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Navier-Stokes System of Equations

Navier-Stokes system

∂v

∂t+ (v · ∇) v +∇p − ν4v = 0

∇ · v = 0 in Ωt

v - fluid velocity

p - pressure

Navier-Stokes system of equations


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Reynolds Averaged Navier-Stokes equations

Navier-Stokes system

∂v

∂t+ v · ∇v +∇p − ν4v = 0

∇ · v = 0 in Ωt

Reynolds Averaging

v = V + v′, p = P + p′, such that v = V

RANS equations

∂Vi

∂t− ν4Vi + (V · ∇) Vi +

∂P

∂xi=

∑j

∂

∂xj

(−v ′i v

′j

),

Reynolds-Stresses σRij = −v ′i v

′j


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Turbulence model - SA model

Reynolds-Stresses approximation σRij = −v ′i v

′j

Reynolds stresses are approximated

σRij =

2

3kδij + νT

(∂Vi

∂xj+∂Vj

∂xi

), νT ≈ ν,

Turbulence Modelling - Spallart-Almaras Model

∂ν

∂t+ (V · ∇)ν =

1

σ

∂

∂xi

((ν + ν)

∂ν

∂xi

)+

cb2

σ(∇ν)2 + G − Y ,


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Comparison of NS and RANS:

Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUT

time/space scales are to small to be correctlyresolved in engineering computations!

Reynolds Averaged Navier-Stokes system

RANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions


Navier-Stokes systemdescribes (turbulent) fluid flowno additional modelling is needed BUTtime/space scales are to small to be correctlyresolved in engineering computations!

Reynolds Averaged Navier-Stokes system

RANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions



Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.

Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions



Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUT

the turbulent stresses requires further modelling(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions



Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling

(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions



Reynolds Averaged Navier-Stokes systemRANS desribes the mean fluid characteristics, thefluctuating part is only modelled.Thus: time/space scales can be correctly resolvedBUTthe turbulent stresses requires further modelling(any) turbulence model is still inexact !!!


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Structure Model

elasticity equation

ρ∂2ui

∂t2−

∑j

∂σij(u)

∂xj= fi

u - structure deflection

special cases: linear stationary elasticity


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Flexibly Supported Airfoil

T

EO

khh

a

h

x'

kaaL t( )

M t( )

U0

x

Airfoil motion equations

system ODEs

mh + Sαα + Khhh = −L

Sαh + Iαα+ Kααα = M3


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Flexibly Supported Airfoil

T

EO

khh

a

h

x'

kaaL t( )

M t( )

U0

x

Airfoil motion equations

system ODEs

mh + Sαα cosα− Sαα2 sinα+ Khhh = −L

Sαh cosα+ Iαα+ Kααα = M3


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Interface Conditions

elastic structure model

interface velocity wI

v = wI , u = wI

equality of fluid/elastic forces

flexibly supported airfoil model

airfoil surface condition

v = wI

aerodynamical fluid forces L - lift and M - torsionalmoment


Svacek, P.

Outline

Introduction

MathematicalModel

General Model

Simplifications

InterfaceConditions


NumericalResults

Conclusions

Goals - revisited

Model

fluid flow

flexibly supported airfoil

Goals

determine the safe region (critical velocity)

simulate post-critical regimes (nonlinear aeroelasticity)

how can we verify our results ?

compare numerical results to experimental data


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization

Spacediscretization ofFluid Model

NumericalResults

Conclusions

Numerical Approximation

time discretization

space discretization

interface conditions


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Computations on Moving Meshes

How to approximat the time derivative ? AVI format


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

How to approximate the time derivative?

Navier-Stokes system in ALE form

DA

Dtv + (v −wg ) · ∇v +∇p − ν4v = 0

∇ · v = 0 in Ωt


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions


Arbitrary Lagrangian-Eulerian method


DA

Dtv + (v −wg ) · ∇v +∇p − ν4v = 0

∇ · v = 0 in Ωt


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions


Arbitrary Lagrangian-Eulerian method

Define ALE mapping At

At : Ωref 7→ Ωt

Domain velocity (grid velocity)

wg (t,Y ) =∂At(Y )

∂t

ALE derivative - time derivative on ALE trajectory

DA

Dtf =

∂f

∂t+ (wg · ∇)f


DA

Dtv + (v −wg ) · ∇v +∇p − ν4v = 0

∇ · v = 0 in Ωt


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Weak Formulation

The ALE derivative is approximated

DAv

Dt≈ 3vn+1 − 4vn + vn−1

2∆t

Weak formulation: find vn+1 = v, p

Weak formulation

(3v

2∆t, ϕ

)+

([(v −wg ) · ∇] v, ϕ

)+ ν(∇v,∇ϕ)

−(p,∇T · ϕ) + (∇ · v, q) =

(4vn − vn−1

2∆t, ϕ

)


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Galerkin formulation

FEM gives unstable results?

Galerkin method is unstable → several sources ofinstabilities

Babuska-Brezzi (inf-sup) condition needs to be satisfied

supvh∈Xh

(qh,∇ · vh)

‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω

very high Reynolds numbers → convection dominatedflows

Re locK =

h‖v‖Kν

> 1


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Galerkin formulation

FEM gives unstable results?

Galerkin method is unstable → several sources ofinstabilities

Babuska-Brezzi (inf-sup) condition needs to be satisfied

supvh∈Xh

(qh,∇ · vh)

‖vh‖1,2,Ω≥ c‖qh‖0,2,Ω

very high Reynolds numbers → convection dominatedflows

Re locK =

h‖v‖Kν

> 1

⇒ use Galerkin/Least-Squares stabilization


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Spatial discretization

Stabilization ψ = (w · ∇)ϕ+∇q

L(U,V ) =∑K

δK

(3

2∆tv − ν4v + (w · ∇) v +∇p, ψ

)K

,

F(V ) =∑K

δK

(4vn − vn−1

2∆t, ψ

)K

,

Stabilized problem

Galerkin terms, GALS stabilization, grad-div stabilization

a(U,V )+L(U,V )+∑K∈τh

τK

(∇ · v,∇ · ϕ

)= f (V )+F(V ).


Svacek, P.

Outline

Introduction

MathematicalModel


TimeDiscretization


NumericalResults

Conclusions

Other topics

solution of nonlinear problem (NS) - linearization

solution of linear problem (UMFPACK)

approximation of ODEs

interface conditions - coupling of fluid-structure models


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Fluid FlowApproximation

FluidApproximationover MovingStructure

AeroelasticSimulations

Conclusions

Numerical Results

Fluid Flow Approximation (fixed structure)

Fluid Flow over Moving structure (validation)

Aeroelastic Simulations


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Numerical Results

Fluid Flow Approximation (fixed structure)

Fluid Flow over Moving structure (validation)

Aeroelastic Simulations compare to NASTRAN


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Boundary layer approximation

Re = 2 · 105, comparison with the analytical Blasius solution


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Approximation of Boundary Layer - Taylor Hood

laminar flow (Re = 2 · 105)

FE Dimension: 16683 x 2 + 4242 = 37608

Nodes: 4242

Elements: 8200

1 1

224

5

3

1


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Blasius solution - Taylor-Hood


FE Dimension: 16683 x 2 + 4242 = 37608

Nodes: 4242 Elements: 8200

Can we improve that?


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Blasius solution - Taylor-Hood


FE Dimension: 16683 x 2 + 4242 = 37608

Nodes: 4242 Elements: 8200

Can we improve that? anisotropic mesh adaptation, use ofhigher order FEs


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Blasius solution - P3/P2 elements


uniform p-distribution

FE Dimension: 4012 x 2 + 1809 = 9833

Nodes: 472

Elements: 866


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Blasius solution - P4/P3 elements


FE Dimension: 2x13235 (Velocity) + 7483 (Pressure) =33953

Nodes: 866, Elements: 1629


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Flow over NACA 632 − 415

Fluid velocity isolines, Re = 5 · 105, AVI format


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


Fluid velocity isolines, Re = 5 · 105 AVI format


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


aerodynamical lift coefficient (time averaged values)

comparison with experimental data for NACA 632 − 415

−8 −6 −4 −2 0 2 4 6 8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

CL

α [°]


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


aerodynamical moment coefficient (time averaged values)


−8 −6 −4 −2 0 2 4 6 8−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

CM

α [°]


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


aerodynamical lift coefficient (time averaged values)


−0.5 0 0.5 1−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

CD

α [°]


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Prescribed vibrations of α, 30 Hz, amplitude 3,2,1degrees.

0 0.02 0.04 0.06 0.08 0.1 0.12−3

−2

−1

0

1

2

3

1

2

3

4

5

6

7

8

9

10

11

12

13

time [sec]

alph

a[de

gree

s]


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Prescribed vibrations of α, 30 Hz, amplitude 1degree.

t[s]

c p

0.2 0.25

-0.5

-0.25

0

0.25

t[s]

c p

0.2 0.25

-0.5

-0.25

0

0.25

Pressure coeficient (up/down) at x/c = 0.15 - dependence ontime.


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Pressure Coefficient

What is pressure coefficient?

cp =p − p012ρU

2∞

for prescribed vibrations

α = α0 · sin(2πft)

the pressure at airfoil surface is expected to behave like

cp = cmeanp + c ′p sin(2πft) + c ′′p cos(2πft)

comparison with experimental dataBenetka, J. et al, Tech. report 3418/02, ARTI, 2002 ,Triebstein, H., 1986., J. Aircraft 23.


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Comparison of cmeanp with experimental data

x/c

c p

0 0.25 0.5 0.75

-0.8

-0.6

-0.4

-0.2

0

0.2

Comparison of cmeanp with experimental data


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Comparison of c ′p with experimental data

x/c

c p’

0 0.25 0.5 0.75

-24

-20

-16

-12

-8

-4

0

Comparison of c ′p with experimental data


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Comparison of c ′′p with experimental data

x/c

c p’’

0 0.25 0.5 0.75

0

4

Comparison of c ′′p with experimental data


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Stall Flutter Approximation

α = (10 + 10 · sin(2πft)),Re = 5000

Naudasher, E., Rockwell, D., Flow-Induced Vibrations, 1994


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Stall Flutter Approximation

α = (10 + 10 · sin(2πft)),Re = 5000

α = 10.859 α = 18.3407 α = 19.9988

α = 19.4553 α = 12.045 α = 8.06Naudasher, E., Rockwell, D., Flow-Induced Vibrations, 1994


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Stall Flutter

Fluid velocity isolines, Re = 5 · 105 AVI(1) format


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Aeroelastic simulations - laminar

t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-60

-40

-20

0

20

40

60

U∞ = 4 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-60

-40

-20

0

20

40

60

U∞ = 8 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-60

-40

-20

0

20

40

60

U∞ = 12 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-60

-40

-20

0

20

40

60

U∞ = 16 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-60

-40

-20

0

20

40

60

U∞ = 18 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0

-60

-40

-20

0

20

40

60

U∞ = 22 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0

-60

-40

-20

0

20

40

60

U∞ = 26 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0-6

-4

-2

0

2

4

6

t [s]h

[mm

]0 0.1 0.2 0.3 0

-60

-40

-20

0

20

40

60

U∞ = 28 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-8

-6

-4

-2

0

2

4

6

8

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-80

-60

-40

-20

0

20

40

60

80

U∞ = 32 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


t [s]

α

0 0.1 0.2 0.3 0.4 0.5 0-8

-6

-4

-2

0

2

4

6

8

t [s]h

[mm

]0 0.1 0.2 0.3 0.4 0.5 0

-80

-60

-40

-20

0

20

40

60

80

U∞ = 34 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


U∞ = 40 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


U∞ = 45 m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Aeroelastic model - Turbulent Flow

flexibly supported airfoil NACA 0012

RANS + Spallart-Almaras turbulence model

NASTRAN computation with the STRIP model criticalspeed U∞ = 37.7m/s

frequencies and damping comparison


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


Solution of the coupled aeroelastic model (h,α), U = 5m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


Solution of the coupled aeroelastic model (h,α), U = 7.5m/s


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions




Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions


U=37 m/s

velocity isolines, AVI format


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Comparison with NASTRAN computation

Frequencies comparison


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults




Conclusions

Comparison with NASTRAN computation

Damping


Svacek, P.

Outline

Introduction

MathematicalModel


NumericalResults

Conclusions

Summary

NS solver numerical results were compared toexperimental/computational data.

NS solver and RANS solver results for aeroelastic problemwere compared each to other.

Performence: RANS x NS (?)

Conclusion

RANS and NS shows good agreement with NASTRANcomputations.

We must provide: careful mesh design, time step value, ...

How to increase reliability?


Svacek, P.

Can the grid motion “pollute” the solution?


Svacek, P.

Can the grid motion “pollute” the solution?

AVI format Test Problem: constant fluid velocity v = (1, 0)on rectangle.


Svacek, P.

Use anallytical derivative of ALE mapping ...

wg =∂x

∂t

Approximately 10 % error (!) AVI format


Svacek, P.

Use finite difference formula ...

wg ≈3x (n+1) − x(n) + x(n−1)

2∆t

Grid Motion Influence on Solution (finite differenceapproximation of grid velocity)AVI format


Svacek, P.

The ALE derivative approximation

use vn−1 - defined on Ωn−1

vn - defined on Ωn

and vn+1 - defined on Ωn+1

DAv

Dt≈ 3vn+1 − 4vn + vn−1

2∆t

where vn and vn−1 lives on Ωn+1


Svacek, P.

Spatial discretization

Stabilization ψ = (w · ∇)ϕ+∇q

L(U,V ) =∑K

δK

(3

2τv − ν4v + (w · ∇) v +∇p, ψ

)K

,

F(V ) =∑K

δK

(4vn − vn−1

2τ, ψ

)K

,

Stabilized problem

Gelhard, T., Lube, G., Olshanskii, M. A., 2004. Stabilizedfinite element schemes with BB-stable elements forincompressible flows. Journal of Computational andApplied Mathematics (accepted).

Gelhard, T., Lube, G., Olshanskii, M. A., 2004. Stabilizedfinite element schemes with LBB-stable elements forincompressible flows. Journal of Computational andApplied Mathematics .

outline on reliability of higher order ﬁnite element...

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