galerkin-characteristics finite element methods for flow … · 2013. 11. 14. · 4 7 finite...

1

Galerkin-Characteristics Finite Element Methods for Flow

Problems, I

Masahisa TabataWaseda University, Tokyo, Japan

1

The 9th Japanese-German International Workshop on Mathematical Fluid Dynamics, November 5-8, 2013, Waseda University, Tokyo, Japan

2

Contents

Galerkin-characteristics FEM and numerical analysis of flow problems

Convergence analysis of

① the scheme for the convection-diffusion equation

② the scheme for the Oseen equations

③ the scheme for the Navier-Stokes equations

④ 2nd order schemes in time and a stabilized scheme for the Navier-Stokes equations

2

Convection-dominated phenomenon

3

4

An example of convection-diffusion problem

Find : such that u R

where

0.1, 0.01, 0.001 Pe 10, 100, 1000

w u u f

0u

(0,0)

(1,1)

(1,0), 1w f 1

i.e., 1u

ux

1 2 1, i.e., ,u x x x

2: :given functionw

, wu v u v dxua

0w

3

5

GK

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 250.5

0.751

210

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 250.5

0.751

0

0

0.25

0.5

0.75

1 0

1

2

110 310

6

T[1977], Pironneau-T[2010]

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 250.5

0.751

210

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 250.5

0.751

310

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

0 250.5

0.751

110

4

7

Finite Element Schemes for Flow ProblemsUpwind approximation:

Characteristic method:

upwind element choice approximati T[1on, 977]mass-conservative upwind approximation, Baba-T[1981]streamline upwind Petrov/Galerkin approximation

first order in t

second order in t

Brooks-Hughes[1982], Johnson, Franca,...

Ewing-Russel[1981]Boukir-Maday-Metivet-Razafindrakoto[1997]

multi-step method:

single-step method: Rui-T[2002]

dicscontinuous Galerkin methodLesaint-Raviart[1974], Johson, Brezzi-Marini-Sulli,...

Notsu-T[2, 009]

Notsta su-bilize T[2d: 013], Douglas-Russel[19Pironne 82],au[1982] Suli[1988]

Pironneau-monot T[20one: 10],

Galerkin-Characteristics FEM Method

8

5

Galerkin Method

9

s.t.: Find f x

0 x 0

f

10; 1, ,h jV j N h V H

Find s.t.h hV

h h h h hdx f dx V

Note. Ritz-Galerkin method, Petrov-Galerkin method

Method of Characteristics

10

Find : 0, ,d T

0, 0 , ,0du f x tt

0where : 0, , : 0, , : .d d d du T f T

Find ( , ) : 0, ,dX T

0 0; , ;X t x t t x

, , , 0dX d

u X t f X t tdt dt

00 00 , 0X x x

0x

0;x X t xt

characteristics

6

Approximation of the Material Derivative

11

: (0, ) dX T R

( ), ( ( ), )D d

X t t X t tDt dt

Material derivative:

( ( ), ) ( ( ), )( ( ), )

D X t t X t t t tX t t

Dt t

• Approximation of material derivative:

( ) ( ( ), ) ,dX

t u X t tdt

),0( Tt

Du

Dt t

Position of a fluid particle:

t X tx

Idea of the Galerkin-Characteristics FEM

12

( ( ; ), ), , ,

D X t t x t t

Dt t

Du

Dt t

, ,dX

u X s s tds

( ; )X t x x

wheret

;X s x

xs

t t

7

13

The pros and cons The matrices are symmetric and independent of time

step for convection-diffusion, Oseen, and Navier-Stokes problems; positive definite for CD.

The schemes are unconditionally stable and convergent for CD and Os, conditionally stable and convergent for NS.

The schemes are robust for convection-dominated problems. c of -estimate is independent of ν for CD.

The attention should be paid for the computation of composite function terms, 1

1n nh hiX dx

1 is not smooth on !n n

h hix t u x x K K

2L

I. Convection-diffusion equation

14

8

15

Convection-Diffusion Equation

0

in (0, )

0 on (0, )

at 0 in

u f Tt

T

t

such that ),0(: Find R T

0T( 2,3), boundedd d R

1, 2 0 20 , ,

du W f L L

where

16

Weak Formulation

0

, , , ,

0

Da f V

Dt

Find : (0, ) such that T V 1

0V H

,f g f g dx

uD Du

Dt Dt t

: material derivative

where

, ,a

9

1st order scheme in time

17

1

0 0

, ( , ) ( , ),n n n

n nh hh h h h h h h

Ph h

Xa f V

t

where ( ) ( )n nX x x u x t

such that ,,...,1 , Find Thnh NnV

tn

tn 1

x

xX n1

tO

:time increment, /Tt N T t : FE-spacehV V

composite function

Note. Backward Euler method

1 11Note. ( ) ( )n n n n

h hX x x u x t

Note. : Poisson projectionPh

18

Theorem

-elementkP

2 2 2, k

h hL Lc t h

where

max ; 0,..., ,nh h TX X

n N

1st order scheme in time (cont.)

: Poisson projection of order hV k

1,

1

W

tu

2

1/22

1 TN n

h hnX Xt

, , 0c u T

10

Framework for the convection-diffusion equation

19

10 , : bdd. , 2,3dV H d

Hilbert space.

FEM space.

Bilinear for ,m. :a V V

,a dx

,h hV QV Q

0:regular, inverse ineq.h h

T 0, , diam, h KK Kh T

: radius of the inscribe ballK 2 12 2, 0, , , diamh c h K c hc c h K T

Poisson projection

20

,:Ph hV V ,ˆP

h h

ˆ , , , hh h hha a V

Ex. : -element, 1h kV P k

1ˆ, 0, k

kh HV

c k c h

1 ,. .,

k

P kh H V

i ce I h

L

11

Plan of the proof

21

Poisson projection

Transformation :nX

Discrete Gronwall's inequality

Error equation in Ph h he

Ph h he I

Transformation

22

n nX x x u x t :nX

Lemma

X x x w x t 1,

0

dw W

1, 1 1 : , 1to 1, ontoW

w t X

1, 2 , 0,1 ,W

w t 1

11

X

x

Ref. Rui-T[Prop. 1, 2002]

12

23

, dist ,x d x x

1, 1 1 : , 1to 1,ontoW

w t X

X x x w x t

w x w y t x

y

X x x w x w y t

x y w t 1,W

x y w t d x X x

2 i i iX x x w x t , , i j ij i jX w x t

, , det deti j ij i j

XX w x t

x

d x

,x X x

A tool for the proof

24

Lemma X x x w x t 1,

0 ,d

w W

221n

LLX c t 0,c

1,0,1 ,W

w t

2

22n n

LX X x dx

2 nxy dy y X x

y

21 c t y dy

2

21

Lc t

13

Error equation in

25

1

satisfies TNn

h ne

1

, , , , n n n

n nh hh h h h h h h

e e Xa e R V

t

0 0he

1 2n n nh h hR R R

1

2 ,n n n

n h hh

XR

t

where

ˆ : 0,h h T

1

1 ,n n n n

nh

D XR

Dt t

Lemma

ˆ ˆ, Ph h h h he

26

0 0 0 0 0ˆ ˆ 0Ph h h h he

1

, , ,n n n

n nh hh h h h

Xa f

t

ˆ, , ,n

n nh h h

Da f

Dt

: solution of CD, Poisson projection

1ˆ ˆn n n nnh

D XR

Dt t

1

1 : truncation errorn n n n

nh

D XR

Dt t

1

2 :projection errorn n n

nh hh

XR

t

1

, , ,n n n

n nh hh h h h h

e e Xa e R

t

14

Estimates of the remainders

27

1 01 ,n

hR c u t

2 02 ,n k

hR c u h

Lemma

28

1

1

n n nnnh

x X xDR x x

Dt t

11

0

1 11 , 1n n n n

ns

x X x x s u x t t s tt t

( ) ( )n nX x x u x t

1

01 , 1n n

t nsu x x s u x t t s t ds

1 ,nh t nR x u x t

1

01 , 1n n

t nsu x x s u x t t s t ds

1

11

1 10 01 1 , 1 1n n

t ns su x x s s u x t t s s t ds

1

1 1

, , ,0 01 2n n n

j k jk j jt tts st s ds u x u x u x

1 1 11 1 , 1 1nnx s s u x t t s s t ds

1 0,n

hR c u t

15

29

1

2

n n nn h hh

XR

t

1

20

11 , 1n n

h h ns

R x x s u x t t s tt

1

01 , 1n n

h t h nsu x x s u x t t s t ds

1/21 2

2 0 01 , 1n n n

h h t h nsR dx u x x s u x t t s t ds

12 2

2 01 , 1n n n

h h t h nsR x u x x s u x t t s t ds

1/21 2

01 , 1n n

h t h nsds u x x s u x t t s t dx

1/2

1 2

0, 1n

h t h ns

xds u x y t s t dy

y

1 ny x s u x t 1

1/22 2

0 0

n

n

t

h t htc u dt

1nt t s t

, kc u h

Discrete energy inequality in

30

0

nhe

2 2

0 0

n nt h hD e e 2 21 2 2

2 20 0, ,n n k

h he c u e c u h t

Lemma

1, , Tn N 0 0he

1

, , , , n n n

n nh hh h h h h h h

e e Xa e R V

t

Substitute into .nh he

1

, , , , 1, , n n n

n n n n nh hh h h h h T

e e Xe a e e R e n N

t

2 2 211 20 0 0 0 0 0

1

2n n n n n n nh h h h h he e X e R R e

t

Proof.

2 2 2 21 2 21 20 0 0 0

2 , ,n n n n kt h h h hD e e e c u e c u h t

16

Discrete Gronwall’s inequality

31

0 1 1 , 1, ,t n n n n nD x y a x a x b n N 0 10 1 1

, , , , , , , 0N N N

n n n n n nn n nx y b a a x y b

Lemma

1 :backward differencen nt n

x xD x

t

0

1(0, ]

2t

a

0 1201 1

, 1, ,n na a n t

n j jj jx t y e x t b n N

Proof.1

0 1 1 , 1, ,n nn n n n

x xy a x a x b n N

t

0 1 11 1 , 1, ,n n n na t x y t a t x b t n N

The result is proved by induction.

Proof of Theorem

32

2 2

0 0

n nt h hD e e

1, , Tn N 0 0he

0

Discrte energy inequality :

Apply the discrte Gronwall's inequality for

1,

1 1

2W

tu

2 2 2

2 2 2 2, ,n kh hl L l L

e e c u T h t

2 2 2, ,n k

h hl L l Le e c u T h t

2 21 2 22 20 0

, ,n n kh he c u e c u h t

1


Problems, II


1


II. Oseen Equations

2

2

3

Oseen Equations

0

( ) in (0, )

0 in (0, )

0 on (0, )

at 0 in

uw u u p f T

tu T

u T

u u t

such that ),0(:, Find RR dTpu


where 1, 2 0 1 0

0 0, , , 0d d d

w W f L u H u

1, 2 , ,

2ji

ijj i

vva u v D u D v D v

x x

4

Weak formulation

0

, , , , ,

, =0,

0

Duv a u v b v p f v v V

Dt

b u q q Q

u u

Find , : (0, ) such that u p T V Q 1 2

0 0,d

V H Q L

, ,b u q u q

where

wDu D u u

w uDt Dt t

3

5


1

0 0

1

, , , , ,

, =0,

,0

n n nn n nh h

h h h h h h h h h

nh h h h

Sh h

u u Xv a u v b v p f v v V

t

b u q q Q

u u

, : FE-spaceh hV V Q Q

:time increment, /Tt N T t

Find , , 1,..., , such that n nh h h h Tu p V Q n N

( ) ( )n nX x x w x t where

6

Theorem

1,

1

W

tw

where

2

1/22

1 TN n

h hnX Xv t v

, , , , 0,c T w u p

/ : Stokes projection of order h hV Q k

max ; 0,..., ,nh h TX X

v v n N


1 2 2, k

h hH Lu u p p c t h

4

Framework for the Oseen equations

7

10 , 2,3, : bdd. dd

V H d 0

2 ; 0q L q x dxQ L

Hilbert spaces.

FEM spaces.


T

Bilinear forms. ,: :a V V b V Q , ,

1,

2ij i j j iD v v v

, ·b v q q dv x

, 2 :a u v D u D v dx

,h hV QV Q

strain rate tensor

0, , diam, h KK Kh T : radius of the inscribe ballK

2 12 2, 0, , , diamh c h K c hc c h K T

Stokes projection

8

,:Sh h hQ QV V ˆ ˆ, ,S

h h hu p u p

ˆ ˆ, , , , ,h h h hh h hha u v b v p a u vb Vv v p ˆ , , , h hh h hb u q b u q q Q

Ex. 1 / : 2 / 1, Taylor-Hood element, 2h hV Q P P k

1ˆ ˆ, ,, 0, k k

kh h V Q H H

u u p pk c uc p h

1 ,. .,

k k

S kh H H V Q

i hce I

L

Ex. 2 / : 1 bubble / 1, MINI element, 1h hV Q P P k Note. / : 1 / 0 does not work. h hV Q P P

5

Stokes projection (cont.)

9

Key condition for the proof: Inf-sup condition.

0 0

,0, inf sup

h h h h

h hq Q v V

h hQ V

b v q

q v

Taylor-Hood element satisfies this condition.

Reference.[1]Brezzi, F., Fortin, M., Mixed and hypbrid finite element methods, Springer, New York, 1991.

MINI element satisfies this condition.

[2]Tabata, M., Numerical analyis of partial differential equations, Iwanami, Tokyo, 2010.

Stokes problem (d=2)

:),( hhhh QVpu spaceselement finite

2:),( Find hhh pu jhh j ,3,2,1;max )(

hh

jK

jh

6

P2/P1 space

hhhhh QVpuu ),,( 21

2d

1hu 2hu hp

2221

2121|2|1 ,,,,,1, xxxxxxuu

jj KhKh 21| ,,1 xxpjKh

spaceelement finite:

Note. Taylor-Hood element, 2k

P1+/P1 space

hhhhh QVpuu ),,( 21

2d

1hu 2hu hp

32121|2|1 ,,,1, xxuujj KhKh 21| ,,1 xxp

jKh

spaceelement finite:

1 2 3Note. MINI element, 1; : bubble functionk

7

Plan of the proof

13

Stokes projection


Transformation :nX

, , ,Sh h h h hu u p p e I u p

Error equation in , , ,Sh h h h he u p u p

Transformation

14

n nX x x w x t

:nX

Lemma

X x x w x t 1,0 ,

dw W

1, 1 1 : , 1to 1, ontoW

w t X

1, 2 , 0,1 ,W

w t 1

11

X

x

1

0 01 ,

LX t w H

1,

0 0,

LX t w W

8

Tools for the proof

15

Korn's inequality

Poincare's inequality

1

0 100, ,

dc D v v c v v H

100 0

0, ,c v c v v H

11 2 1 2 01 10, 0, ,

dc c c v D v c v v H

Error equation 1 in

16

ˆ ˆ ˆ ˆ, , , , , , satisfies Sh h h h h h h h he u u p p u p u p

,h he

1

, , , , , n n n

n n nh hh h h h h h h h h

e e Xv a e v b v R v v V

t

, =0, , 1, ,nh h h h Tb e q q Q n N

0 0

10,S

h he p

1 2,n n nh h hR R R

1

1 ,n n n n

nh

Du u u XR

Dt t

1

2 ,n n n

n h hh

XR

t

where

ˆ : 0, ,dh hu u T n nX x x wt x

9

17

ˆ, = , , =0 -0=0.n n nh h h h h hb e q b u q b u q

0 0 0 0 0 0 0

1 1 1ˆ ,0 , 0,S S S

h h h h h he u u u u p p

1

, , , ,n n n

n n nh hh h h h h h

u u Xv a u v b v p f v

t

ˆ ˆ, , , ,n

n n nh h h h

Duv a u v b v p f v

Dt

, : solution of NS, Stokes projectionu p

1ˆ ˆn n n nnh

Du u u XR

Dt t

1


nh

Du u u XR

Dt t

1


nh hh

XR

t

Error equation 2 in

18


,h he

1

3, , , , , n n

n n n nh hh h h h h h h h h h

e ev a e v b v R R v v V

t

, =0, , 1, ,nh h h h Tb e q q Q n N

0 0

10,S

h he p

1 2 ,n n nh h hR R R

1

1 ,n n n n

nh

Du u u XR

Dt t

1

2 ,n n n

n h hh

XR

t

where

ˆ : 0, ,dh hu u T n nX x x wt x

1 1

3

n n nn h hh

e e XR

t

10


19

1 0,n

hR c w u t

2 0, , ,n k

hR c w u p h

Lemma

13 0 0

n nh hR c w e

20

1

1

n n nnnh

u x u X xDuR x x

Dt t

11

0

1 11 , 1n n n n

ns

u x u X x u x s w x t t s tt t

( ) ( )n nX x x w x t 1

01 , 1n n

t nsw x u u x s w x t t s t ds

1 ,nh t nR x w u u x t

1

01 , 1n n

t nsw x u u x s w x t t s t ds

1

11

1 10 01 1 , 1 1n n

t ns sw x u u x s s w x t t s s t ds

1 1 11 1 , 1 1nnx s s w x t t s s t ds

1 0,n

hR c w u t

1

1 1

, , ,0 01 2n n n

j k jk j jt tts st s ds w x w x u w x u u

11

21

1

2 2 :projection errorn n n

n nh hh h

XR R

t

1

20

11 , 1n n

h h ns

R x x s w x t t s tt

1

01 , 1n n

h t h nsw x x s w x t t s t ds

1/21 2

2 0 01 , 1n n n

h h t h nsR dx w x x s w x t t s t ds

12 2

2 01 , 1n n n

h h t h nsR x w x x s w x t t s t ds

1/21 2

01 , 1n n

h t h nsds w x x s w x t t s t dx

1

1/22 2

0 0

n

n

t

h t htc w dt

1nt t s t , , , kc w u p h

1/2

1 2

0, 1n

h t h ns

xds w x y t s t dy

y

1 ny x s w x t

1 1

3 ,n n n

n h hh

e e XR

t

1

3 0 0L

n n nh hR c w e


22

0

nhD e

Lemma

1, , Tn N

2 2

00

1

2n n

t h t hD D e D e

21 2 2

1 20

, , ,n khc w D e c w u p h t

0 0

0

kh k

D e c p h

12

23

Substitute into in the error equation 2. nt h hD e v

3, , ,n n n n n n nt h t h h t h h h t hD e D e a e D e R R D e

Proof.

ˆ, , , 0 0 0, 0n n n n n nh h h h h hb e b u b u n

22

1 2 30 0 0 0 00

n n n n n nt h t h h h h t hD e D D e R R R D e

, 1n

2 22 1 2 21 20 0 0

1, , ,

2n n n k

t h t h hD e D D e c w D e c w u p h t

0 0

10,S

h he p 0 0

1

kh k

e c p h

0

10,S

h I p

0 0

0

kh k

D e c p h

Proof of Theorem

24

1, , Tn N

Discrte energy inequality :

Apply the discrte Gronwall's inequality.

2 22

2 2 2 21, , , ,

2n k

h t h l Ll LD e D e c w u p T h t

2 2

00

1

2n n

t h t hD D e D e

21 2 2

1 20

, , ,n khc w D e c w u p h t

0

0

khD e c p h

2 22

1, , , ,

2n k

h t h l Ll LD e D e c w u p T h t

13

25

3

1sup , , ,

h h

n n n nv V h h h t h h h h

h V

c R R v D e v a e vv

Estimate of the pressure.

0

,sup

h h

nh hn

h v Vh V

b vc

v

30 0 0 02n n n n

h h t h hc R R D e D e

1

0 0 0, , , 2k n n n

h t h hc w u p h t e D e D e

0, , , k n

t hc w u p h t D e

2 2 2 2, , , , k n

h t hL Lc w u p T h t D e

, , , , kc w u p T h t

1


Problems, III


1


III. Navier-Stokes Equations

2

2

3

Navier-Stokes Equations

in 0at

),0(on 0

),0(in 0

),0(in )(

0 tuu

Tu

Tu

Tfpuuut

u



where 2 0 1, 00, , 0

d df L u W u

1, 2 , ,

2ji

ijj i

vva u v D u D v D v

x x

4

Weak formulation

0

, , , , ,

, =0,

0


Dt

b u q q Q

u u


0 0,d

V H Q L

, ,b u q u q

where

,u wDu D u D u u

w uDt Dt Dt t

3

5


1 1

0 0

1

, , , , ,

, =0,

,0

n n nn n nh h h

h h h h h h h h h

nh h h h

Sh h


t

b u q q Q

u u

, : FE-spaceh hV V Q Q

:time increment, /Tt N T t


1 1( ) ( )n nh hX x x u x t

where

Note. Pironneau[1982]

6

Theorem/4

0 0 0 0, 0, , dh c h h t c h

where

2

1/22

1 TN n

h hnX Xv t v

Note. Süli[1988]

, , , 0,c T u p


max ; 0,..., ,nh h TX X

v v n N


1 2 2, k

h hH Lu u p p c t h

4

Framework for the Navier-Stokes equations

7

10 , 2,3, : bdd. dd

V H d

02 ; 0q L q x dxQ L

Hilbert spaces.

FEM spaces.


T

Bilinear forms. ,: :a V V b V Q , ,

1,

2ij i j j iD v v v

, ·b v q q dv x


,h hV QV Q

strain rate tensor



Stokes projection

8


h h hu p u p

ˆ ˆ, , , , ,h h h hh h hha u v b v p a u vb Vv v p ˆ , , , h hh h hb u q b u q q Q

Ex. 1 / : 2 / 1, Taylor-Hood element, 2h hV Q P P k

1ˆ ˆ, ,, 0, k k

kh h V Q H H

u u p pk c uc p h

1 ,. .,

k k

S kh H H V Q

i hce I

L

Ex. 2 / : 1 bubble / 1, MINI element, 1h hV Q P P k

5

Plan of the proof

9

Stokes projection



1,

1Induction is employed to evaluate and nh W

t u

1 at each step. n

h Lu


New terms to be evaluated

10

1The transformation depends on the function .nhu

1 1= n nh hX x x u xt

1,

1The estimate is requied. nh W

t u

1 appears as coefficients in the error equation.nhu

1The estimate is requied. nh L

u

6

Tools for the proof

11

Korn's inequality

Poincare's inequality 1

0 100, ,

dc D v v c v v H

100 0

0, ,c v c v v H 1

1 2 1 2 01 10, 0, ,

dc c c v D v c v v H

Inverse inequality / /, 0, ,q p

d q d ph h h hL L

c p q v ch v v V 1 p q

Sobolev's imbedding ,

,0, ,q m p

m p

L Wc v c v v W ,

dpq

d mp

Lagrang interpolation operator :h hV V 1

10, ,k

k kh V H

c v v ch v v H

1, 1,

1,0, , ,h W Wc v c v v W

,h L L

v v v L

Error equation1 in

12


,h he

1 1

, , , , , n n n

n n nh h hh h h h h h h h h

e e Xv a e v b v R v v V

t

0 0

10,S

h he p

1 2 3,n n n nh h h hR R R R

1 1

1

n n n nnh

Du u u XR

Dt t

1 1

2 ,n n n

n h h hh

XR

t

1 1 1 1

3

n n n nn hh

u X u XR

t

where

ˆ : 0, ,dh hu u T 1 1n nX x x u xt

, =0, , 1, ,nh h h h Tb e q q Q n N

7

13


0 0 0 0 0 0 0

1 1 1ˆ ,0 , 0,S S S


1 1

, , , ,n n n

n n nh h hh h h h h h

u u Xv a u v b v p f v

t

ˆ ˆ, , , ,n

n n nh h h h

Duv a u v b v p f v

Dt


1 1ˆ ˆn n n nn hh

Du u u XR

Dt t

1 1


nh

Du u u XR

Dt t

1 1


nh h hh

XR

t

1 1 1 1

3 :perturbation errorn n n n

nhh

u X u XR

t

Error equation2 in

14


,h he

1

4, , , , , n n



t

, =0, , 1, ,nh h h h Tb e q q Q n N

0 0

10,S

h he p


1 1

1

n n n nnh

Du u u XR

Dt t

1 1

2 ,n n n

n h h hh

XR

t

where

ˆ : 0, ,dh hu u T 1 1 n nX x x ut x

1 1 1

4

n n nn h h hh

e e XR

t

1 1 1 1

3

n n n nn hh

u X u XR

t

8


15

1 0

nhR c u t

13 0 0

,n n kh hR c u p e h

Lemma

12 0

, , 1n n kh h L

R c u p u h

1 14 0 0

L

n n nh h hR c u e

1,

1 1nh W

u t

16

1 1

1

n n nnnh

u x u X xDuR x x

Dt t

1

1 1 1

0

1 11 , 1n n n n

ns

u x u X x u x s u x t t s tt t

1 1( ) ( )n nX x x u x t

1 1 1

01 , 1n n

t nsu x u u x s u x t t s t ds

112 ,n n

h t nR x u x u u x t

1 1 1

01 , 1n n

t nsu x u u x s u x t t s t ds

1

11 1 11 10 0

1 1 , 1 1n nt ns s

u x u u x s s u x t t s s t ds

1

1 1 1 1 1, , ,0 0

1 2n n nj k jk j jt tts s

t s ds u x u x u u x u u

11 1 11 1 , 1 1n

nx s s u x t t s s t ds 12 0

nhR c u t

11, ,n n n

h nR x u x u x u x t 1 11 12n n nh h hR x R x R x

11 0

nhR c u t

9

17

1 1

2 2 :projection errorn n n

n nh h hh h

XR R

t

11

20

11 , 1n n

h h h ns

R x x s u x t t s tt

1 1 1

01 , 1n n

h h t h h nsu x x s u x t t s t ds

1/21 21 12 0 0

1 , 1n n nh h h t h h ns

R dx u x x s u x t t s t ds

12 21 12 0

1 , 1n n nh h h t h h ns

R x u x x s u x t t s t ds

1/21 21 1

01 , 1n n

h h t h h nsds u x x s u x t t s t dx

1/2

1 21

0, 1n

h h t h ns

xds u x y t s t dy

y

11 nhy x s u x t

1

1/22 21

0 0

n

n

tnh h t hL t

c u dt

1nt t s t

1,S

t h h t tt I u p t 1, , 1n kh L

c u p u h

18

1 1 13 0 0

L

n n n nh hR c u u u

1 1 1 1

3 3 :perturbation errorn n n n

n nhh h

u X u XR R

t

1 1 1 1 1

0 0ˆ ˆ

L

n n n n nh h hc u u u u u

1

0, k n

hc u p h e

1 1

4 4 :convection termn n n

n nh h hh h

e e XR R

t

1 14 0 0

L

n n nh h hR c u e

10


19

0

nhD e

Lemma

2 2

00

1

2n n

t h t hD D e D e

03

02 k

hD e c p h

21 1 1 2 2

1 20

, , , , , ,n n n kh h hL L

c u p u D e c u p u h t

1,

11 1, , 1nh TW

t u n N

20

1

4, , , , , n n



t

, =0, , 1, ,nh h h h Tb e q q Q n N

Substitute into .nt h hD e v

4, , ,n n n n n n nt h t h h t h h h t hD e D e a e D e R R D e

Proof of (1).

ˆ, , , 0 0 0, 0n n n n n nh h h h h hb e b u b u n

22

1 2 3 40 0 0 0 0 00

n n n n n n nt h t h h h h h t hD e D D e R R R R D e

, 1n

22 2 2 2 2

1 2 3 40 0 0 0 00

1

2n n n n n n

t h t h h h h hD e D D e R R R R

Error equation2:

11

21

1 1 1

0 0 0'n n n

h h he c e c D e

22 2 2 2 2

1 2 3 40 0 0 0 00

1

2n n n n n n

t h t h h h h hD e D D e R R R R

2 22 1 2 1 2

0, , 1 ,n k n k

h hLc u t c u p u h c u p e h

2 21 1

0

n nh hL

c u e

21 1 1 2 2

1 20

, , , , ,n n n kh h hL L


21 1 1 2 2

1 20

, , , , , ,n n n kh h hL L


0 0

1k

kh H

e c p h

0 0 0

1 12 0, 0,S S

h h he p I p

0 0

0, k

h kD e c p h

Proof of Theorem

22

We show the following by induction, 0, , :Tn N

From the discrete energy inequality,

2 21 1 1 2 21 21 1

; , , ; , ,n n n n kt h h h hL L

D e c u u p e c u u p h t

2 20 2 21 21 1

1 exp 1 1n kh hL L L Ln

e c u n t e c u h t n t

1,

1 1, , 1 implies nh TW

t u n N

1 2where , :monotone increasing w.r.t. the first variable. c c

1,2 n

hn Wt u

3 1nh L Ln L

u u

/40 0 0 0Choose small , 0. , .dh c h h t c h

12

23

1 1 1

General step, 1. Assume 1 , 2 and 3 are valid.n n n

n

1

1 From the discrete energy inequality and 3 , we obtain n n

1,

1 implies the existence of , .n n nh h hW

t u u p

2 21 1 1 2 21 21 1

; , , ; , ,n n n n kt h h h hL L

D e c u u p e c u u p h t

21 2 21 21

1 1 .n khL L

c u e c u h t

Applying the discrete Gronwall's inequality, we obtain

2 20 2 21 21 1

exp 1 1 .n kh hL L L L


Initial step, 0.n

01 LHS=RHS

1,

0

02 . Proved similarly to that of (2) .h nW

t u

0

03 1. Proved similarly to that of (3) .h nL LL

u u

24

1, 1, 1,2 n n n n

h h h hn W W Wu u u u

1 1,

/220 21

d n n nh h H W

c h u u c u

1,

/220 211 1 1

ˆ ˆd n n n n n n nh h h h W

c h u u u u u u c u

1,

/220 3 22 23 21

d k k k n

Wc h c t h c h c h c u

1,

/224 21

d k n

Wc h t h c u

1, 1,

/224 21

n d k nh W W

t u t c h t h c u

1,

/4 /40 24 0 24 21

k d d n

Wc c c c h c h u

1,

/2 2 /224 21

d k d n

Wc h t h t c t u

/4

0dt c h

1,

/4 /40 24 0 24 0 21 0

k d d n

Wc c c c h c h u

13

25

3 n n n nh h h hn L L L

u u u u

6

/630

d n n nh h L L

c h u u u

/630 1 1 1

ˆ ˆd n n n n n n nh h h h L

c h u u u u u u u

/630 3 31 32

d k k k n

Lc h c t h c h c h u

/6 /634

d k d n

Lc h t h u

/12 /634 0

d k d n

Lc c h h u

1 n

Lu

/40

dt c h

/12 /634 0 0 0

d k d n

Lc c h h u

26

2 20 2 21 21 1

1 exp 1 1n kh hL L L Ln


1,2 n

hn Wt u

3 1nh L Ln L

u u

The induction, 0, , ,Tn N

has been completed.

31, , , ,n k

he c T u p h t n

1 3 , , , .kh H

e c T u p h t

14

27

Now the discrete energy inequaliy

2 2

00

1

2n n

t h t hD D e D e

21 1 1 2 2

1 20

, , , , , ,n n n kh h hL L


can be written as

2 2

00

1

2n n

t h t hD D e D e

21 2 2

4 50

, , , , , , .n khc u p T D e c u p T h t

Applying the discrete Gronwall's inequaliy, we obtain

2 2 6 , , , .kt h L

D e c u p T h t

0Estimate of n

t hD e

28

4

1sup , , ,

h h

n n n nv V h h h t h h h h

h V

c R R v D e v a e vv

Estimate of the pressure.

0

,sup

h h

nh hn

h v Vh V

b vc

v

40 0 0 02n n n n

h h t h hc R R D e D e

1 1

0 0 0, , , 2n k n n n

h h t h hLc u p u h t e D e D e

0, , k n

t hc u p h t D e

2 2 2 2, , , k n

h t hL Lc u p T h t D e

, , , kc u p T h t

1


Problems, IV


1


IV. 2nd Order Schemes in Time and a Stabilized Scheme

2

2

2nd Order Schemes in Time

3

2nd order Galerkin-characteristics schemes

4

2step scheme

Singlestep scheme

1 2, 2, ,n n nh h h Tu u u n N 1 0, : initial functionsh hu u

1 2, ,n nh h Tu u n N 0 : initial functionhu

23 ( ) 4 ( ) ( 2 )f t f t t f t t dft O t

t dt

2( ) ( )

2

f t f t t df tt O t

t dt

Note. ( ) ( )f t f t t dft O t

t dt

3

5

Idea of 2nd order two-step scheme

1 2

21 23 4( )

2

nn n n n nX Xx u x O t

t t

1 ( ) ( )n nX x x u x t where

2 ( ) 2 ( )n nX x x u x t tn

tn 1

x

2n t 1 ( )nX x

2 ( )nX x

1 2*( ) 2 ( ) ( )n nu x u x u x

or 1 ( ) *( )nX x x u x t

2 ( ) 2 *( )nX x x u x t

Ewing-Russell[1981]

2Note. * ( ) ( ) ( )nu x u x O t

6


in 0at

),0(on 0

),0(in 0

),0(in )(

0 tuu

Tu

Tu

Tfpuuut

u



where 2 0 1, 00, , 0

d df L u W u

4

1, 2 , ,

2ji

ijj i

vva u v D u D v D v

x x

7

Weak formulation

0

, , , , ,

, =0,

0


Dt

b u q q Q

u u


0 0,d

V H Q L

, ,b u q u q

where

,u wDu D u D u u

w uDt Dt Dt t

8

2nd order two-step scheme

1 1 2 11 2

1

0 0

1

3 4, , , , ,

2

, =0,

: *

,0

n n n n nn n nh h h h h

h h h h h h h h h

nh h h h

h

Sh h

u u X u Xv a u v b v p f v v V

t

b u q q Q

u

u u

,h hV V Q Q :time increment, /Tt N T t


1 * * 1 21 ( ) ( ) , 2n n nh h h h hX x x u x t u u u

where

1 *2 ( ) 2 ( )n

h hX x x u x t

1 1 2Note. should be found by another method. khu u O t h

tn

tn 1

x

2n t 1

1 ( )nhX x

12 ( )n

hX x

5

9

Theorem/6

0 0 0 0, 0, , dh c h h t c h

where

2

1/22

1 TN n

h hnX Xv t v

Note. Boukir et al.[1997]

, , , 0,c T u p


max ; 0,..., ,nh h TX X

v v n N

2nd order two-step scheme (cont.)

1 2 2

2, kh hH L

u u p p c t h

Error equation1 in

10


,h he

1 1 2 1

1 23 4, , , , ,


h h h h h h h h h

e e X e Xv a e v b v R v v V

t

, =0, , 1, ,nh h h h Tb e q q Q n N

0 0

10,S

h he p


1 1 2 11 2

1

3 4n n n n n nnh

Du u u X u XR

Dt t

1 1 2 1

1 22

3 4n n n n nn h h h h hh

X XR

t

1 1 1 1 2 1 2 11 1 2 2

3 4n n n n n n n n

n h hh

u X u X u X u XR

t t

where

ˆ : 0, ,dh hu u T

1 2 khe O t h

1 * * 1 21 ( ) ( ) , 2n n nX x x u x t u u u

1 *2 ( ) 2 ( )nX x x u x t

6

11


0 0 0 0 0 0 0

1 1 1ˆ ,0 , 0,S S S


1 1 2 1

1 23 4, , , ,


h h h h h h

u u X u Xv a u v b v p f v

t

ˆ ˆ, , , ,n

n n nh h h h

Duv a u v b v p f v

Dt


1 1 2 11 2ˆ ˆ ˆ3 4n n n n n n

n h hh

Du u u X u XR

Dt t

1 1 2 1

1 21

3 4: truncation error

n n n n n nnh

Du u u X u XR

Dt t

1 1 2 1

1 22

3 4:projection error

n n n n nnh h h h hh

X XR

t

1 1 1 1 2 1 2 1

1 1 2 234 :perturbation error

n n n n n n n nnh hh

u X u X u X u XR

t t

Error equation2 in

12


,h he

1 2

4

3 4, , , , ,

n n nn n n nh h h

h h h h h h h h h h

e e ev a e v b v R R v v V

t

, =0, , 1, ,nh h h h Tb e q q Q n N

0 0

10,S

h he p


1 1 2 11 2

1

3 4n n n n n nnh

Du u u X u XR

Dt t

1 1 2 1

1 22

3 4n n n n nn h h h h hh

X XR

t

1 1 1 1 2 1 2 11 1 2 2

3 4n n n n n n n n

n h hh

u X u X u X u XR

t t

where

1 2 khe O t h

1 1 1 2 2 11 2

4 4n n n n n n

n h h h h h hh

e e X e e XR

t t

7


13

21 0

nhR c u t

13 0 0

,n n kh hR c u p e h

Lemma

12 0

, , 1n n kh h L

R c u p u h

1 14 0 0

L

n n nh h hR c u e

1,

1 1nh W

u t

14

1, 1, 1,2 n n n n

h h h hn W W Wu u u u

1 1,

/220 21

d n n nh h H W

c h u u c u

1,

/220 211 1 1

ˆ ˆd n n n n n n nh h h h W

c h u u u u u u c u

1,

/2 220 3 22 23 21

d k k k n

Wc h c t h c h c h c u

1,

/2 224 21

d k n

Wc h t h c u

1, 1,

/2 224 21

n d k nh W W

t u t c h t h c u

1,

2 /3 /60 24 0 24 21

k d d n

Wc c c c h c h u

1,

/2 3 /224 21

d k d n

Wc h t h t c t u

/6

0dt c h

1,

2 /3 /60 24 0 24 0 21 0

k d d n

Wc c c c h c h u

for the 2nd order scheme1,

Estimate of nh W

u

8

15

3 n n n nh h h hn L L L

u u u u

6

/630

d n n nh h L L

c h u u u

/630 1 1 1

ˆ ˆd n n n n n n nh h h h L

c h u u u u u u u

/6 230 3 31 32

d k k k n

Lc h c t h c h c h u

/6 2 /634

d k d n

Lc h t h u

2 /6 /634 0

d k d n

Lc c h h u

1 n

Lu

/60

dt c h

2 /6 /634 0 0 0

d k d n

Lc c h h u

for the 2nd order scheme

Estimate of nh L

u

16

2nd order one-step scheme

tn1

2n t

x

1n t

, 0, , 0f x t T xt

10, , ,f H

t

1

1/2 1/2 10, , ,

n nn nf H

t

1

1 11, , ,

2 2

n nn n n nh h

h h h h h h hf f Vt

2Crank-Nicolson scheme: t

9

17

Convection-Diffusion Equation

0

in (0, )

0 on (0, )

at 0 in

u f Tt

T

t

such that ),0(: Find R T


1, 2 0 20 , ,

du W f L L

where

18

Pure Crank-Nicholson approximation

1

1 12 1, , ,

2 2

n n nn n n nh h

h h h h h h h

Xf f

t

where

hh V

tt

xuxuxxX nnn

2)()( 2/1

2

00 hh


tn

tn 1

x

xX n2

12 ( ) ( )

2n n n nt

X x x u x u x tu x

Heun method

or 2nd order Runge-Kutta method

10

Pure Crank-Nicholson approximation (cont.)

19

)(2

)(2 tOxxXx n

The total accuracy is ! )( tO

1 1/ 2 2

2n n nx x O t

22/11

2

1tOxfxff nnn

1/ 21

22 2 ( )( ) ( )

2

nn n n nX x X xx u O t

t t

However, tn

tn 1x xX n

2

x

tO

This scheme is not of .2( ) O t

2O t

20

A Second Order Characteristic FEM (Rui-T[2002])

where

j

ni

ijn

x

uJ

][

hh V

txuxxX nn )()(1

tt

xuxuxxX nnn

2)()( 2/1

2

00 hh


tn

tn 1

x

xX n2 xX n

1

2tO

12 ( ) ( )

2n n n nt

X x x u x u x tu x

or

112

1

1 11

11

, ,2

, div ,2

1,

2

n n nn n nh h

h h h h

n n n n nh h h h

n n nh h h

XX

t

tJ X u

f f X

11

A Second Order Characteristic FEM (Cont.)

21

element-kP

where

2 1

2

1/221' 1

1

2T

n n nN

H nL

Xt

Theorem (Rui-T[2002])

k

HhLh htc

2'

122 ,

2 2 max ; 0,..., ,n

h h TL Ln N

, , 0c u T

Key Expression

22

11

1( )

2n nX

1

1 21

, 1

1

2 2

n ndjn n

i j i j i

utX O t

x x x

1 1/ 2 21 1

1 1+( )

2 2n n n n nX x X O t

tn

tn 1

x

xX n2 xX n

1

2tO

12

23

Example 1 (rotating Gaussian hill)

T ,)5.0,5.0()5.0,5.0(

2 10, ,u u x xt

2

1 2

| ( ) |( , , ) exp

4 4cx t x

x x tt t

3444 100.1,100.5,105.2,1025.1

)cossin,sincos()( 2121 txtxtxtxtx 0.01),0,25.0( cx

Exact solution:

Rotating Gaussian Hill

24

1x

2x

, : 0t t

( 0.5, 0.5)

(0.5,0.5)

13

25

Example 1 (cont.)

)5/(2 ,/2 NTtNh

Pure Crank-Nicolson scheme:

hn

hn

hhnh

nhh

nnh

nh ff

t

X ,

2

1,

2, 112

1

triangles2 ofunion a into divided is 2N

P1 element

O h t O t

2nd order scheme:

2h t , 32,48,64,80N

2 2O h t O t

112

1

1 11 1

, ,2

1, ,

2 2

n n nn n nh h

h h h h

n n n n n nh h h h h

XX

t

tJ X f f X

26

41025.1 4105.2 4100.5 3100.1

2 -error vs.L t : Pure Crank-Nicholson scheme: 2nd order characteristics scheme

2E2E

2E2E

t t t t

2Lh

14

27


in 0at

),0(on 0

),0(in 0

),0(in )(

0 tuu

Tu

Tu

Tfpuuut

u



where 2 0 1, 00, , 0

d df L u W u

28

2nd order single-step scheme, Notsu-T[2009]

1 11 12

1 1

1 1 1 11

( , ) 1, , ,

2

1 1, , ,

2 2

, =0,

n n n n nn n n n n nh h h h

h h h h h h h

T Tn n n n n n nh h h h h h h h h h

nh h

u u X u uv D u D u X D v v p p X

t

t J u J u p J u J v f f X v v V

b u q

0 0

h h

h h

q Q

u u

,h hV V Q Q :time increment, /Tt N T t


11 ( ) ( ) ,n n

hX x x u x t

where

2 , ( )2

n tX w u x x w x u x u x

tn

tn 1

x

xX n2 xX n

1

15

29

2nd order single-step scheme (cont.)

1 11 1

12

1 1

1 1 1 11

( , ) 1, , ,

2

1 1, , ,

2 2

, =0,

n n nn n n nh h

h h h h h

T Tn n n n n n nh h h h h h

hh h

h h h h

hh

hu X uv D D u X D v v p X

t

t J u J u p J u J v f f X v v V

b q

ww

wr

w

0 0

h h

h h

q Q

u u

0 1Internal iteration: nh hw u

Find , , 1,..., , such that l lh h h h Tw r V Q N

1, , ,n nh h h h hw w r u p

1Note. This scheme can be used to get in two-step method.hu

A stabilized Galerkin-characteristics scheme for the Navier-Stokes Equations

30

16

Stabilized P1/P1 scheme

31

P1/P1 element for NS equations.

Stabilizing term is necessary.P1/P1 element does not satisfy the inf-sup condition.

Especially useful for 3D computation.

Stabilized P1/P1 Galerkin-characteristics FE scheme.symmetric matrix, small DOF

less computation timefiner subdivision for 3D problems

32


in 0at

),0(on 0

),0(in 0

),0(in )(

0 tuu

Tu

Tu

Tfpuuut

u



where 2 0 1, 00, , 0

d df L u W u

17

1, 2 , ,

2ji

ijj i

vva u v D u D v D v

x x

33

Weak formulation

0

, , , , ,

, =0,

0


Dt

b u q q Q

u u


0 0,d

V H Q L

, ,b u q u q

where

,u wDu D u D u u

w uDt Dt Dt t

34

P1/P1 stabilized scheme

1 1

0 0

1

, , , , ,

, , =0,

,0

n n nn n nh h h

h h h h h h h h h

n nh h h h h h h

Sh h


t

b u q p q q Q

u u

-C

, : P1 FEspaceh hV V Q Q :time increment, /Tt N T t


1 1( ) ( )n nh hX x x u x t

where

Note. Notsu-T[2008]

20, , = ,h

h K Kkp q h p q

CT

18

35

Theorem/4

0 0 0 0, 0, , dh c h h t c h

where

2

1/22

1 TN n

h hnX Xv t v

Ref. Notsu-T[2013]

0, , , , 0,c T u p

/ : P1/P1 elementh hV Q

max ; 0,..., ,nh h TX X

v v n N

P1/P1 stabilized scheme (cont.)

1 2 2,h hH L

u u p p c t h

Framework for the Navier-Stokes equations

36

10 , 2,3, : bdd. dd

V H d 0

2 ; 0q L q x dxQ L

Hilbert spaces.

FEM spaces.


T

Bilinear forms. : :, : , hVV b QV QQa C

, ·b v q q dv x


,h hV QV Q



20, , = ,h

h K Kkp q h p q

CT

19

Stokes projection

37


h h hu p u p

ˆ ˆ, , , , ,h h h hh h hha u v b v p a u vb Vv v p ˆ ˆ, , , , ,h h h h h h h hh hb u q p q b u qp Qq q -C -C

Ref. Brezzi-Douglas[1988]

2 1ˆ0, ˆ, ,h h V Q H Hu u p p p hc uc

2 1 ,. ., S

h H H V Qi e cI h

L

Stokes projection (cont.)

38

Key condition for the proof: Stability inequality.

0 0,

, 0,

, , , ,inf sup

, ,h h h h h h h h

h h h h h h h h hu p V Q v q V Q

h h h hV Q V Q

a u v b v p b u q p q

u p v q

-C

Ref. Brezzi-Douglas[1988], Franca-Stenberg[1991]

Note.

0 0

,0, inf sup

h h h h

h hq Q v V

h hQ V

b v q

q v

Inf-sup condition,

does not hold for the P1/P1 element.

20

Outline of the proof

39

Stokes projection of the stabilized type



1,

1Induction is employed to evaluate and nh W

t u

1 at each step. n

h Lu


Refer to Notsu-T[2013] for the details.

Estimates of the term h C

Schemes free from numerical integration error

Mass lumping technique:

Pironneau-T[2010]

Finite difference method:

Notsu-Rui-T[2013]

Exact integration for approximated velocity:

T-Uchiumi[2013]

40

esitmateL

2discrete esitmateL

21

Galerkin-characteristics FEM of lumped mass type

41

11

, ( , ) ( , ),n n n

h h h n nh h h h h

I XI f

t

where

hh V

1 ( ) ( )n nX x x u x t

0 0h hI


:time increment, /Tt N T t : 1 FEM spacehV V P

: , , : node, interpolationh h hI C V I v P v P P

2: , , : node, lumpingh h h PV L v x v P x D P

A First Order Characteristic FEM (cont.)

42

Theorem

1-element, weakly acute type triangulationP

2

,h h L

hI c h t

t

Note. max ; 0,...,nh h TX X

n N

0,1 , 2; 0, 3d d

1, ,h h L

I c h t h

Pironneau-T[2010]

22

Application to a Two-Fluid Flow Problem

43

44

Multiphase flow problems with interface tension(1)

NS eqns

Interface condition: 0,i

u

0 t BC slip

1

2

TD u u u

Fluids 0,1, ..,m

,i i

0 0,

i t

n

2i

ipn D u n n

: coefficienti: curvature

or 0u

1,...,i m

1 t

m t i t

Note. i t 1,...,i m

23

45

Multiphase flow problems with interface tension(2)

fpIuDuut

ukkk

2)(

0 u

, satisfies NS eqns. in each domain , 0,...,ku p t k m

0Interface conditions on i it t t

,0u ,,0on conditionsBoundary T

0, 0 or 0u n D u n n u

0at conditions Initial t,0uu 00 , 0,...,k k k m

2i

ipn D u n n

Evolution of , ; [0,1)i it s t s

, , 1, ,iiu t i m

t

t�0.00000

46

Different density fluids in an “Hourglass”

32,N 1

8t

0.5,0

0.5, 2

60T

1

0nonslip BC

1 1 1, , 100,0.2,0.1

0 0, 1,1

0

2f

2 2 2 2, , 120,0.2,0.1

Ref. T[2010]

24

References M. Tabata. A finite element approximation corresponding to the upwind finite differencing.

Memoirs of Numerical Mathematics, 4(1977), 47-63.

R. E. Ewing and T. F. Russell. Multistep Galerkin methods along characteristics for convection-diffusion problems. In R. Vichnevetsky and R. S. Stepleman, editors, Advances in Computer Methods for Partial Differential Equations, 4, 28-36, IMACS, 1981.

O. Pironneau. On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numerische Mathematik, 38(1982), 309-332.

F. Brezzi and J. Jr. Douglas. Stabilized mixed methods for the Stokes problem. NumerischeMathematik, 53(1988), 225-235.

E. Suli. Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numerische Mathematik, 53(1988), 459-483.

L. P. Franca and R. Stenberg. Error analysis of some Galerkin least squares methods for the elasticity equations. SIAM Journal on Numerical Analysis, 28(1991), 1680-1697.

K. Boukir, Y. Maday, B. Metivet, and E. Razafindrakoto. A high-order characteristics/finite element method for the incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 25(1997), 1421-1454.

H. Rui and M. Tabata. A second order characteristic finite element scheme for convection-diffusion problems. Numerische Mathematik, 92(2002), 161-177.

H. Notsu and M. Tabata. A single-step characteristic-curve finite element scheme of second order in time for the incompressible Navier-Stokes equations. Journal of Scientific Computing, 38(2009), 1-14.

47

O. Pironneau and M. Tabata. Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type. International Journal for Numerical Methods in Fluids, 64(2010), 1240-1253.

M. Tabata. Numerical simulation of fluid movement in an hourglass by an energy-stable finite element scheme. In M. N. Hafez, K. Oshima, and D. Kwak, editors, Computational Fluid Dynamics Review 2010, 29-50. World Scientific, Singapore, 2010.

H. Notsu, H. Rui, and M. Tabata. Development and L2-analysis of a single-step characteristics finite difference scheme of second order in time for convection-diffusion problems. Journal of Algorithms & Technology, 7(2013), 343-380.

H. Notsu and M. Tabata. Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. WIAS-DP-2013-001, Waseda Univ., 2013.

H. Notsu and M. Tabata. Error estimates of a pressure-stabilized characteristics finite element scheme for the Navier-Stokes equations. WIAS-DP-2013-002, Waseda Univ., 2013.

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