Numerical Methods for Problems in Unbounded Domains Weizhu Bao Department of Mathematics & Center for Computational Science and Engineering National University.

Numerical Methods for Problems in Unbounded Domains

Weizhu Bao

Department of Mathematics& Center for Computational Science and Engineering

National University of SingaporeEmail: [email protected]

URL: http://www.math.nus.edu.sg/~bao

Collaborators:– H. Han, Z. Huang, X. Wen

Outline

MotivationDifferent approachesFor model problemsNew `optimal’ error estimatesExtension of the resultsApplication to Navier-Stokes equationsConclusion & Future challenges

Motivation

Problems in unbounded domains– Potential flow

– Wave propagation & radiation

– Linear/nonlinear optics

u

beam

Motivation

– Coupling of structures with foundation

– Fluid flow around obstacle or in channel

– Quantum physics & chemistry

u

dam

)(yu

Motivation

Numerical difficulties– Unboundedness of physical domain– Others

Classical numerical methods– Finite element method (FEM)– Finite difference method (FDM)– Finite volume method (FVM)

Linear/nonlinear system with infinite unknowns

i

Different Approaches

Integral equation– Boundary element method (BEM): Feng, Yu, Du, …

– Fast Multipole method (FMM): Roklin & Greengard, …

Infinite element method: Xathis, Ying, Han, …

Domain mappingPerfect matched layer (PML): Beranger

FEM with two different types basis functions:Spectral method: Shen, Guo, …

i

Artificial Boundary Conditions

Artificial boundary conditions (ABCs)– Introduce an artificial boundary– Engineers use

• Dirichlet or Neumann boundary condition on it

– Better way:• Solve on analytically• Design high-order ABC ( DtN ) on based on transmission conditions,

i.e. establish

– Reduce to – Solve the reduced problem by a classical method– How to design high-order ABCs & do error analysis ???

e

)()(:)( 2/12/1ee HHKuK

n

uee

ee

i

i

e

i R

e

Artificial Boundary Conditions

1&2D wave equation: Engquist & majda, 77’ 3D case Teng, 03’

Helmholtz equation in waveguides: Goldstein, 82’

Elliptic equations: Bayliss, Gunzburger & Turkel, 82’

Helmholtz equation (local ABC): Feng 84’

Laplace & Navier system: Han & Wu, 85’ & 92’, Yu 85’

Elliptic equations in a cylinder: Hagstrom & Keller, 86’

Linear advection diffusion equation: Halpern, 86’

Helmholtz equation (DtN): Givoli & Keller, 95’

Artificial Boundary Conditions

Stokes system: Bao & Han, 97’

Navier-Stokes equations: Halpern 89’; Bao, 95’, 97’, 00’

Linear Schrodinger equation: Arnold, 99’; NLS Besse 02’

Ginburg-Landau equation: Du & Wu, 99’

New `optimal’ error estimates: Bao & Han, 00’, 03’

Flow around a submerged body: Bao & Wen, 01’

Shrodinger-Poisson: Ben Abdallah 98’

Landau-Lipschitz: Bao & Wang,

Artificial Boundary Conditions

Types of artificial boundary– Circle– Straight line– Segments– Polygonal line– Elliptic curve

Artificial Boundary Conditions

Types of ABCs– Local: Dirichlet or Neumann

– Nonlocal: DtN boundary condition

– Discrete:

0or

ee n

uuu

eee

uKuKn

uN

mm xx

xx

xx

xx

u

u

A

n

u

n

u

11

e

Model Problem (I)

1D problem:

– Assume that:– Artificial boundary:– Exterior problem:

ruu

rrfrur

m

dr

rdur

dr

d

r

when0,0)0(

0),()()(1

2

2

0when0)(,0 0 rrrfm

0rRr

ruRu

rRrur

m

dr

rdur

dr

d

r

when0,given)(

,0)()(1

2

2

Model Problem (I)

– Exact solution:– Transmission conditions:

– Exact boundary condition:

– Reduced problem:

RrrRRuru m )/()()(

)()()()( RuRuRuRu

RRumRu /)()(

RRumRuu

Rrrfrur

m

dr

rdur

dr

d

r

/)()(,0)0(

0),()()(1

2

2

Model Problem (II)

2D problem:

– Assume: – Artificial boundary: – Exterior problem:

xrugu

Qfu

i

c

whenbounded,

in

)0()(supp0

Ri Bf

0with20|),( RRRR

ruRu

Rru

whenboundedgiven,),(

,0

Ri

Model Problem (II)

– Exact solution:

– Transmission conditions:

– Exact boundary condition:

dnRubdnRua

Rrnbnar

Raru

nn

nnn

n

sin),(1

,cos),(1

,)sincos(2

),(

2

0

2

0

1

0

20),(),(),(),(

Rr

uR

r

uRuRu

2 2

1 10 0

1/ 2 1/ 2

1 1( , ) ( , ) cos ( ) ( , )sin ( )

: ( ( , )) ( ) , : ( ) ( )R

n n

R R

u uR n u R n d R n d

r R R

K u R K u K H H

Model Problem (II)

– Approximate ABCs:

– Reduced problem:

)BCNeumann (0),(0

020)(cos),(1

:)(),(1

2

0

Rr

uN

NdnRunR

uKRr

u N

nN R

20)(),(,

in

RiuKR

r

ugu

fu

N

i

i

Ri

Model Problem (II)

Variational formulation:

– With exact BCs: Find s.t.

– With approximate ABCs: Find s.t.

gVuVvvFvuBvuA ),(),(),(

2

0

2

0

2

01

2

0

2

0

2

0

2

01

2

0

,2,1

1

]sin),(sin),(cos),(cos),([),(

]sin),(sin),(cos),(cos),([),(

)(),(

:normwith 0|)(

dnRvdnRudnRvdnRunvuB

dnRvdnRudnRvdnRunvuB

xdvfvFxdvuvuA

vvvHvV

N

nN

n

Vi

ii

ii

gN Vu VvvFvuBvuA NNN ),(),(),(

Model Problem (II)

Finite element approximation: Find s.t.

Properties of

Properties of

Vuh

g

h

N

hhhhh

NNhh

NVvvFvBvA uu ),(),(),(

),(&),( vuBvuB N

),( vuA

),(),(2

21 vvAMvuMvuA vVVV

),(),(0

),(),( 33

vvBvvB

vuMvuBvuMvuB

N

VVNVV

Model Problem (II)

Existing error estimates (Han&Wu, 85’, Yu, 85’, Givoli & Keller 89’)

Deficiency– N=0: no convergence, but numerically gives– How does error depend on R?

Find new error estimates depend on– h, N & R ?????

N

kp

pp

V

h

N R

R

NhuRC

NhuRCu u 0

)1(

1),(or

)1(

1),(

New `Optimal’ Error Estimate

N=0, convergence linearly as Fixed N, ,convergence as Tradeoff between N and RIn practice, (Bao & Han, SIAMNA 00’)

00 ,2/1

1

0,10

,1||

)1(

1|| p

N

ppph

Nu

R

R

NuhCu

ii

u

R

0RR

0 R R

N

10~5:,0 NRR

New `Optimal’ Error Estimate

Ideas (Bao & Han, SIAMNA, 00’):

– Use an equivalent norm on V:

– Analysis B(u,v) carefully

– Notice u satisfying Laplacian when

i

vvvAv

,2,1

2/1

*),(

),(),(2

***vvAvuvuA v

0Rr

1

0

2

*,2/1

22

1

22

1

2

)sincos(2

),(

R/),()()(),(

nnn

nn

nn

N

nnN

ndncc

Rv

vvvBnnvvB vdcdc RR

Numerical example

Poisson equation outside a disk with radius 0.5:– Choose f and g s.t. there exists exact solution– piecewise linear finite element subspace

Test cases:– Mesh size h effect:– N effect: – R effect: varies

:hV

101,10 NRR

10 RR

0RR next

h effect

Conclusion:

070.0139.0275.0517.0-u

4836.63728.22081.12175.4-u

42.024E48.097E33.24E21.306E-umax

/8h/4h/2h0.31416hMesh

i

i

Ω1,

h

N

Ω0,

h

N

h

N

0000

u

uu

EEEE

)(-umax)(-u

)(-u

2h

N

2

Ω0,

h

N

Ω1,

h

N

uu

u

i

i

hOhO

hO

back

h & N effect

Conclusion: 1

0

,10

,,

)1(

1

N

R

N

R

pk

h

N

h

R

RORR

NO

iiuuuu

back

Extension of the Results

Yukawa equation (Bao & Han, SIAMNA, 00’):

– Assumption: – Exact & approximate ABCs:– Error bounds:

rukn

ugu

inxfxuxxux

NDwhen0,,

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

000 ||when,0)(,0)(,0)( Rxrxxxf

)()1(

)()1(

001

0100

,1

RKN

RKhCu

Np

Nph

Ni

u

D

N

Extension of the Results

Problem in a semi-infinite strip (Bao & Han, SIAMNA,00’):

– Assumption: – Exact & approximate ABCs:– Error bounds:

B.C.

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

inxfxuxxux

000 when,0)(,0)(,0)( dxxxxf

eubndd

pph

N NhCu

i

/)1()(

0,1

0

)1(

1

i

dx

0y

by

Extension of the Results

Exterior Stokes Eqs. (Bao, IMANA, 03’)

– Exact & approximate nonlocal/local ABCs:– Difficulty: Constant in inf-sup condition– Error bounds:

rpu

u

ufpu

i

when0bounded

on0

in,0div,grad

i

}1,1max{

00 )1(

1N

kkh

NhN R

R

NhCppuu

Extension of the Results

Exterior linear elastic Eqs. (Bao & Han, Math. Comp. 01’)

– Exact & approximate nonlocal/local ABCs:– Difficulty: Constant in Korn inequality– Error bounds:

ru

u

fuu

i

whenbounded

on0

in,divgrad)(

i

}1,1max{

00 )1(

1N

kkh

N R

R

NhCuu

High-order Local ABCs

Poisson Eq.

Exact BC

Approximate s.t. correct for first N terms

xrugu

Qfu

i

c

whenbounded,

in

1

0

1

)sincos(2

),(

),()sincos(1

),(

nnn

nnn

nbnaa

Ru

RuLnbnanR

Rr

u

NLL

N

m

N

m

mm

mN

m

N

m

mN NnnnRu

RRuL

1

)(22

2

1

)(,,1,),()1(

1),(

High-order Local ABCs

N=1:

Finite element approximation:Error bounds: (Bao & Han, CMAME, 01’)

20),(1

),(2

2

Ru

RR

r

u

1

0,

,1

Np

uN

h

N R

RhCu

iu

For Navier-Stokes Eqs. (Bao, JCP,95’,97’,00’)

Two types exterior flows: around obstacles & in channel

0

\inRe

1)( 2

u

Rupuu i

u )(yu

Ideas

Introduce two lines and set

Introduce a segment and set

Lxx 22 &0

1,01

2

2

1,012,02 0

Re

1222

xx

u

x

uu LxLxLx

bx 1

Lxuxu bx 20)(1

Ideas

Introduce a segment and design ABCs– Linearize NSEs on by Oseen Eq.– Solve Oseen Eq. on analytically by given– Use transmission conditions

– Design ABCs on

cx 1

c

),( 2xcu

c

),(),(),(),( 2222 xcxcxcuxcu nn

c

LxxcpxcuTpuTxc Ncxcxn 2222 0),(),,((),(),(11

Ideas

Reduction

Solve the reduced problem

.0)),(),,((),(,),(

,0,0

in0,Re

1)(

22222

1,02,02

122

LxxcpxcuTxcuxbu

cxbux

uu

uupuu

Nn

LxLx

T

i

Well-posedness

Variational formulation:

with

)(0),(

)]([)(),(),(),,(),(2

21*21

TN

TNNNNNN

LWqquB

HVvvFpvBvuAvuuAvuA

]cos),(cos),([),(

),(2

)(),(

))(())((2

1),,(

Re2

1)()(

Re

2),(

20

2212

10

2212

0 221

2

1

2

1,

2

1,

dxL

xmxcvdx

L

xmxcuvuA

dxxcva

vFxdvqqvB

xdvwuwvuwvuA

xdx

v

x

v

x

u

x

uxdvuvuA

LN

m

LN

L

ji i

j

j

i

i

j

j

i

jiijij

T

T

TT

Well-posedness

Well-posedness: – There exists solution of the reduced problem– When Re is not too big, uniqueness

Error estimates for N-S Eqs.

Error estimates for Oseen Eqs.c

uN

Cppuu NN

,22/3)1(

0

0

,22/3

))(1(

)1( cu

N

eCppuu

ccNNN

Finite Element Approximation

FEM approximation:

Error estimates for N-S Eqs.

Error estimates for Oseen Eqs.

hhhNh

hhhNhhh

Nh

Nh

Nh

Nhh

Nh

WqquB

VvvFpvBvuAvuuAvuA

0),(

)(),(),(),,(),( 21

cTT

uN

CpuhCppuumm

mNh

Nh

,22/3

2,,11 )1(

0

0

,22/3

))(1(2

,,11 )1( cTTu

N

eCpuhCppuu

ccN

mm

mNh

Nh

Examples

Backward-facing step flow:– Streamfuction & vorticity

Flow around rectangle cylinder :– Velocity field & near obstacle

Flow around circular cylinder :– Velocity field & near obstacle

next

Flow in Channel

back

Flow in Channel

back

Flow around cylinder

Re=100

Re=200

Re=400

back

Flow around cylinder

Re=100

Re=200

Re=400

back

Flow around cylinder

Re=100

Re=200

Re=400

back

Flow around cylinder

Re=100

Re=200

Re=400

back

Conclusions & Future challenges

Conclusions:– New `optimal’ error estimates– New high-order local B.C.– Application to N-S Eqs.

Future challenges– 3D problems– Nonlinear problems– Coupling system