numerical methods for problems in unbounded domains weizhu bao department of mathematics &...
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Numerical Methods for Problems in Unbounded Domains
Weizhu Bao
Department of Mathematics& Center for Computational Science and Engineering
National University of SingaporeEmail: bao@math.nus.edu.sg
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
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