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Homogenisation Theory for PDEs
Homogenisation for Advection-Diffusion Equations
Starting point
Homogenisation
fu Α
Homogenised equation
fu Α
uu ?0
Starting point
Review- Steady state heat conduction
xxfx
xuxa
x jij
i
),())(
)((
xxu ,0)(
Goal: Homogenised equation
fu Α
several assumptions…
),(),(),()( 22
10
x
xux
xux
xuxu
macroscopic scalex
x
y microscopic scale
Independent
Variables
Multiple Scale Method
Assume the expansion Validity?
fu Α
00 u0Α
01 uu 10 ΑΑ
fuuu 012 210 ΑΑΑ
Solvability condition for : fu 0Α
0(y)d yfY
By inserting the expansion into , we obtain
),(),(),()( 22
10
x
xux
xux
xuxu
Instead of , we now havefu Α
00 u0Α
01 uu 10 ΑΑ
fuuu 012 210 ΑΑΑ
)(),(0 xuyxu
Homogenised
Equation
Cell problem:assume that
j
j
x
xuyyxu
)(
),(1 i
ijj
y
yay
)(
0A
)()(2
xfxx
xua
jjij
, where
Y
k
j
ikijij yy
yyayaa )
)(
)()()((
Starting point
Steady state heat conductionfu Α
xxfx
xuxa
x jij
i
),())(
)((
xxu ,0)(
Homogenised equation fu Α
xxf
xx
xua
jjij ),(
)(2
xxu ,0)(
There exists a unique solution of
Existence, uniqueness and convergence
xxfx
xuxa
x jij
i
),())(
)((
xxu ,0)(
uu and
u
There exists a unique solution ofu
weakly in .)(H10
)(Hf -1
)]([)A( yay ij
xxf
xx
xua
jjij ),(
)(2
xxu ,0)(
fu Α
fu Α
)Y,,(M)A( per y
Remark (validity of the expansion))(),(0 xuyxu
j
j
x
uxyxu
),(1
ji
ij
xx
uxyxu
2
2 ),(
i
ijj
y
yay
)(
)(0A
)()(0 yby ijij A
k
ikj
k
j
ikijijij y
yya
y
yyayaayb
))()(()()()()(
21
)(H2
21 1
)],(),()([)( Cxxuxxuxuxu
cell problem
higher order
cell problem
),(),(),()( 22
10
x
xux
xux
xuxu
, solution of the homogenised problem
Under certain conditions, we get the estimate
higher order
cell problems
Advection-Diffusion equations
),,()),(),(.(),(
txutxutxat
txu
IRtIRx d ,
),()0,( xuxu in dIRx0
0),(. txa
)(xaa
incompressible:
given, 1-periodic and sufficiently smooth
a
passive tracer
Linear Transport equations ( )
Where is ?
0
,0),().(),(
txuxa
t
txu IRtIRx d ,
),()0,( xuxu in dIRx
L
lL
l
New variables
,0),().(),(
txuxa
t
txu IRtIRx d ,
),()0,( xuxu in dIRx
xX
tT
,0),(.),(
txux
at
txu
IRtIRx d ,
),()0,( xuxu in dIRx
0Goal:
Rescaling
New formulation of the problem
))2sin(),2(sin()( 12 yyya
Multiple Scale Method
),,(),,(),( 10 tx
xutx
xutxu
By substituting the expansion in the equation, we obtain
000 uL
0110 uLuL
yyaL ).(0
xyat
L
).(1
Problem!!!
),(0 txuu
Example:
)2cos()2cos()( 210 yyyu
where
If}yin constants{}).({ 0 yyaLN
Then indeed
000 uL ),(0 txuu
By computing
yuLyuLYY
dd 0110
we obtain the homogenised equation
0),(.),(
txua
t
txux Y
yyaa d)(, where
0110 uLuL
)(xaa ergodic
Advection-Diffusion equations ( )
),,()),(),(.(),(
txutxutxat
txu
IRtIRx d ,
),()0,( xuxu in dIRx
0
)(xaa
incompressible:
Given, 1-periodic and sufficiently smooth
a
passive tracer
0d)( yyaY
),,(),().(),(
txutxuxat
txu
0),(. txa
),,()),().(1),(
txutxux
at
txu
IRtIRx d ,
),()0,( xuxu in dIRx
xX
tT 2
New variables
Rescaling
New formulation of the problem
0Goal:
IRtIRx d ,
),()0,( xuxu in dIRx
),,(),().(),(
txutxuxat
txu
By substituting the expansion in the equation, we obtain
Multiple Scale Method
),,(),,(),,(),( 22
10 tx
xutx
xutx
xutxu
000 uL
0110 uLuL
yy yaL ).(0
xyx yakL ).(1
021120 uLuLuL
tkL x
2
where
Solvability Condition
fuL 0
Integrate over Y
0d0 Y yuL
Solvability condition
0d)( Y yyf
yy yaL ).(0
smooth 1-periodic function
0),(. txa
)(xaa
uL0
First step ( )
000 uL ),(0 txuu
Y yyY
yuyauuyuLu d)).((d0
In fact,
Y y yuu d
0
0d)().()( Y y yyuyayu
0
000 uL
Y y yu d
2
0uu
0110 uLuL
Solvability condition
0d01 yuLY
0d)( yyaY
Separation of variables
),().(),,(1 txuytyxu x
Using this in , we obtain the cell problem0110 uLuL
)()().()( yayyay jj
yj
y
Second step ( )
021120 uLuLuL
Solvability condition
0d0211 yuLuLY
Leads to the homogenised equation
d
ji ji xx
u
t
u
1,
2
ijK
Y
jiij yyyaκ d)()( ijK
Effective Diffusivity:
Third step ( )
where ji ,][ ijKK matrix
Effective Diffusivity
For every vector we havedIR
2
1,
d
jijiijK
The homogenisation procedure enhances diffusion; the effective diffusivity is always greater than the molecular diffusivity in the following sense:
Summary
Steady state heat conduction (Review)
Multiple Scale Method
Existence, uniqueness and convergence
Remark (validity of the expansion)
Advection-Diffusion equations
00
(linear transport equation)
