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Geometric (Classical) MultiGrid. Grid:. x=0. x=1. x 0. x 1. x 2. x i. x N-1. x N. local averaging. Let. Linear scalar elliptic PDE (Brandt ~1971). 1 dimension Poisson equation Discretize the continuum. Linear scalar elliptic PDE. 1 dimension Laplace equation - PowerPoint PPT Presentation
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Geometric (Classical) MultiGrid
Linear scalar elliptic PDE (Brandt ~1971)
1 dimension Poisson equation
Discretize the continuum
LU )(xx F)(U 10 x
0)U()U( 10
x0 x1 x2 xi xN-1 xN
x=0 x=1h
Grid: ihxN
h i ,1
Ni 0
h
Let ihi FF local
averaging),( ixU )( ixFi
hi UU
Linear scalar elliptic PDE 1 dimension Laplace equation
Second order finite difference approximation
=> Solve a linear system of equationsNot directly, but iteratively=> Use Gauss Seidel pointwise relaxation
LU 0 )(U x 10 x
0)U()U( 10
hi
hUL 0UUU
211 2
hiii 11 Ni
00 NUU
Influence of (pointwise) Gauss-Seidelrelaxation on the error
Poisson equation, uniform grid
Error of initial guess Error after 5 relaxation
Error after 10 relaxations Error after 15 relaxations
The basic observations of ML Just a few relaxation sweeps are needed to
converge the highly oscillatory components of the error
=> the error is smooth Can be well expressed by less variables Use a coarser level (by choosing every other
line) for the residual equation Smooth component on a finer level becomes
more oscillatory on a coarser level=> solve recursively The solution is interpolated and added
TWO GRID CYCLE
Approximate solution:hu~
hhh u~UV hhh RVL
hhhh u~LFR
Fine grid equation: hhh FUL
2. Coarse grid equation: hhh RVL 22
hh2
h2v~~~ hold
hnew uu h
h2
Residual equation:Smooth error:
1. Relaxation
residual:
h2v~Approximate solution:
3. Coarse grid correction:
4. Relaxation
TWO GRID CYCLE
Approximate solution:hu~
hhh u~UV hhh RVL
hhhh u~LFR
Fine grid equation: hhh FUL
2. Coarse grid equation: hhh RVL 22
hh2
hold
hnew uu h2v~~~ h
h2
Residual equation:Smooth error:
1. Relaxation
residual:
h2v~Approximate solution:
3. Coarse grid correction:
4. Relaxation
1
2
34
5
6
by recursion
MULTI-GRID CYCLE
Correction Scheme
interpolation (order m)of corrections relaxation sweeps
residual transfer
ν ν enough sweepsor direct solver*
.. .
*
1ν
1ν
1ν
2ν
2ν
2ν
Vcyclemultigrid
h0
h0/2
h0/4
2h
h
V-cycle: V
Hierarchy of
graphs
Apply grids in all scales: 2x2, 4x4, … , n1/2xn1/2
Coarsening Interpolate and relax
Solve the large systems of equations by multigrid!
G1
G2
G3
Gl
G1
G2
G3
Gl
Linear (2nd order) interpolation in 1D
x1 x2x
F(x)
)()()( 212
11
12
2 xFxx
xxxF
xx
xxxF
i
S(i)
(Ulb,Vlb)
(Urt,Vrt)(Ult,Vlt)
(Urb,Vrb)
(x2,y2)(x1,y2)
(x2,y1)(x1,y1)
(x0,y0)
Bilinear interpolation
C(S(i))={rb,rt,lb,lt}
i
S(i)
(Ulb,Vlb)
(Urt,Vrt)(Ult,Vlt)
(Urb,Vrb)
(x2,y2)(x1,y2)
(x2,y1)(x1,y1)
(x0,y0)
lbltlrbrtr UUUUyy
yyU
yy
yyU ......;
12
02
12
10
(Ul,Vl) (Ur,Vr)
lr Uxx
xxU
xx
xxyxU
12
02
12
1000 ),(
From (x,y) to (U,V) by bilinear intepolation
])~~(
)~~[(),(
])()[(),(
))((
2
))((
2
))(())((,
22
,
jscpjpjpj
iscpipipi
jscpjpjpj
iscpipipi
jiij
jijiji
ij
VyVy
UxUxaVUE
yyxxayxE
hi
ii
hj
hi
jiij
hhh uluuquEVUu ,
)(]|[
The fine and coarse LagrangiansFor each square k add an equi-density constraint
eqd(k) = current area + fluxes of in/out areas –
allowed area = 0
is the bilinear interpolation from grid 2h to grid h
At the end of the V-cycle interpolate back to (x,y)
)()( kii
ki buakeqd )()(),( keqduEuL
k
hk
hhhhh hhI2
)()()( 22
hhh
oldhnewh uIuu
)(),( 2222222 KeqduLuuQuLK
hK
I
hII
IJ
hJ
hIIJ
hhh