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