Geometric Classical MultiGrid

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Geometric (Classical) MultiGrid
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Linear scalar elliptic PDE (Brandt 1971)

• 1 dimension Poisson equation
• Discretize the continuum

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Linear scalar elliptic PDE

• 1 dimension Laplace equation
• Second order finite difference approximation
• gt Solve a linear system of equations
• Not directly, but iteratively
• gt Use Gauss Seidel pointwise relaxation

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The basic observations of ML

• Just a few relaxation sweeps are needed to
  converge the highly oscillatory components of the
  error
• gt 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
• gt solve recursively
• The solution is interpolated and added

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TWO GRID CYCLE
Fine grid equation
1. Relaxation
Approximate solution
Smooth error
Residual equation
residual
2. Coarse grid equation
Approximate solution

3. Coarse grid correction
4. Relaxation
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TWO GRID CYCLE
MULTI-GRID CYCLE
Fine grid equation
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1. Relaxation
Approximate solution
Smooth error
Residual equation
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residual
2. Coarse grid equation
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Approximate solution
by recursion


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h
h
3. Coarse grid correction
u
u
old
new
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4. Relaxation
Correction Scheme
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V-cycle V(n1,n2)
residual transfer
enough sweeps or direct solver
relaxation sweeps

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G1
G1
Apply grids in all scales 2x2, 4x4,
, n1/2xn1/2
G2
G2
Solve the large systems of equations by multigrid!
G3
G3
Gl
Gl
Hierarchy of graphs
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Linear (2nd order) interpolation in 1D
F(x)
x1
x2
x
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Bilinear interpolation
(Urt,Vrt)
(Ult,Vlt)
(x2,y2)
(x1,y2)
i
(x0,y0)
S(i)
(x2,y1)
(x1,y1)
(Ulb,Vlb)
(Urb,Vrb)
C(S(i))rb,rt,lb,lt
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(Urt,Vrt)
(Ult,Vlt)
(x2,y2)
(x1,y2)
i
(Ul,Vl)
(Ur,Vr)
(x0,y0)
S(i)
(x2,y1)
(x1,y1)
(Ulb,Vlb)
(Urb,Vrb)
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From (x,y) to (U,V) by bilinear intepolation
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The fine and coarse Lagrangians

• For 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)