Introduction to Scientific Computing II From Relaxation to Multigrid Dr. Miriam Mehl
Jan 19, 2016
Introduction to Scientific
Computing II
From Relaxation to Multigrid
Dr. Miriam Mehl
Relaxation Methods
problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)
sequentially place peas on the line between two neighbours
we get a smooth curve instead of a straight line global error is locally (almost) invisible
Relaxation Methods – Gauss-Seidel
Relaxation Methods – Jacobi
place peas on the line between two neighbours in parallel
we get a high plus a low frequency oscillation these fequencies are locally (almost) invisible
Relaxation Methods – Properties
• convergence depends on
– method
– frequency of the error
– stepsize h
Jacobi – Details
• fast for
– middle frequencies
• slow for
– high and low frequencies
Gauss-Seidel – Details
• fast for
– high frequencies
• slow for
– low frequencies
Multigrid – Principle
• fine grid
– eliminate high frequencies
• coarse grids
– eliminate low frequencies(!)
– equation for the error(!)
– error smooth => representable
Multigrid – Algorithm
• iterate (GS) on the fine grid
• restrict residual to the coarse grid
• solve coarse grid equation for the error
• interpolate error to the fine grid
• correct fine grid solution
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Presmoothing
Gauss Seidel
Multigrid Methods – Residual
Almost zero neglected in following slides
Multigrid Methods – Restriction
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarsest Grid
Multigrid Methods – Coarsest Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Coarse Grid
Multigrid Methods – Postsmoothing
Multigrid Methods – Postsmoothing
Multigrid Methods – Postsmoothing
Multigrid Methods – Postsmoothing
Multigrid Methods – Postsmoothing
Multigrid Methods
Multigrid
• remember: Gauss Seidel
error
afterbefore smoothing 10 iterations
Multigrid
• fine grid
reduce high frequencies
error
afterbefore smoothing smoothing
Multigrid
• switch to coarse grid
restrict residual
residual
before restriction restrictionafter
Multigrid
• solve coarse grid equation
recursive call of multigrid
coarse grid solution
Multigrid
• solve coarse grid equation
recursive call of multigrid
fine grid errorcoarse grid solution
Multigrid
fine grid errorinterpolated coarse grid solution
• switch to fine grid
– interpolate coarse grid solution
Multigrid
• switch to fine grid
apply coarse grid correction
fine grid error
before correction after correction
Multigrid
• fine grid
eliminate new high frequencies
fine grid error
before smoothing after smoothing
Multigrid
• comparison Gauss-Seidel – multigrid
error
after 10 Gauss-Seidel iterations after 1 multigrid iteration
Multigrid – Cycles
• V-cycle: one recursive call
• W-cycle: two recursive calls
• F-cycle: V-cycle on each level