Introduction to Scientific

Computing II

From Relaxation to Multigrid

Miriam Mehl, Michael Bader

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