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)
Relaxation Methods – Gauss-Seidel 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 – 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 before smoothing after 10 iterations
Multigrid fine grid reduce high frequencies error before smoothing after smoothing
Multigrid switch to coarse grid restrict residual residual before restriction after restriction
Multigrid solve coarse grid equation recursive call of multigrid coarse grid solution
Multigrid solve coarse grid equation recursive call of multigrid coarse grid solution fine grid error
Multigrid switch to fine grid interpolate coarse grid solution interpolated coarse grid solution fine grid error
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