Introduction to Scientific Computing II

1
Introduction to Scientific Computing II

• Multigrid

2
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

3
Multigrid Degrees of Freedom

• smoother
• relation step sizes coarse fine
• grid transfer operators
• restriction
• interpolation
• processing order of grid levels

4
Multigrid Convergence

• two grid analysis
• h-independent convergence
• for good components

5
Two Grid Multigrid

• Example 2D Poisson 5-point-stencil

h two-grid analysis V-cycle
1/32 0.042 0.042
1/64 0.042 0.044
1/128 0.042 0.044
1/256 0.042 0.043
1/512 0.042 0.043
1/1024 0.042 0.042
1/2048 0.042 0.042
6
Multigrid Some Rules

• smoother
• optimal smoothing
• not!!! optimal convergence
• small number of smoothing iterations!

7
Multigrid Some Rules

• grid coarsening
• standard doubling of h
• exceptions
• unisotropic operators
• adaptively refined grids
• unstructured grids/general SLEs

8
Multigrid Some Rules

• restriction/interpolation
• order of restriction order of interpolation
  gt
  order of discretisation

9
Multigrid Some Rules

• V-cycle
• faster
• W-cycle
• more robust