1. Introduction to Scientific Computing II
From Relaxation to Multigrid
Dr. Miriam Mehl

2. Relaxation Methods problem: order an amount of peas on a straight line
(corresponds to solving uxx=0)

3. 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

4. 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

5. Relaxation Methods – Properties
convergence depends on
method
frequency of the error
stepsize h

6. Jacobi – Details fast for middle frequencies slow for
high and low frequencies

7. Gauss-Seidel – Details
fast for
high frequencies
slow for
low frequencies

8. Multigrid – Principle fine grid eliminate high frequencies
coarse grids
eliminate low frequencies(!)
equation for the error(!)
error smooth => representable

9. 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

10. Multigrid Methods – Presmoothing
Gauss Seidel

11. Multigrid Methods – Presmoothing
Gauss Seidel

12. Multigrid Methods – Presmoothing
Gauss Seidel

13. Multigrid Methods – Presmoothing
Gauss Seidel

14. Multigrid Methods – Presmoothing
Gauss Seidel

15. Multigrid Methods – Presmoothing
Gauss Seidel

16. Multigrid Methods – Presmoothing
Gauss Seidel

17. Multigrid Methods – Presmoothing
Gauss Seidel

18. Multigrid Methods – Presmoothing
Gauss Seidel

19. Multigrid Methods – Presmoothing
Gauss Seidel

20. Multigrid Methods – Presmoothing
Gauss Seidel

21. Multigrid Methods – Presmoothing
Gauss Seidel

22. Multigrid Methods – Presmoothing
Gauss Seidel

23. Multigrid Methods – Presmoothing
Gauss Seidel

24. Multigrid Methods – Presmoothing
Gauss Seidel

25. Multigrid Methods – Presmoothing
Gauss Seidel

26. Multigrid Methods – Presmoothing
Gauss Seidel

27. Multigrid Methods – Presmoothing
Gauss Seidel

28. Multigrid Methods – Presmoothing
Gauss Seidel

29. Multigrid Methods – Presmoothing
Gauss Seidel

30. Multigrid Methods – Presmoothing
Gauss Seidel

31. Multigrid Methods – Residual
Almost zero  neglected in following slides

32. Multigrid Methods – Restriction

33. Multigrid Methods – Coarse Grid

34. Multigrid Methods – Coarse Grid

35. Multigrid Methods – Coarse Grid

36. Multigrid Methods – Coarse Grid

37. Multigrid Methods – Coarse Grid

38. Multigrid Methods – Coarse Grid

39. Multigrid Methods – Coarse Grid

40. Multigrid Methods – Coarse Grid

41. Multigrid Methods – Coarse Grid

42. Multigrid Methods – Coarsest Grid

43. Multigrid Methods – Coarsest Grid

44. Multigrid Methods – Coarse Grid

45. Multigrid Methods – Coarse Grid

46. Multigrid Methods – Postsmoothing

47. Multigrid Methods – Postsmoothing

48. Multigrid Methods – Postsmoothing

49. Multigrid Methods – Postsmoothing

50. Multigrid Methods – Postsmoothing

51. Multigrid Methods

52. Multigrid remember: Gauss Seidel error before smoothing after
10 iterations

53. Multigrid fine grid reduce high frequencies error before smoothing
after
smoothing

54. Multigrid switch to coarse grid restrict residual residual before
restriction
after
restriction

55. Multigrid solve coarse grid equation recursive call of multigrid
coarse grid solution

56. Multigrid solve coarse grid equation recursive call of multigrid
coarse grid solution
fine grid error

57. Multigrid switch to fine grid interpolate coarse grid solution
interpolated coarse grid solution
fine grid error

58. Multigrid switch to fine grid apply coarse grid correction
fine grid error
before correction
after correction

59. Multigrid fine grid eliminate new high frequencies fine grid error
before smoothing
after smoothing

60. Multigrid comparison Gauss-Seidel – multigrid error
after 10 Gauss-Seidel iterations
after 1 multigrid iteration

61. Multigrid – Cycles V-cycle: one recursive call
W-cycle: two recursive calls
F-cycle: V-cycle on each level