1. Introduction to Scientific Computing II
Institut für
Informatik
Scientific Computing
In Computer Science
Introduction to Scientific Computing II
Relaxation Methods
Dr. Miriam Mehl

2. Gauss-Seidel – Convergence
twice as fast as Jacobi (in our case!!!)
number of iterations:
O(1/h)2

3. Jacobi/GS – Costs per Iteration
A sparse
O(1) nonzero entries per line
O(1/h)2 operations in 2D
O(1/h)3 operations in 3D

4. Jacobi/GS – Costs number of iterations O(1/h)2 (both 2D and 3D)
costs per iteration
O(1/h)2 in 2D
O(1/h)3 in 3D
total O(1/h)4 or O(1/h)5

5. Implementational Aspects
Gauss-Seidel:
no memory overhead
sequential
Jacobi:
memory overhead
parallel
Red-Black GS

6. Successive Overrelaxation (SOR)
start from Gauss-Seidel
introduce w:

7. Successive Overrelaxation (SOR)
optimal w:
convergence?

8. SOR – Convergence
worse for small h
number of iterations:
O(1/h)

9. SOR – Costs number of iterations O(1/h)2 (both 2D and 3D)
costs per iteration
O(1/h)2 in 2D
O(1/h)3 in 3D
total O(1/h)4 or O(1/h)5

10. Towards Multigrid optimal costs O(1/h)2 or O(1/h)3 Gauss-Seidel:
reduction of different frequencies
multigrid idea:
high frequencies => fine grids
low frequencies => coarse grids