1. Introduction to Scientific Computing II
Multigrid & Steepest Descent
Dr. Miriam Mehl

2. Multigrid – Some Rules smoother optimal smoothing
not!!! optimal convergence
small number of smoothing iterations!

3. Multigrid – Some Rules grid coarsening standard: doubling of h
exceptions:
unisotropic operators
adaptively refined grids
unstructured grids/general SLEs

4. Multigrid – Some Rules restriction/interpolation
order of restriction + order of interpolation > order of discretisation

5. Multigrid – Some Rules
V-cycle
faster
W-cycle
more robust

6. Steepest Descent – Basic Idea
solution of SLE
minimization
iterative
one-dimensional minima
direction of steepest descent?

7. Steepest Descent – Principle

8. Steepest Descent – Principle

9. Steepest Descent – Algorithm

10. Steepest Descent – Algorithm II

11. Steepest Descent – Example
initial error
after 1 iteration
after 10 iterations

12. Steepest Descent – Example
1/128
1/64
1/32
1/16 h
48,629
11,576
2,744
646
iterations