1. University of Colorado
Multigrid Methods
Shijie Zhong
Dept. of Physics
University of Colorado
Boulder, Colorado
Workshop for Advancing Numerical Modeling of Mantle Convection and Lithospheric Dynamics
July 2008, UC-Davis

2. Numerical modeling Kd=F A scientific problem 
Partial differential equations
within a domain 
Discretize PDE using FE, FD,
FV, … on a certain grid 
a matrix equation:
Kd=F
f=ma

3. A toy problem: 1-D heat conductionx

4. Discretize with FE
x=0
x=1 e

5. e 1

6. Kd=F

7. Iterative Solvers Kd=F A matrix equation: Iterative solvers:
memory usage ~ N (# of unknowns in d),
# of flops ~ N (e.g., for multigrid solver),
suitable for parallel computing.

8. Jacobi and Gauss-Seidel methods
Matrix Equation:
Rewrite matrix K:
Jacobi method:
Start with a guessed solution d(0), then update d iteratively to
get d(1), … until residual e=||Kd(n)-F|| is less than some tolerance.
Gauss-Seidel method:

9. Jacobi method

10. Gauss-Seidel Method

11. The idea behind multigrid
Gauss-Seidel

12. A road map

13. A road map continued but reversed

14. Different cycles
W-cycle
V-cycle n
n-1 1

15. THE method for elliptic equations (i.e., “diffusion” like problems)

16. Execution Time vs Grid Size N for Multi-grid Solvers in Citcom
t ~ N-1
FMG: Zhong et al. 2000
MG: Moresi and Solomatov, 1995