Post on 01-Jan-2016
description
Multigrid Methods
Shijie Zhong
Dept. of PhysicsUniversity of Colorado
Boulder, Colorado
Workshop for Advancing Numerical Modeling of Mantle Convection and Lithospheric Dynamics
July 2008, UC-Davis
Numerical modeling
Discretize PDE using FE, FD,
FV, … on a certain grid
a matrix equation:
Kd=F
A scientific problem
Partial differential equations
within a domain
f=ma
A toy problem: 1-D heat conduction
0 1 x
Discretize with FE
x=0 x=1
e
e1
0
Kd=F
Iterative Solvers
A matrix equation:
Kd=F
Iterative solvers: memory usage ~ N (# of unknowns in d), # of flops ~ N (e.g., for multigrid solver), suitable for parallel computing.
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 =||Kd(n)-F|| is less than some tolerance.
Gauss-Seidel method:
Jacobi method
Gauss-Seidel Method
The idea behind multigridGauss-Seidel
A road map
A road map continued but reversed
Different cycles
V-cycle
n
n-1
1
W-cycle
THE method for elliptic equations (i.e., “diffusion” like problems)
Execution Time vs Grid Size N for Multi-grid Solvers in Citcom
FMG: Zhong et al. 2000MG: Moresi and Solomatov, 1995
t ~ N-1