Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado
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Transcript of Multigrid Methods Shijie Zhong Dept. of Physics University of Colorado Boulder, Colorado
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