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