Chemistry Reference
In-Depth Information
-4
-8
-12
50
100
150
200
Iterations
Figure 5
Convergence rates for a simple one-dimensional Poisson problem in a finite
domain with 65 points. The potential was fixed at 0 on the boundaries, and two discrete
charges were placed in the domain. The figure plots the number of fine-grid relaxation
iterations vs. the log(10) of the residual. The top curve (crosses) is for Gauss-Seidel
relaxation on the fine grid alone. The lower curve (
x
's) is for repeated cycling through
MG V-cycles with 6 levels. The apparent ''stalling'' after 40 iterations for the MG
process occurs because machine-precision zero has been reached for the residual.
computer time required scales linearly with system size, which is the best
we can do for an electrostatic problem. We do point out, though, that
on uniform grids, FFT methods for solving the Poisson problem scale as
N
g
log
N
g
, and the log term grows very slowly with system size. So FFT
and MG are competitive methods for obtaining the Poisson potential
over the whole domain of a uniform grid.
58
Figure 6
Full multigrid cycle (FMG); see also Figure 4. Instead of starting with an initial
guess on the finest level and performing V-cycles from that point, iterations are begun on
the coarsest level and the solver progresses toward the finest level. By the time the left
side of the finest-level V-cycle has been reached, an excellent initial approximation to the
function has been obtained for little computational cost.
