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5. Solution of Linear Equation Systems
Table 5.5. Numbers of iterations required by various solvers to reduce the L1
residual norm below
for the 3D Poisson equation with Neumann boundary
conditions
Grid GS
SIP ICCG CGSTAB FMG-GS FMG-SIP
where x* = x / X , y* = y/Y, z* = z / Z , and X , Y, Z are the dimensions of
the solution domain. The equation is discretized using FV method. The sum
of the source terms over the domain is zero, and Neumann boundary conditions (zero gradient normal to boundary) were specified a t all boundaries.
In addition to GS, SIP and ICCG solvers introduced above, we used also
the CGSTAB method with incomplete Cholesky preconditioning. The initial
solution is zero. The numbers of iterations required t o reduce the normalized
sum of absolute residuals four orders of magnitude are presented in Table
5.5.
The conclusions reached from this exercise are similar t o those drawn
from 2D problems with Dirichlet boundary conditions. When an accurate
solution is required, GS and SIP become inefficient on fine grids; conjugate
gradient solvers are a better choice, and multigrid methods are best. The
FMG strategy, in which the solution on a coarse grid provides the initial
solution for the next finer grid, is better than straight multigrid. FMG with
ICCG or CGSTAB as a smoother requires even fewer iterations (three t o four
on the finest grid), but the computing time is higher than for MG-SIP. The
FMG principle can be applied to other solvers as well.