The Poisson differential operator can be split in a natural way as the sum of two operators:
Now let 
, 
be discretized representations of 
, 
. Based on the observation that
, iterative schemes
such as
with suitable choices of 
and 
have been proposed.
This alternating direction implicit, or ADI, method was first
proposed as a solution method for parabolic equations. The 
are then approximations on subsequent time steps. However, it can also
be used for the steady state, that is, for solving elliptic equations.
In that case, the become subsequent iterates;
see D'Yakonov [82],
Fairweather, Gourlay and Mitchell [97],
Hadjidimos [119], and
Peaceman and Rachford [173].
Generalization
of this scheme to variable coefficients or fourth order elliptic
problems is relatively straightforward.
The above method is implicit since it requires systems solutions, and it
alternates the 
and 
(and if necessary 
) directions. It is
attractive from a practical point of view (although mostly on tensor
product grids), since solving a system with, for instance,
a matrix 
entails only a number of uncoupled tridiagonal
solutions. These need very little storage over that needed for the
matrix, and they can be executed in parallel , or one can vectorize
over them.
A theoretical reason that ADI preconditioners are of interest is that they can be shown to be spectrally equivalent to the original coefficient matrix. Hence the number of iterations is bounded independent of the condition number.
However, there is a problem of data distribution. For vector
computers, either the system solution with or with will
involve very large strides: if columns of variables in the grid are
stored contiguously, only the solution with will involve
contiguous data. For the the stride equals the number of
variables in a column.
On parallel machines an efficient solution is possible if the
processors are arranged in a 
grid. During, e.g., the
solve, every processor row then works independently of other
rows. Inside each row, the processors can work together, for instance
using a Schur complement method. With sufficient network bandwidth
this will essentially reduce the time
to that for solving any of the subdomain
systems plus the time for the interface system. Thus, this method will
be close to optimal.