Any (partial) differential equation must be *discretized* before it can
be treated computationally.
The most common techniques are

- finite differences
- finite volumes
- finite elements

For the
multigrid workbench
the
example problem
is discretized by *five point differences*.
Here the (square) domain is replaced by an equidistant mesh with
gridlines.

Each node is represented by an unknown

and the partial derivatives are replaced by

so that Laplace's equation becomes a system of linear equations of the form

This *banded system* may be solved by direct elimination techniques,
however, much faster are
(linear) iterative methods,
like the
multigrid method.

Ulrich Ruede , Thu Feb 2 21:05:32 MEZ 1995

Updated by Craig C. Douglas