* Laplace's equation*,

is elliptic since
the discriminant, , is negative.
Laplace's equation occurs in numerous physically based
simulation models and is usually associated with a diffusive
or dispersive process in which the state variable, is
in an equilibrium condition. For example, could
represent an equilibrium temperature in a two dimensional
thermodynamic model based on Fick's Law. Of interest to the
computational scientist is the fact that solutions of
Laplace's equation, and elliptic equations in general, can
support large gradients only in response to external
stresses manifested as a source/sink term (**g** in equation
(7))
or as an abrupt change in type of or value of a
boundary condition. Almost invariably the computational
analysis of an elliptic equation reduces to a linear algebra
problem of solving a system of diagonally dominant linear
equations. Armed with this knowledge, the computational
scientist has apriori knowledge of the types of
algorithms and architectures that may provide an
efficient numerical solution of an elliptic equation of
the form (7).