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3 Classification of Linear PDEs in Two Independent Variables     continued...

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).