### Short introduction to elliptic PDE

In this section, we discuss in more details numeric methods for solving elliptic PDE (for hyperbolic and parabolic problems the situation is quite similar – at least for our solver the computation schema is practically the same).

First of all, let’s note that any elliptic equation can be converted to its canonical form:

where is Laplace operator (for 2-dimension problem we have ∆ =).

In case of σ = 0 and f = 0 we have a Laplace equation. If the right hand side of the equation f = f(x, y) is not zero, then it is called Poisson equation.

The typical problem, governed by Poisson equation, is the problem of calculating distribution of electrical field potential u = u(x, y) in the given region, with the given distribution of electric charges f = f(x, y).

The case σ = 0 corresponds to vacuum. For heterogeneous environment function σ = σ(x, y) define conduction of the field in the given region.

Functions f and σ are considered to be known, function u is unknown.

For numerical solving of elliptic equations, we apply discretization, based on finite differences method.
For simplicity we discuss only regular grid with equal step.

1. ##### Discretization on a regular grid
We begin with applying FDM approach to elliptic problems