# Finite Difference Method Elliptic Pdes

An **elliptic partial differential equation** is a general partial differential equation of second order of the form

that satisfies the condition

Just as one classifies conic sections and quadratic forms based on the discriminant , the same can be done for a second-order PDE at a given point. Discriminant in a PDE is given by due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

, and . This resembles the standard ellipse equation:

In general, if there are *n* independent variables *x*_{1}, *x*_{2} , ..., *x _{n}*, a general linear partial differential equation of second order has the form

, where L is an elliptic operator.

For example, in three dimensions (x,y,z) :

which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives

This can be compared to the equation for an ellipsoid;

**The Finite Difference Equations**:

Presume the rectangle is described by:

R = {a ≤ x ≤ b, c ≤ y ≤ d}.

We will divide R in sub-rectangles. If we have m subdivisions in the x direction as well as n subdivisions in the y direction then the step size in the x and y directions respectively are

h = (b – a)/m and k= (d – c)/ n

We acquire the finite difference equations for by replacing u_{xx} and u_{yy} by their central differences to acquire:

(u_{i+1,j}− 2u_{ij}+ u_{i−1,j})/ h^{2} + (u_{i,j+1}− 2u_{ij}+ u_{i,j−1})/ k^{2}= f(x_{i,} y_{j}) = f_{ij}

for 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1. The boundary conditions are commence by:

u_{0,j}= g(a, y_{j}), u_{m,j}= g(b, y_{j}), u_{i,0} = g(x_{i}, c), and u_{i,n}= g(x_{i}, d).

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