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

(Assuming implicitly that . )

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:

, which becomes (for : ) :

, and . This resembles the standard ellipse equation:

In general, if there are n independent variables x₁, x₂ , ..., 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)/ h² + (u_i,j+1− 2u_ij+ u_i,j−1)/ k²= 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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