Finite Difference Method Elliptic Pdes


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

matlab101

that satisfies the condition

matlab102

(Assuming implicitly thatmatlab103 . )

Just as one classifies conic sections and quadratic forms based on the discriminantmatlab104 , the same can be done for a second-order PDE at a given point. Discriminant in a PDE is given bymatlab105 due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

matlab106, which becomes (for :matlab107 ) :

matlab108, and matlab109. This resembles the standard ellipse equation:matlab110

In general, if there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form

matlab1014, where L is an elliptic operator.

matlab1111

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

1012

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

matlab1013

This can be compared to the equation for an ellipsoid;matlab1014

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 uxx and uyy by their central differences to acquire:

(ui+1,j− 2uij+ ui−1,j)/ h2 + (ui,j+1− 2uij+ ui,j−1)/ k2= f(xi, yj) = fij

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

u0,j= g(a, yj), um,j= g(b, yj), ui,0 = g(xi, c), and ui,n= g(xi, d).

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