Scheme for solving Laplace's (Poisson's) equation
In order to illustrate how we can transform the last equations into a
linear algebra problem of the type
Ax=w, with
A a matrix and
x and
w unknown and known
vectors respectively, let us also for the sake of simplicity assume
that the number of points
n=3. We assume also that
u(x,y)=g(x,y) on the border
δΩ.
The inner values of the function u are then
given by
4u11−u21−u01−u12−u10=−˜ρ114u12−u02−u22−u13−u11=−˜ρ124u21−u11−u31−u22−u20=−˜ρ214u22−u12−u32−u23−u21=−˜ρ22.