I'm using numerical method to solve a coupled system. To be more precise, I'm using finite differences solve a electrostatics problem (let's call this system $A$), and using a function of the solution in $A$ to improve the boundary condition (system $B$).
Now, if I:
(a) update $B$ after $A$ has converged, and finish after B does no varies more, the error is a little high.
(b) update $B$ after a fixed number if iterations of $A$, I get a lower error, but sometimes $B$ diverges!
So, I'm missing something. something fundamental. This is consistent, I've gone through the code and modified it in different ways, and always end with a situation like this. So I'm certain there is something I'm not taking into account.
So, I'm searching for theory on coupled system. Does anybody around has any idea about where would be a good place to start? let it be books, papers...
To clarify things more, in $A$ I'm using over-relaxation to solve a finite differences system given by $\nabla\cdot(\epsilon\nabla\phi)=\rho$, which, if $\epsilon$ is constant, turns into Poisson's equation. It's a squared grid.
In $B$, I'm integrating over a closed surface, in a similar way to Boundary Elements method, to approximate the value at the boundary using Green's second identity.