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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.

thanks!

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I assume that your system $A$ looks something like $\nabla^2\phi(\bf x) = \rho(\bf x)$ where $\phi$ is a potential and $\rho$ is a charge density? Can you give a little more detail about how exactly you modify the boundary conditions in response to the solution of this equation? –  Chris Taylor Nov 10 '11 at 11:44
    
Now if only you had mentioned the precise systems that you have... –  J. M. Nov 10 '11 at 11:44
    
You are both right. Sorry. I'm editing the question to specify the system. –  jbcolmenares Nov 10 '11 at 16:17
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