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I wish to clear up a slight confusion that I have. This isn't very complicated but I'm just a bit stumped.

we know the the Robin Boundary Condition states:

$$ au+b\frac{\partial u}{\partial n} = g\hspace{15pt}on\hspace{15pt}\partial\Omega$$

My question is, assuming a=b=1, what's the difference between

$$u=g \biggr\rvert_{\frac{\partial u}{\partial n}=0},$$or $$\frac{\partial u}{\partial n}=g \biggr\rvert_{u=0},$$or $$ u+\frac{\partial u}{\partial n} = g$$ ?

In my case, a=b=1, and my Neumann and Dirchilet Boundaries are

$$u=1, \hspace{15pt} \frac{\partial u}{\partial n} =0$$

Which yields

$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} \frac{\partial u}{\partial n}=0$$

However, this can just as easily be

$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} u=0,\frac{\partial u}{\partial n}=1$$ or even

$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} u=0.5,\frac{\partial u}{\partial n}=0.5$$

Clearly, each of these possibilities are mathematically/physically distinct. How is this reconciled?

RRR
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  • Assuming you are using standard notation, for example $u=g\bigg|_{\frac{\partial u}{\partial n}}$ means you evaluate $g$ at points that have zero normal derivative. Also, Robin boundary conditions doesn't tell you anything about Dirichlet/Neumann BCs. – Chee Han May 22 '16 at 07:31
  • Ok. Thanks for the input. Clearly I'm rather new with numerical methods. I'm trying to impose a Dirichlet and Neumann BC at the exact same node under a finite difference scheme. How do I go about deriving the finite difference equation for that node? I was under the impression that Robin BCs were the way to go. Cheers – RRR May 22 '16 at 07:37
  • I am not familiar with numerical methods, sorry ): I suggest you post that as another question (: – Chee Han May 22 '16 at 07:45

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