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I was wondering where I could get a detailed account (either in print or online) on applying a Neumann/mixed Boundary condition along the $r=0$ axis in an axially symmetric geometry. Though this is a very common task I have always struggled to find a reasonable account of it. Piecing together sparce and poorly referenced notes I believe that in 1D polar coordinates where:

$$ \nabla^2 u = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} $$

one gets, by Taylor expansion at the origin,:

$$ \nabla^2 u \rvert_{r=0} \simeq 2\frac{\partial^2 u}{\partial r^2}\rvert_{r=0} $$

when $u$ is symmetric about the $r=0$ axis.

I would like to more information on higher order derivatives, best practice, non-symetric distributions ... etc. Also I would be very keen to find some info on best practice for applying this boundary condition in FEM analysis.

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This is the wrong boundary condition. If your solution is smooth, then the correct condition at $r=0$ is $$ \frac{du}{dr} = 0. $$ You can see this by thinking about what would happen if you cut a line through the origin through the entire domain, i.e., you look at the solution not only for $r\ge 0$ but also for $r\le 0$. To the left of $r=0$ you of course have a mirror image of the solution at the right. If you don't want the solution to have a kink at the origin, you need to require the condition above.

This is, of course, also the natural boundary condition so it is easy to implement.

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Yes this is the bc but to apply it with 2nd order accuracy in FD/FV methods one typically introduces ghost points outside the domain, assigns $u_{-1} = u_1$, and subs back into the discretised expression, solving for $u_0$. This procedure fails in cylindrical geometry at $r=0$, due to the $1/r$ term. In such cases I believe one uses Taylor expan, as above, to reformulate the eqn in the limit $r\rightarrow 1$, and then applies the ghost point procedure. I was wondering what the equivalent 2nd order procedure was in FEM, but I think your right, in FEM it appears to be simply the natural bc. – dmon Dec 12 '12 at 10:56

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