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 sparse 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-symmetric distributions ... etc. Also, I would be very keen to find some info on best practice for applying this boundary condition in FEM analysis.


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.

  • $\begingroup$ 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. $\endgroup$ – dmon Dec 12 '12 at 10:56

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