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I am working to solve Poisson's equation in 2D axisymmetric cylindrical coordinates using the Jacobi method. The $L^2$ norm decreases from $\sim 10^3$ on the first iteration (I have a really bad guess) to $\sim 0.2$ very slowly. Then, the $L^2$ norm begins to increase over many iterations.

My final matrix is weakly diagonally dominate, except for the 2nd order Neumann condition at $r = 0$.

Can I make a small tweek to make this work, is it numeric or do I need a new algorithm?

My geometry is parallel plates with sharp points at $r = 0$ on both plates.

My boundary conditions are $$\left. \frac{\partial V}{\partial r} \right|_{r=0} = 0$$

Although I would like my second radial BC to be $$\left. \frac{\partial V}{\partial r} \right|_{r=\infty} = 0$$ I settled for $$\left. \frac{\partial V}{\partial r} \right|_{r=a} = 0$$

Then Dirichlet conditions at the upper and lower boundaries $$V(r, L(r) ) = V_0$$ $$V(r, U(r) ) = V_L$$

where $$L(r) = \begin{cases} & 0 \text{ if } r \geq R_L \\ & H_L (1 - \frac{r}{R_L} ) \text{ if } r \leq R_L \end{cases}$$

and

$$U(r) = \begin{cases} & H \text{ if } r \geq R_U \\ & H + H_U (\frac{r}{R_U} - 1 ) \text{ if } r \leq R_U \end{cases}$$

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    $\begingroup$ What's your damping factor? Are you sure it's small enough? $\endgroup$ – Wolfgang Bangerth Sep 1 '15 at 15:22
  • $\begingroup$ outside of the location "R=0" what can you tell us about your spatial discretization? Why is there two "plates"? $\endgroup$ – EngrStudent Sep 10 '15 at 15:40
  • $\begingroup$ It's a uniform grid spatially (so I'm stair stepping the features). There are two plates because that's the system. $\endgroup$ – user1543042 Sep 10 '15 at 23:44

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