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 $$U(r) = \begin{cases} & H \text{ if } r \geq R_U \\  & H + H_U (\frac{r}{R_U} - 1 ) \text{ if } r \leq R_U \end{cases}$$
Jacobi method converging then diverging
pde;algorithms;finite difference;numerics;boundary conditions