# Numerically solving the polar poisson equation [closed]

I want to solve the Poisson equation for a 2D polar system:

$$\Delta_r f(r) = u(r)$$ with the Laplace operator: $\Delta_r f(r) = \frac{1}{r}\frac{\partial}{\partial r} \left[r \frac{\partial}{\partial r} f(r) \right]$

I have $u(r)$ given as a vector for a non-uniform (quasi-logarithmic) grid. Any ideas how to solve this?

• What are the boundary conditions? What numerical methods are you familiar with? Finite differences? Spectral methods? Finite element? As Bill Barth says there are many ways to tackle this problem. Please provide more information so that the community can help you. – James May 8 '14 at 13:44

$$\frac{\partial^2 }{\partial r^2}u+\frac{1}{r}\frac{\partial}{\partial r}u=f$$
Define the mesh points in the $r-\theta$ as $r=i \delta r$ so now at point $(i)$ the the equation is approximated as
$$\frac{u_{i+1}-2u_{i}+u_{1-i}}{(r(i+1)-r(i-1))^2}+\frac{1}{r(i)} \frac{u_{i+1}-u_{i-1}}{2(r(i+1)-r(i))} =f(i)$$
• Doug, you are correct. When I get the chance later I can rewrite the FD equation with explicit differencing in the denominators. Also, yes I see now that the OP's post has no $\partial\theta$ term. – user7257 May 8 '14 at 18:05