I would like to hear some suggestions on how to numerically solve a 4d Poisson equation with two singular sources, which can be brought to the following form (thanks to symmetries)

\begin{equation} \partial_z^2\varphi(z,\rho)+\frac{1}{\rho}\partial_\rho^2\left(\rho\varphi(z,\rho)\right)=-\frac{\delta(\rho)}{4\pi \rho^2}\left(\delta(z-z_0)+\delta(z+z_0)\right) \end{equation}

with boundary condition

$\varphi(z,\rho)\rightarrow 0$ as $z,\rho\rightarrow \infty$


  • This is actually a warm-up version of the problem that I want to solve. I chose this particular PDE because it has some resemblance with the problem that I want to solve and the exact solution can be found analytically, i.e. \begin{equation} \phi(z,\rho)=\frac{1}{4\pi^2}\left(\frac{1}{\rho^2+(z-z_0)^2}+\frac{1}{\rho^2+(z+z_0)^2}\right) \end{equation} So it is good for testing numerical schemes.

  • The actual problem is nonlinear, and so I would like to solve the above Poisson equation with a method that also works for non-linear problems.

  • It is important for me to get the asymptotic behavior near the singularities right.

Some attempts and why they do not seem to work:

  • Use Mathematica's built in DiracDelta[x]: the effect of the Dirac deltas is hardly felt.
  • Regularize the Dirac delta with a Gaussian function: either the effect of Dirac deltas is not felt or solution appears wrinkly.
  • Integrate by hand around the vicinity of the sources to get boundary conditions near the sources: there is no guarantee that the solution close to each source is circularly symmetric, so this isn't helpful.
  • Just to see what happens, I tried removing the sources and use the exact analytical solution (in the presence of sources) at some finite rectangular boundary as a boundary condition: the solution does not exhibit and singular-like behaviour near the sources.
  • $\begingroup$ This is a linear PDE. Do you, by any chance mean the operator $\partial_z^2 \varphi + \rho^{-1} \partial_\rho ( \rho \, \partial_\rho \varphi)$? That would be the Laplace operator in polar coordinates. $\endgroup$ – Henrik Schumacher Mar 30 '19 at 17:06
  • $\begingroup$ @HenrikSchumacher Yes this is a linear PDE. I want to solve it using non-linear methods because the full problem that I want to solve is more complex and is non-linear. I chose to study this linear PDE because I know it's exact analytical solution. Please read my problem fully. $\endgroup$ – Unobviously Obvious Mar 30 '19 at 17:11
  • $\begingroup$ Have you tried FEM in Cartesian coordinates? Or course, due to the singularities, the solutions is not very smooth and hence the approximation will be only rather coarse. In general, it would be a good idea to show what you have already tried. $\endgroup$ – Henrik Schumacher Mar 30 '19 at 17:14
  • $\begingroup$ Thinking of it, any answerer is doomed to get a "Yes, but..."-reply from you as this appears to be an X-Y-question. Moreover, there is no code and apparently, the formula for your PDE is wrong. In total, you don't make it easy for others to help you. $\endgroup$ – Henrik Schumacher Mar 30 '19 at 17:20
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    $\begingroup$ @UnobviouslyObvious It makes no sense to solve your problem in the form as you formulated it here. A nonlinear problem may require a different algorithm. $\endgroup$ – Alex Trounev Mar 30 '19 at 19:04