# How is Poisson's Equation solved numerically?

This question is of pure interest. I would like to know, how a mixed boundary value problem like the following can be solved numerically:

Lets say I have two conducting plates (not necessarily parallel) in two dimensions (infinite in one, so 3D reduces to 2D). On the plates are total charges $$Q_1$$ and $$Q_2$$. Now I place a point charge $$q$$ somewhere between. How is such problem solved? I mean numerically, not analytically.

If potentials $$\Phi_1$$ and $$\Phi_2$$ on the plates would be given, and there is no additional charge, I know, that this can be solved by relaxation methods on a grid.

But in this situation the potentials on the plates are not known in advance, the only thing is to know, that they are constant on each plate and that the surface integral of the fields directly at the plates must yield charges $$Q_1$$ and $$Q_2$$. So I'm not able to start some kind of iteration with fixed values at the plates. Additionally, how are point charges considered in such case? The surface integral around the point charge must yield $$q$$, but how can this be transformed into a numerical method?

What literature can be recommended to learn such things? I have good books on classical electrodynamics (like Jackson, ...) but here the scope is to solve analytically.

Of course, there is FEM software available. But I want to know HOW available software solves such problems in principle. I don't need it for an urgent solution.

• Check out the Fenics tutorial. – thermomagnetic condensed boson Feb 2 at 8:13
• Thanks. Seems to be on a level which is easily understandable for non-mathematicians and is not too shallow on the other hand. I will start reading it NOW ;-) Could be exactly what I looked for... – michael Feb 2 at 8:35
• There are many ways to set this up. One would be: divide the plates into $n$ small segments, which an unknown amount of charge on each segment. For each segment, set up an equation for the potential field caused by all the unknown charges plus the point charge. This will give you $n$ equations in $n$ unknowns which you can solve. Of course you need to do this numerically for a realistically large number of segments (say of the order of 100 or 1,000). Knowing the charge distribution on the plates, you can then calculate the electric field anywhere in space. – alephzero Feb 2 at 9:40
• I answered basically this same question with respect to the Laplace equation here: physics.stackexchange.com/q/310447/25301 – Kyle Kanos Feb 2 at 13:26