I forgot to say that the potential is not given in the entire volume but only in fractions of it.

Assuming I correctly understand your question, Yes.  It's called the boundary Integral (or boundary element) method.  Essentially you apply the divergence theorem to volumetric terms to reduce them to intergrals on the surface (boundary).

Here's a good place to start - http://www.boundary-element-method.com/

Thanks Adam. But as far as I can see, the BEM also requires the potential to be given on the boundary. The potential is then determined for all points of interest in the enclosed volume. My problem aims the opposite, i.e. I have to determine the boundary potential whille the potential is given in a fraction of the volume enclosed by the boundary. Let me give you an example. An experiment with charged particles may require a certain potential distribution, i.e. a certain electric field geometry. Then a number of electrodes is positioned and the problem arises which voltage to apply to them to produce the required field. I know that an exact solution might not exist. I would be grateful to know the electrode voltages that generate a field that resembles the required filed in a sense...

ah ok.  In that case you need to take the free space green function for the sources that you have and add them up (it's a linear equation so that's ok). Then you just evaluate the resulting sum of functions at any point you like in the domain.  I.e. Take the appropriate phi(x-x') for any inclusions that you have and make a sum of these functions by evaluating them at the point x' where the particular source is located.  Then you just get left with a big sum which is a function of x. You can evaluate that at any point x to get the resulting field. You can choose x' to be any point on the boundary and you're done.  

NB. If any of your sources happen to be on the boundar itself then you need to adjust the weight to take account of the boundary shape at that particular point.

hope this helps? It's basically the same underlying maths as the boundary integral, but usually you solve a problem for boundary conditions and then use the sum of sources to interpolate the solution into the domain itself. 

thanks, I have to think about your suggestion... btw, is this "reverse" BVP something common in the area of potential theory? I did not see any hint in the literature I looked through so far. The highlight would be to have a methode that tells us not only the electrode potential but also the position where to place electrodes in order to generate a desired electric field!

Dear Adam, I think I have been misunderstood... There are no sources in the volume. I know the formula you are talking about. But in the underlying case the domain is source free. An electric field is given in a fraction of the domain. The problem is now to determine the necessary voltage of some electrodes that generate such a field. The position, shape and the number of electrodes is also given. The particles I mentioned do not play a role in the problem. Simply forget them...

Ok. So define the number of electrodes and shapes that you have and solve the field produced by one electrode - you might be able to find an analytical solution for this based on combinations of greens functions as discussed before. Write this in terms of the variables you want to optimise e.g. Position and voltage.

next add up all these sub solutions to give you the total induced field as a function of the unknowns that you want to optimise

now define a difference between the field produced and the field that you want. Take the modulus of this or square it so that all deviations are positive. This is your point error measure

now integrate over the domain where you know the field - lots of ways to do this either analytical or just taking some representative points and adding them up

now you just need to minimise the error quantity with respect to all of the variables you have simultaneously. Again lots of ways to do this - try numerical methods in c by press et al. for example

Alternatively you could look at something like flexPDE, which should be able to do this kind of problem if I remember correctly.