How to solve Poissons equation using FEM with integral BC?

Question

I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$
Insulator (Neumann BC)
Electrode set at some potential (Dirichlet BC).
Additionally I have some electrodes $E_l$ which are set to have a fixed current $I_l$, so on these the boundary condition is $\int_{E_l} \sigma(x,y,z)(du/dn) dA=I_l$.

How would I go about incoprorating this third type of BC into FEM? Any description or references is what im looking for.

Additionaly or alternatively, I would like to know if there is some way to obtain accurate flux computations over surfaces in FEM. The normal way to solve Poisson equation using FEM does not give very accurate fluxes, however commercial software using FEM does, does anyone know how they do it? I am looking for references on how to compute such accurate fluxes.

Answer 1

You must specify a function $g$ and a nonhomogeneous Neumann boundary condition of the type $\frac{\partial u}{\partial n} = g$ so that $g$ satisfies $\int_{E_l} \sigma g \,d\mathrm{A} = I_l$. Otherwise the problem is not well posed. In physical terms you must specify the shape of the current profile. I think people working with EIT typically choose a function which is zero outside the electrode and has constant value at the electrode. The constant value is chosen such that the given integral constraint is satisfied.

I suppose that you can get more accurate fluxes by using a mixed method together with, e.g., Raviart-Thomas elements. The idea is to use the fluxes as degrees-of-freedom. There are lots of references on mixed methods, see for example the reference list in R. Stenberg - A Nonstandard Mixed Finite Element Family. I have not used them as means of obtaining accurate fluxes but I would guess that they are suitable for this purpose as well.