Neumann Problem

In the Neumann problem the current density on the boundary is prescribed. Given $g\in H^{-1/2}(\partial \Omega )$, find $u$ with

$$\nabla \cdot (\gamma \nabla u)=0\text{in }\Omega ,\gamma {\partial }_{\nu }u=g\text{on }\partial \Omega ,$$

where $\nu$ is the outward unit normal. Here $g$ is the current flowing into the body per unit boundary length or area: with $\mathbf{J}=-\gamma \nabla u$, it equals $-\mathbf{J}\cdot \nu$.

Compatibility. Integrating the equation over $\Omega$ and using the divergence theorem shows that a solution can only exist if

$${\int }_{\partial \Omega }g\mathrm{d}s=0,$$

that is, the injected current equals the extracted current (charge conservation).

Non-uniqueness up to constants. If $u$ solves the problem, so does $u+c$ for every $c\in \mathbb{R}$. Only potential differences are physical. Uniqueness is restored by grounding: fix ${\int }_{\partial \Omega }u\mathrm{d}s=0$, or fix $u$ at one point, or work in $H^1(\Omega )/\mathbb{R}$.

Weak form. Find $u\in H^1(\Omega )/\mathbb{R}$ such that

$${\int }_{\Omega }\gamma \nabla u\cdot \nabla v\mathrm{d}x={\int }_{\partial \Omega }gv\mathrm{d}s\forall v\in H^1(\Omega ).$$

Coercivity on $H^1(\Omega )/\mathbb{R}$ follows from the Poincaré–Wirtinger inequality (see Sobolev and Trace Spaces and Lax-Milgram Theorem). The derivation is in Weak Formulation of the Conductivity Equation. How the null space is handled discretely is described in Null Space of the Neumann Problem.

The map $g\mapsto u{|}_{\partial \Omega }$ is the Neumann-to-Dirichlet Map.

In ModularEIT.jl: forward_neumann.

References

1. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019
2. M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Review 41(1), 85–101. doi:10.1137/S0036144598333613