Grounding of the Potential

The solution of the Neumann Problem is unique only up to an additive constant: if $u$ solves it, so does $u+c$. Only potential differences are physical. A grounding condition is one linear functional $\ell$ with $\ell (1)\ne 0$ that selects one representative:

$$\ell (u)=0.$$

Common choices

grounding	continuous	discrete ($w^{\top }\mathbf{u}=0$)
zero mean over the domain	${\int }_{\Omega }u\mathrm{d}x=0$	$w=M1$
zero mean on the boundary	${\int }_{\partial \Omega }u\mathrm{d}s=0$	$w=M_{\Gamma }1$
zero sum of boundary nodal values	–	$w_i=1$ for boundary DOFs, else $0$
zero sum of all nodal values	–	$w=1$ (Euclidean minimum norm)
one grounded node / electrode	$u(x_0)=0$	$w=e_{i_0}$
CEM electrode voltages	${\sum }_{\ell }{U}_{\ell }=0$	see Complete Electrode Model

The boundary sum and the boundary mean agree, up to a factor, on a uniform boundary mesh. In general they differ by the weights of the boundary mass matrix: $w=M_{\Gamma }1$ has the entries $w_i={\int }_{\partial \Omega }{\phi }_i\mathrm{d}s$, and $w^{\top }\mathbf{u}={\int }_{\partial \Omega }{u}_h\mathrm{d}s$ holds exactly.

Mesh independence. The boundary mean is a functional of the potential alone. The nodal sum depends on how the boundary nodes are distributed: refining the mesh near one electrode adds nodes there and pulls the reference potential towards the local voltage. Solutions on different meshes then differ by a constant that has nothing to do with discretisation accuracy. This matters whenever voltages from different meshes are compared, as in convergence studies, adaptive refinement, or data simulated on a fine mesh and inverted on a coarse one. The boundary mean is therefore the robust choice. The nodal sum is a convenient approximation of it on uniform boundary meshes.

Changing the grounding afterwards. All solutions differ by constants, so any solution $\tilde{\mathbf{u}}$ can be regrounded with the (oblique) projection

$$\mathbf{u}=\tilde{\mathbf{u}}-1\frac{w^{\top }\tilde{\mathbf{u}}}{w^{\top }1}.$$

For a general null space with basis $V$ and $k$ grounding functionals $W$ this becomes $\mathbf{u}=\tilde{\mathbf{u}}-V(W^{\top }V{)}^{-1}{W}^{\top }\tilde{\mathbf{u}}$, which requires $W^{\top }V$ to be invertible. A solver can therefore work with whatever representative is most convenient, for example the one orthogonal to the null space (see Projected Conjugate Gradient) or the one with a pinned node (see Projected Cholesky Factorization), and ground at the end.

Why the boundary grounding is natural for EIT.

• The Neumann-to-Dirichlet Map is defined with ${\int }_{\partial \Omega }u=0$: only boundary voltages are measured, so the reference should be fixed by boundary values, not by the (unmeasured) interior.
• The boundary misfit $\Vert u{|}_{\partial \Omega }-f{\Vert }^2$ is not invariant under $u\mapsto u+c$. Minimising it over the constant gives ${\int }_{\partial \Omega }(u+c-f)=0$. If the measured voltages $f$ have zero boundary mean, grounding $u$ with zero boundary mean automatically picks the best-fitting constant, so the misfit only measures the physically meaningful voltage differences. Discretely, the constant that is removed from the voltage error should be taken with the same weights as the grounding: the $w$-weighted mean $\mathbf{e}-1w^{\top }\mathbf{e}/w^{\top }1$. With unit weights this reduces to the zero sum of boundary nodal values and the plain mean.
• Grounding with the interior mean, or with the full nodal sum, adds a constant offset that depends on the conductivity in the interior. That spurious offset then appears in the data misfit.

In ModularEIT.jl: boundary_grounding, reground.

References

1. P. Bochev, R. B. Lehoucq (2005). On the Finite Element Solution of the Pure Neumann Problem. SIAM Review 47(1), 50–66. doi:10.1137/S0036144503426074
2. E. Somersalo, M. Cheney, D. Isaacson (1992). Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography. SIAM J. Appl. Math. 52(4), 1023–1040. doi:10.1137/0152060
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344