Conditions

Tested Dirichlet BC, which just enforces given potential (zero) at some surfaces Air cube for boundaries was increased to 0.5 meter in size, with the 4 spheres (0.09 meter radius) being in the middle of it Meshes are coarse: 5 mm max element size

4 spheres in air, realistic radii and conductivities

Tested positions:

Electrode in the skin
Electrode in the skin + air cube having enforced 0 on all boundaries
Electrode in the brain (deep) + air cube having enforced 0 on all 6 sufaces
air cube having enforced 0 on all 6 surfaces

Dirichlet BC Monopoles

Monopole at the center

All curves are re-grounded at z=-0.05

boundary without electrode explodes into 1e9 range inside of the spheres, and quickly decays to 0 towards the boundaries of the domain

Top of the sphere:

re-grounded at 0.08

Boundary only BC, without ground electrode has potential of gigavolts inside the sphere, and has a different decay, symmetrical, unlike others, due to no 0 enforement inside of the spheres, but also shape is slightly different from other MFEM solutions. More than that MFEM solution with only far boundary BC follows free space kCSD curve EXACTLY until the boundary of the materials!!!

Grounding electrode on the sphere introduces artifacts. Let's look at them:

1 Amper point source, location: just below the skull in the brain: 0,0,0.0785

Grounding electrode in the skin

(domain spans from -0.1 to 0.2)

far boundary and electrode in the skin

far boundary and electrode in the brain tissue

only far boundary on the cube

free space potential (sigma=0.33)

Just to compare:

Dirichlet BC Dipoles

Dipole at 0,0,0.3

All curves are re-grounded at z=-0.04

Most likely due to having GIANT monopole potentials inside the 4 spheres the boundary only BC potential for dipoles has floating point quantization issues (dipole is simulated using 2 point sources)