Dear Frodo,
I have written a version with Laplacian and boundary condition E\times n =0
EMtet.edp (7.8 KB)
I get for the 10 first eigenvalues
78.88776433 79.02651853 79.02651853 153.9973688 163.2695866
163.7718604 163.7718604 174.020952 175.1561748 197.8465549
It seems that there is a first eigenvalue with triple multiplicity, then an eigenvalue 154 (your fundamental mode ?), then again an eigenvalue with triple multiplicity.
You have to run the code EMtet.edp to plot (built-in with medit, but also possible with paraview with file E.pvd).
I use SLEPc as eigenvalue solver.
About the method: I use a Lagrange multiplier formulation (to avoid large coefficients in the matrix if we use penalty), using that
I fix the value E=0, p=0, phi=0 on all 6 edges.