For example, consider the problem of finding the electric potential (Laplace problem) inside 2 concentric dielectric spheres of different conductivities ε1 and ε2 outside and inside respectively (homogeneous, isotropic within each sphere). We know the potential (Dirichlet condition) at each point on the outer surface of the outer-most sphere.

The surface of the inner sphere forms the boundary between 2 dielectrics. How do I specify the Neumann condition at this boundary (reference: equation 20 in Electrostatic Boundary Conditions - Ximera):

- First case: when there is no free charge at the interface
- Second case: when there is free charge density of known magnitude at the surface.

A snippet or tutorial demonstrating the specification of the grid functions and operators at the boundaries would be very helpful !!