Restrict Function Space to an Internal Boundary

1

Hi, I want to solve a Poison problem in two subdomains (with different permittivity) separated by an interface.

The only free charge present is the surface charge density at the interface between the two subdomains. This surface charge density depends (non-linearly) itself on the electric fields at both of the subdomains.

My idea is to build a mixed function space with the potential and the surface charge, but I do not know how to restrict a function space only at an Internal Boundary.

I’m aiming for something like this:

mesh = Mesh(“Mesh_new.xml”)
boundaries = MeshFunction(“size_t”, mesh, “Mesh_facet_region_new.xml”)
domains = MeshFunction(“size_t”, mesh, “Mesh_physical_region_new.xml”)

# FUNCTION SPACES #
potential = FiniteElement(‘P’,mesh.ufl_cell(), 2)
SurfaceCharge = FiniteElement(‘P’,mesh.ufl_cell,1)[boundaries][‘1’]
W = FunctionSpace(mesh, U*SurfaceCharge)

Where I have the interface marked in the function “boundaries” with the marker “1”. (This is obviously not correct)

2

You might take a look at Francesco Ballarin’s library multiphenics:

In particular, Tutorial 3 on Lagrange multipliers illustrates a mechanism for restricting functions to subdomains, including an internal boundary.