Thermal Contact Resistance and Hybrid DG

Dear community,

I am looking to solve a heat conduction problem with thermal contact resistance over an interface in the interior of my domain. For simplicity, let us consider a rectangle split in two by an interface named Gamma. The temperature can (and should) be discontinuous across the interface, while the heat flux cannot. The heat flux across the interface is given by a known function depending on the local temperature on either side of the interface; let us denote them T+ and T-.

An interface condition like this can be readily expressed using a standard DG approach, but I am now attempting to use hybrid DG instead, due to e.g. this issue [1]. However, I struggle to find the appropriate syntax for setting up my problem using HDG.

The root cause of the issue is that the intermediate temperature T* has no natural interpretation in my model. My envisioned remedy is to enforce T-=T* and then use T* in place of T- in the model describing the heat flux across the interface. The problem then is figuring out how to access T+ and T- independently such that I can achieve this goal.

Equivalently, for the first equation under the heading Hybrid DG in Tutorial 2.8 [2], how can one split the sum in the rightmost term into one sum over F+ and another over F-, such that different integrands can be used for the two sides of the interface?

Do you have any recommendations?

[1]  github.com/NGSolve/ngsolve/issues/63
[2]  docu.ngsolve.org/latest/i-tutorials/unit-2.8-DG/DG.html

Best,
Sindre