I am not able to solve a transmission problem for the Laplace problem using the hybridized mixed method with static condensation. I wonder if the “element_boundary” method gives a normal vector with the same orientation from both sides of the interface.

Programming the same problem using the “skeleton” NGSolve style gives the right answer.

in an element-boundary integral, the normal vector is always pointing outside of the element.

The problem with your ‘wrong solution’ is that a discontinuous H(div) has no Trace anymore.

To build the right hand side with the jump1 - term, you can do element-boundary integrals everywhere, and multiply with a selection function set to 1 at the interface.