As you want your solution to be discontinuous, you need to use the discontinuous galerkin methods. See for instance http://www.maths.nuigalway.ie/~eoghan/GroupPresentations/Roisin-27Oct2017.pdf
Alternatively, I believe you can use @cdaversin framework for mixed dimensional finite element methods, see: https://arxiv.org/pdf/1911.01166.pdf