I am interested in solving a Laplace problem in a domain made of two sub domains (transmission problem). The solution is regular in each subdomain but the jump of the trace of the solution on the interface is no zero. So we cannot use finite element space P1 in the whole domain. I would have to use P1dc and use a Lagrange multiplier to specify that the jump is zero in each edges except in the interface.
Has anyone done this before? If so, do you have any tips?