Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants

We consider a simplified scalar model problem related to Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic constants. We build two variational formulations equivalent to the model problem. We show that, under some suitable conditions, both formulations are well-posed since they fit into the coercive plus compact framework. Advantages over previous studies is the validity of the formulations in the general case of Lipschitz interface between the two media and L^~ dielectric and magnetic constants. An interesting feature of these formulations is that they allow a simple finite element numerical implementation.