The strategies for coupling the Finite and Boundary Element Methods for second order diffusion problems appeared in the engineering literature in the seventies. Already at the end of this decade, Zienkiewicz was surveying the contributions to the subject and naming this first coupling of methods a "marriage à la mode". Claes Johnson and Jean-Claude Nédélec gave the first theoretical proof of convergence of this method. However, they found a serious drawback: because they were using discrete Fredholm theory, a particular integral operator that appears in the method had to be assumed to be compact. This is not the case if the coupling interface is a polygon, so they just took a smooth enough interface and dealt with the consistency error of having to fit finite elements near this interface. In the case of the elasticity system, not even the smoothness of the interface was enough to allow the argument hold.

This led to a theoretical bottleneck that was partially solved by the introduction of the symmetric or two-equation coupling methods by Martin Costabel and Houde Han. This type of methods has never been very popular in the engineering community, in particular because no convergence problems are seen because of the lack of smoothness of the coupling interface.

In this talk I will explain how the smoothness of the interface was, after all a purely theoretical problem. I will try to give some hints at several interesting phenomena on the problem and on situations where the simple non-symmetric coupling might not work (as numerical evidence suggests) and something else has to be done. Sometimes we will be forced to move the computational interface away from the physical interface and to wait for the stable regime. In other cases, such as coupling with Discontinuous Galerkin methods or coupling in the time domain, symmetric coupling methods seem to be the only practicable solution and some energy arguments can be invoked to support this option.