Variational formulations of transmission problems via FEM, BEM and DtN mappings Original Research Article

The purpose of this work is to review some of the main results on the weak solvability and Galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics. Both linear and nonlinear model problems are included in our presentation. We consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or Dirichlet-to-Neumann (DtN) mappings. We provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated Galerkin schemes. Some open questions are also mentioned.