The authors introduce a new unifying framework for hybridization of finite element methods for second order elliptic problems. The framework covers mixed methods using Raviart-Thomas or Brezzi-Douglas-Marini finite elements, the classical continuous finite element method, some nonconforming finite elements and a large variety of discontinuous Galerkin methods. For the methods in this framework the only globally coupled degrees of freedom are those of an approximation of the solution on the boundary of the elements. For the discontinuous Galerkin methods this approach allows to significantly reduce the number of globally coupled degrees of freedom.
Moreover using this approach it is possible to use different methods in different subdomains and automatically couple them. This framework also allows to devise novel methods using new mortaring techniques. The authors describe the general structure of the hybridized finite element methods, given also some implementation details, and construct several examples of hybridizable finite element methods.