2019 Feb 1, Bernardo Cockburn, University of Minnesota, Variational principles for hybridizable discontinuous Galerkin methods: A short story

Bernardo Cockburn, University of Minnesota
Variational principles for hybridizable discontinuous Galerkin methods: A short story  
The so-called hybridizable discontinuous Galerkin (HDG) methods were originally introduced about a decade ago in the framework of steady-state diffusion. The guiding principle was to make it sure that the only globally-coupled unknowns were those associated with the approximation in the internet-element boundaries. Here, we present a different point of view and show how these methods can be obtained by the variational principles which were essential in the early 60s for the devising of finite element methods for solid mechanics: The principle of minimal potential energy, and the principle of minimal complementary energy. We describe these two principles and then show that each of them is associated with four types of numerical methods. We describe the various corresponding finite element methods and show how to obtain the HDG methods. Finally, using this framework, we briefly discuss several historical approaches leading to the HDG methods.