Discontinuous Galerkin Methods
Discontinuous Galerkin (DG) methods are a powerful class of techniques for approximating solutions to partial differential equations. My interest in DG stems from its strong mathematical theory and robust capabilities in the context of time dependent PDEs.
Classical DG methods give rise to an abundance of degrees of freedom, which poses major challenges for implicit PDEs. The Hybridizable discontinuous Galerkin (HDG) method remedies this by introducing additional Lagrange multiplier unknowns on the element boundaries. With a judicious choice of numerical flux and trace, the HDG method is capable of statically condensing its interior unknowns. This results in a smaller globally coupled linear system in terms of the additional hybrid unknowns only.
A depiction of the Hybridizable discontinuous Galerkin method is given below.