Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection

In this work we construct and analyze a Discontinuous Galerkin method to solve advection-diffusion-reaction PDEs with anisotropic and semi-definite diffusion.
The method is designed so as to automatically detect the so-called elliptic-hyperbolic interface without requiring any further intervention.
The key idea of the method is the use of weighted average and jump operators to ensure consistency.
The error analysis provides optimal estimates in the broken graph norm and is consistent with well-known results when the problem is either hyperbolic or uniformly elliptic.
The theoretical results are supported by numerical evidence.