In this talk, I present an overview of some recent developments in Discontinuous Galerkin Finite Element Methods (DGFEEMs) and their applications in physics and engineering.

DGFEEMs are a class of finite element methods that allow for discontinuities in the finite element solutions across faces and edges. This lack of continuity introduces extra flexibility that can overcome limitations of standard finite element methods and their implementation.

After a short discussion on how easy is for DGFEEM to accommodate non-standard finite element shapes and non-standard shape functions, I describe how to incorporate automatic mesh-adaptivity in such methods. Mesh adaptivity is becoming a very useful tool to improve the accuracy of finite element simulations.

The general idea behind mesh adaptivity is to enrich the finite element space where it is necessary to reduce the error and introducing the smallest number of extra degrees of freedom. In many cases, the most advanced adaptive techniques can achieve exponential convergence to the right solution.

If time allows, I will introduce further extensions of DGFEMs to multi-level methods and geometric multi-level preconditioners.