DG methods have in particular received considerable interest for problems with a dominant first-order part, e. Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as finite element analysis pdf technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation. The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually.

DG methods for elliptic problems were already developed in a paper by Baker in the setting of 4th order equations in 1977. A more complete account of the historical development and an introduction to DG methods for elliptic problems is given in a publication by Arnold, Brezzi, Cockburn and Marini. A number of research directions and challenges on DG methods are collected in the proceedings volume edited by Cockburn, Karniadakis and Shu. Lecture Notes in Computational Science and Engineering, 11. On a finite element method for solving the neutron transport equation. Springer Texts in Applied Mathematics 54. Springer Verlag, New York, 2008.