DG methods have in particular received considerable interest for problems with a dominant what is finite element analysis pdf-order part, e. Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a 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. SIAM Frontiers in Applied Mathematics, 2008. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.