Discontinuous Galerkin method

Discontinuous Galerkin methods (DG methods) in mathematics form a class of numerical methods for solving partial differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic and parabolic problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.

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. However, among the early influential contributors were Babuška, J.-L. Lions, Nitsche and Zlamal. Interestingly 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.

See also

• Galerkin method

References

• D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39(5):1749-1779, 2002.
• G. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59.
• B. Cockburn, G. E. Karniadakis and C.-W. Shu (eds.), Discontinuous Galerkin methods. Theory, computation and applications, Lecture Notes in Computational Science and Engineering, 11. Springer-Verlag, Berlin, 2000.
• D.A. Di Pietro and A. Ern, "Mathematical Aspects of Discontinuous Galerkin Methods". Mathématiques et Applications, Vol. 69, Springer-Verlag, Berlin, 2011.
• J.S. Hesthaven and T. Warburton, "Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications". Springer Texts in Applied Mathematics 54. Springer Verlag, New York, 2008.
• CFD Wiki http://www.cfd-online.com/Wiki/Discontinuous_Galerkin
• W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.