Discontinuous Galerkin method

Discontinuous Galerkin methods, usually shortened to DG methods, are a class of numerical methods for solving differential equations that combine features of the finite element and the finite volume frameworks. Like the continuous Galerkin finite element method they start from a weak formulation of the model problem, but they search for the solution in a space of functions that are only piecewise continuous, with polynomial pieces defined on each element of a mesh, so the numerical solution may jump across element boundaries. This makes them well suited to problems with a dominant first-order part, such as those in electrodynamics, fluid mechanics and plasma physics, where solutions can have strong gradients or even discontinuities that defeat classical finite element methods while finite volume methods stay limited to low order. The first DG method was introduced in 1973 by Reed and Hill for the hyperbolic neutron transport equation, and variants for elliptic problems developed gradually through the 1970s onward.