Discontinuous Galerkin method

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In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form 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 Miloš Zlámal. 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.



Much like the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method is a finite element method formulated relative to a weak formulation of a particular model system. Unlike traditional CG methods that are conforming, the DG method works over a trial space of functions that are only piecewise continuous, and thus often comprise more inclusive function spaces than the finite-dimensional inner product subspaces utilized in conforming methods.

As an example, consider the continuity equation for a scalar unknown   in a spatial domain   without "sources" or "sinks" :


where   is the flux of  .

Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain   restricted to a discrete triangulation  , written as


for   the space of polynomials with degrees less than or equal to   over element   indexed by  . Then for finite element shape functions   the solution is represented by


Then similarly choosing a test function


multiplying the continuity equation by   and integrating by parts in space, the semidiscrete DG formulation becomes:


Scalar hyperbolic conservation lawEdit

A scalar hyperbolic conservation law is of the form


where one tries to solve for the unknown scalar function  , and the functions   are typically given.

Space discretizationEdit

The  -space will be discretized as


Furthermore, we need the following definitions


Basis for function spaceEdit

We derive the basis representation for the function space of our solution  . The function space is defined as


where   denotes the restriction of   onto the interval  , and   denotes the space of polynomials of maximal degree  . The index   should show the relation to an underlying discretization given by  . Note here that   is not uniquely defined at the intersection points  .

At first we make use of a specific polynomial basis on the interval  , the Legendre polynomials  , i.e.,


Note especially the orthogonality relations


Transformation onto the interval  , and normalization is achieved by functions  


which fulfill the orthonormality relation


Transformation onto an interval   is given by  


which fulfill


For  -normalization we define  , and for  -normalization we define  , s.t.


Finally, we can define the basis representation of our solutions  


Note here, that   is not defined at the interface positions.

Besides, prism bases are employed for planar-like structures, and are capable for 2-D/3-D hybridation.


The conservation law is transformed into its weak form by multiplying with test functions, and integration over test intervals


By using partial integration one is left with


The fluxes at the interfaces are approximated by numerical fluxes   with


where   denotes the left- and right-hand sided limits. Finally, the DG-Scheme can be written as


See alsoEdit