Discontinuous Galerkin method
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, Joachim Nitsche and Miloš Zlámal. DG methods for elliptic problems were already developed in a paper by Garth 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. Overview. 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 formula_1 in a spatial domain formula_2 without "sources" or "sinks" : where formula_4 is the flux of formula_1. Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain formula_2 restricted to a discrete triangulation formula_7, written as The index formula_13 should show the relation to an underlying discretization given by formula_14. Note here that formula_9 is not uniquely defined at the intersection points formula_16. At first we make use of a specific polynomial basis on the interval formula_17, the Legendre polynomials formula_18 , i.e., Note especially the orthogonality relations Transformation onto the interval formula_21, and normalization is achieved by functions formula_22 which fulfill the orthonormality relation Transformation onto an interval formula_10 is given by formula_26 which fulfill For formula_29-normalization we define formula_30, and for formula_31-normalization we define formula_32, s.t. Finally, we can define the basis representation of our solutions formula_34 Note here, that formula_34 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. DG-scheme. 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 formula_39 with where formula_41 denotes the left- and right-hand sided limits. Finally, the "DG-Scheme" can be written as Scalar elliptic equation. A scalar elliptic equation is of the form This equation is the steady-state heat equation, where formula_44 is the temperature. Space discretization is the same as above. We recall that the interval formula_45 is partitioned into formula_46 intervals of length formula_47. We introduce jump formula_48 and average formula_49 and The  linear forms formula_51 and formula_52 are and The penalty parameter formula_55 is a positive constant. Increasing its value will reduce the jumps in the discontinuous solution. The term formula_56 is chosen to be equal to formula_57 for the symmetric interior penalty Galerkin method; it is equal to formula_58 for the non-symmetric interior penalty Galerkin method. Direct discontinuous Galerkin method. The direct discontinuous Galerkin (DDG) method is a new discontinuous Galerkin method for solving diffusion problems. In 2009, Liu and Yan first proposed the DDG method for solving diffusion equations. The advantages of this method compared with Discontinuous Galerkin method  is that the direct discontinuous Galerkin method derives the numerical format by directly taking the numerical flux of the function and the first derivative term without introducing intermediate variables. We still can get a reasonable numerical results by using this method, and the derivation process is more simple, the amount of calculation is greatly reduced. The direct discontinuous finite element method is a branch of the Discontinuous Galerkin methods. It mainly includes transforming the problem into variationally form, regional unit splitting, constructing basis functions, forming and solving discontinuous finite element equations, and convergence and error analysis. For example, consider a  nonlinear diffusion equation, which is one-dimensional: Space discretization. Firstly, define formula_61, and formula_62. Therefore we have done the space discretization of formula_63. Also, define formula_64. We want to find an approximation formula_65 to formula_66 such that formula_67, formula_68, formula_69, formula_70 is the polynomials space in formula_71 with degree at formula_72 and lower than formula_72. Formulation of the scheme. Flux: formula_74. formula_75: the exact solution of the equation. Multiply the equation with a smooth function formula_76 so that we obtain the following equations: formula_77, formula_78 Here formula_79 is arbitrary, the exact solution formula_66 of the equation is replaced by the  approximate solution formula_65, that is to say, the numerical solution we need is obtained by solving the differential equations. The numerical flux. Choosing a proper numerical flux is critical for the accuracy of DDG method. The numerical flux needs to satisfy the following conditions: ♦ It is consistent with formula_82 ♦ The numerical flux is conservative in the single value on formula_83. ♦ It has the formula_84-stability; ♦ It can improve the accuracy of the method. Thus, a general scheme for numerical flux is given: In this flux, formula_72 is the maximum order of polynomials in two neighboring computing  units. formula_87 is integral function. Note that in non-uniform grids, formula_63 should be formula_89 and formula_90 in uniform grids. Error estimates. Denote that the error between the exact solution formula_66 and the numerical solution formula_65 is formula_93 . We measure the error with the following norm: formula_94 and we have formula_95,formula_96