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. Indeed, the solutions of such problems may involve strong gradients (and even discontinuities) so that classical finite element methods fail, while finite volume methods are restricted to low order approximations.

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

edit

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 law

edit

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 discretization

edit

The  -space will be discretized as

 

Furthermore, we need the following definitions

 

Basis for function space

edit

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.

DG-scheme

edit

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

 

Scalar elliptic equation

edit

A scalar elliptic equation is of the form

 

This equation is the steady-state heat equation, where   is the temperature. Space discretization is the same as above. We recall that the interval   is partitioned into   intervals of length  .

We introduce jump   and average   of functions at the node  :

 

The interior penalty discontinuous Galerkin (IPDG) method is: find   satisfying

 

where the bilinear forms   and   are

 

and

 

The linear forms   and   are

 

and

 

The penalty parameter   is a positive constant. Increasing its value will reduce the jumps in the discontinuous solution. The term   is chosen to be equal to   for the symmetric interior penalty Galerkin method; it is equal to   for the non-symmetric interior penalty Galerkin method.

Direct discontinuous Galerkin method

edit

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.[1][2] 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 variational 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:

 , in which  

Space discretization

edit

Firstly, define  , and  . Therefore we have done the space discretization of  . Also, define  .

We want to find an approximation   to   such that  ,  ,

 ,   is the polynomials space in   with degree at most  .

Formulation of the scheme

edit

Flux:  .

 : the exact solution of the equation.

Multiply the equation with a smooth function   so that we obtain the following equations:

 ,

 

Here   is arbitrary, the exact solution   of the equation is replaced by the approximate solution  , that is to say, the numerical solution we need is obtained by solving the differential equations.

The numerical flux

edit

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  

♦ The numerical flux is conservative in the single value on  .

♦ It has the  -stability;

♦ It can improve the accuracy of the method.

Thus, a general scheme for numerical flux is given:

 

In this flux,   is the maximum order of polynomials in two neighboring computing units.   is the jump of a function. Note that in non-uniform grids,   should be   and   in uniform grids.

Error estimates

edit

Denote that the error between the exact solution   and the numerical solution   is   .

We measure the error with the following norm:

 

and we have  , 

See also

edit

References

edit
  1. ^ Hailiang Liu, Jue Yan, The Direct Discontinuous Galerkin (DDG) Methods For Diffusion Problems,SIAM J. NUMER. ANAL. Vol. 47, No. 1, pp. 675–698.
  2. ^ Hailiang Liu, Jue Yan, The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections, Commun. Comput. Phys. Vol. 8, No. 3, pp. 541-564.