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, 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.

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 $\rho$ in a spatial domain $\Omega$ without "sources" or "sinks" :

{\frac {\partial \rho }{\partial t}}+\nabla \cdot {\mathbf {J}}=0,

where $\mathbf {J}$ is the flux of $\rho$ .

Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain $\Omega$ restricted to a discrete triangulation $\Omega _{{h}}$ , written as

S_{{h}}^{{p}}(\Omega _{{h}})=\{v_{{|\Omega _{{e_{{i}}}}}}\in P^{{p}}(\Omega _{{e_{{i}}}}),\ \ \forall \Omega _{{e_{{i}}}}\in \Omega _{{h}}\}

for $P^{{p}}(\Omega _{{e_{{i}}}})$ the space of polynomials with degrees less than or equal to $p$ over element $\Omega _{{e_{{i}}}}$ indexed by $i$ . Then for finite element shape functions $N_{{j}}\in P^{{p}}$ the solution is represented by

\rho _{{h}}=\sum _{{j=1}}^{{dofs}}\rho _{{j}}^{{i}}(t)N_{{j}}^{{i}}({\boldsymbol {x}}),\quad \forall {\boldsymbol {x}}\in \Omega _{{e_{{i}}}}.

Then similarly choosing a test function

\phi _{{h}}({\boldsymbol {x}})=\sum _{{j=1}}^{{dofs}}\phi _{{j}}^{{i}}N_{{j}}^{{i}}({\boldsymbol {x}}),\quad \forall {\boldsymbol {x}}\in \Omega _{{e_{{i}}}},

multiplying the continuity equation by $\phi _{h}$ and integrating by parts in space, the semidiscrete DG formulation becomes:

{\frac {d}{dt}}\int _{{\Omega _{{e_{{i}}}}}}\rho _{{h}}\phi _{{h}}d{\boldsymbol {x}}+\int _{{\partial \Omega _{{e_{{i}}}}}}\phi _{{h}}{\mathbf {J}}_{{h}}\cdot {\boldsymbol {n}}d{\boldsymbol {x}}=\int _{{\Omega _{{e_{{i}}}}}}{\mathbf {J}}_{{h}}\cdot \nabla \phi _{{h}}d{\boldsymbol {x}}.

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.
B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM Frontiers in Applied Mathematics, 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.

Numerical partial differential equations by method

Finite difference

Parabolic	Forward-time central-space (FTCS) Crank–Nicolson

Hyperbolic	Lax–Friedrichs Lax–Wendroff MacCormack Upwind Method of characteristics

Others	Alternating direction-implicit (ADI) Finite-difference time-domain (FDTD)

Finite volume

Finite element

Meshless/Meshfree

Smoothed-particle hydrodynamics (SPH)
Moving Particle Semi-implicit Method (MPS)
Material point method (MPM)

Domain decomposition

Others

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Discontinuous Galerkin method

Overview

See also

References