Discontinuous Galerkin method
Encyclopedia
Discontinuous Galerkin methods (DG methods) in mathematics
form a class of numerical
methods for solving partial differential equation
s. They combine features of the finite element
and the finite volume
framework and have been successfully applied to hyperbolic
, elliptic and parabolic
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 Zlamal. Interestingly 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
form a class of numerical
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
methods for solving partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s. They combine features of the finite element
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
and the finite volume
Finite volume method
The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]....
framework and have been successfully applied to hyperbolic
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
, elliptic and parabolic
Parabolic partial differential equation
A parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
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
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
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
Ivo Babuška
Ivo M. Babuška is a Czech-American mathematician, noted for his studies of the finite element method and the proof of the Babuška-Lax-Milgram theorem in partial differential equations. One of the celebrated result in the finite elements is the so-called Babuska-Brezzi condition, which provides...
, J.-L. Lions
Jacques-Louis Lions
Jacques-Louis Lions ForMemRS was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John Von Neumann prize in 1986. Lions is listed as an ISI highly cited researcher.-Biography:After...
, Nitsche and Zlamal. Interestingly 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.