On discontinuous Galerkin methods

Discontinuous Galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present. Several alternatives for treating the diffusion and advective fluxes have been introduced. This report summarizes some of the methods that have been proposed. Several numerical examples are included in the paper. These present discontinuous Galerkin solutions of one-dimensional problems with a scalar variable. Results are presented for diffusion–reaction problems and advection–diffusion problems. We discuss the performance of various formulations with respect to accuracy as well as stability of the method. Copyright © 2003 John Wiley & Sons, Ltd.

[1]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[2]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[3]  George Em Karniadakis,et al.  A Discontinuous Galerkin Method for the Viscous MHD Equations , 1999 .

[4]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[5]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[6]  O. C. Zienkiewicz,et al.  Finite element methods for second order differential equations with significant first derivatives , 1976 .

[7]  Yoshio Ando,et al.  A new variational functional for the finite-element method and its application to plate and shell problems , 1972 .

[8]  Pigache.  A SEMITOROIDAL REFLEX DISCHARGE AS A PROPULSION DEVICE. Technical Report No. 0253-4 , 1963 .

[9]  W. H. Reed,et al.  TRIPLET: a two-dimensional, multigroup, triangular mesh, planar geometry, explicit transport code , 1973 .

[10]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[11]  Bernardo Cockburn.  An introduction to the Discontinuous Galerkin method for convection-dominated problems , 1998 .

[12]  J. Oden,et al.  A discontinuous hp finite element method for convection—diffusion problems , 1999 .

[13]  D. Arnold,et al.  Discontinuous Galerkin Methods for Elliptic Problems , 2000 .

[14]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[15]  E Rivoalen,et al.  Numerical Simulation of Axisymmetric Viscous Flows by Means of a Particle Method , 1999 .

[16]  George Em Karniadakis,et al.  Regular Article: A Discontinuous Galerkin ALE Method for Compressible Viscous Flows in Moving Domains , 1999 .

[17]  Bernardo Cockburn,et al.  Discontinuous Galerkin Methods for Convection-Dominated Problems , 1999 .

[18]  J. Tinsley Oden,et al.  A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems , 2000 .

[19]  Theodore H. H. Pian,et al.  Basis of finite element methods for solid continua , 1969 .

[20]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .