Abstract
We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D.N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19: 742–760, 1982.
I. Babuška. The finite element method with penalty. Math. Comp., 27: 221–228, 1973.
I. Babuška and M. Zlámal. Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal., 10: 863–875, 1973.
G.A. Baker. Finite element methods for elliptic equations using nonconforming elements. Math. Comp., 31: 45–59, 1977.
G.A. Baker, W.N. Jureidini, and O.A. Karakashian. Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal., 27: 1466–1485, 1990.
F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131: 267–279, 1997.
F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini. A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In R. Decuypere and G. Dibelius, editors, 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics,pages 99–108, Antwerpen, Belgium, March 5–7 1997. Technologisch Instituut.
C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. in press, special issue on Spectral, Spectral Element, and hp Methods in CFD, edited by G.E. Karniadakis, M. Ainsworth and C. Bernardi.
C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for the Navier-Stokes equations. In 10th. International Conference on Finite Element in Fluids, 1998.
R. Becker and P. Hansbo. A finite element method for domain decomposition with non-matching grids. Technical Report 3613, INRIA, 1999.
F. Brezzi, D. Marini, P. Pietra, and A. Russo. Discontinuous finite elements for diffusion problems. In Atti Convegno in onore di F. Brioschi, Istituto Lombardo, Accademia di Scienze e Lettere,1997. to appear.
F. Brezzi, D. Marini, P. Pietra, and A. Russo. Discontinuous finite elements for diffusion problems. Numerical Methods for Partial Differential Equations, 1999. submitted.
P. Castillo. An optimal error estimate for the local discontinuous Galerkin method. In this volume, pages —, 1999.
B. Cockburn, G.E. Karniadakis, and C.-W. Shu. The development of discontinuous Galerkin methods. In this volume, pages —, 1999.
B. Cockburn and C.W. Shu. The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal., 35: 2440–2463, 1998.
J. Douglas, Jr. and T. Dupont. Interior penalty procedures for elliptic and parabolic Galerkin methods, volume 58 of Lecture Notes in Physics. Springer-Verlag, Berlin, 1976.
M. Dubiner. Spectral methods on triangles and other domains. J. Sci. Comp., 6: 345–390, 1991.
I. Lomtev and G.E. Karniadakis. A discontinuous Galerkin method for the Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 29: 587–603, 1999.
J.A. Nitsche. Über ein Variationsprinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind. Abh. Math. Sem. Univ. Hamburg, 36: 9–15, 1971.
J.T. Oden, Ivo Babuška, and C.E. Baumann. A discontinuous hp finite element method for diffusion problems. J. Comput. Phys., 146: 491–519, 1998.
W.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73–479, Los Alamos Scientific Laboratory, 1973.
B. Rivière and M.F. Wheeler. A discontinuous Galerkin method applied to nonlinear parabolic equations. In this volume, pages —, 1999.
B. Rivière and M.F. Wheeler. Part I. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Technical Report 99–09, TICAM, 1999.
T. Rusten, P.S. Vassilevski, and R. Winther. Interior penalty preconditioners for mixed finite element approximations of elliptic problems. Math. Comp., 65: 447–466, 1996.
E. Siili, Ch. Schwab, and P. Houston. hp-DGFEM for partial differential equations with non-negative characteristic form. In this volume, pages —, 1999.
E. Siili and Ch. Schwab and P. Houston. hp-finite element methods for hyperbolic problems. In J. R. Whiteman, editor, The Mathematics of Finite Elements and Applications, MAFELAP X. Springer Verlag, June 1999. to appear.
M.F. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal., 15: 152–161, 1978.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D. (2000). Discontinuous Galerkin Methods for Elliptic Problems. In: Cockburn, B., Karniadakis, G.E., Shu, CW. (eds) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59721-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-59721-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64098-8
Online ISBN: 978-3-642-59721-3
eBook Packages: Springer Book Archive