Abstract
We propose a modified local discontinuous Galerkin (LDG) method for second–order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincaré–Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
F. Brezzi, M. Manzini, D. Marini, P. Pietra, A. Russo, Discontinuous Finite Elements for Diffusion Problems. Atti del Convegno in Memoria di F. Brioschi (Istiuto Lombardo di Scienze e Lettere, Milano, 1997)
D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44 (Springer, Berlin/New York, 2013)
P. Castillo, B. Cockburn, I. Perugia, D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (5), 1676–1706 (2000) (electronic)
B. Cockburn, C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (6), 2440–2463 (1998) (electronic)
D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques & Applications, vol. 69 (Springer, Berlin/New York, 2012)
T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79 (272), 2169–2189 (2010)
J.-C. Nédélec, Mixed finite elements in R 3. Numer. Math. 35 (3), 315–341 (1980)
Acknowledgements
The work of the third author was partially supported by the NSF grant DMS–1417980 and the Alfred Sloan Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
John, L., Neilan, M., Smears, I. (2016). Stable Discontinuous Galerkin FEM Without Penalty Parameters. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_17
Download citation
DOI: https://doi.org/10.1007/978-3-319-39929-4_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39927-0
Online ISBN: 978-3-319-39929-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)