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Interior Penalty Continuous and Discontinuous Finite Element Approximations of Hyperbolic Equations

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posted on 2023-06-08, 10:08 authored by Erik Burman, Alfio Quarteroni, Benjamin Stamm
In this paper we present in a unified setting the continuous and discontinuous Galerkin methods for the numerical approximation of the scalar hyperbolic equation. Both methods are stabilized by the interior penalty method, more precisely by the jump of the gradient across element faces in the continuous case whereas in the discontinuous case the stabilization of the jump of the solution and optionally of its gradient is required to achieve optimal convergence. We prove that the solution in the case of the continuous Galerkin approach can be considered as a limit of the discontinuous one when the stabilization parameter associated with the penalization of the solution jump tends to infinity. As a consequence, the limit of the numerical flux of the discontinuous method yields a numerical flux for the continuous method as well. Numerical results will highlight the theoretical results that are proven in this paper.

History

Publication status

  • Published

ISSN

0885-7474

Publisher

SPRINGER/PLENUM PUBLISHERS, 233 SPRING ST, NEW YORK, NY 10013 USA

Issue

3

Volume

43

Page range

293-312

Presentation Type

  • paper

Event name

International Conference on Recent Developments of Numerical Schemes for Flow Problems

Event location

Fukuoka, JAPAN, JUN 27-29, 2007

Event type

conference

Department affiliated with

  • Mathematics Publications

Notes

SOURCE: JOURNAL OF SCIENTIFIC COMPUTING

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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