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The dynamical behavior of the discontinuous Galerkin method and related difference schemes


Authors: Donald J. Estep and Andrew M. Stuart
Journal: Math. Comp. 71 (2002), 1075-1103
MSC (2000): Primary 65L07
DOI: https://doi.org/10.1090/S0025-5718-01-01364-3
Published electronically: November 21, 2001
MathSciNet review: 1898746
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Abstract: We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.


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Additional Information

Donald J. Estep
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: estep@math.colostate.edu

Andrew M. Stuart
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
Email: stuart@maths.warwick.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-01-01364-3
Keywords: Attractors, contractivity, discontinuous Galerkin method, dissipativity, dynamical system, existence, initial value problems, quadrature
Received by editor(s): May 24, 1999
Received by editor(s) in revised form: September 12, 2000
Published electronically: November 21, 2001
Additional Notes: The research of the first author was partially supported by the National Science Foundation, DMS 9805748.
The research of the second author was partially supported by the Office of Naval Research under grant No. N00014-92-J-1876 and by the National Science Foundation under grant No. DMS-9201727.
Article copyright: © Copyright 2001 American Mathematical Society