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A variational H (div) finite-element discretization approach for perfect incompressible fluids

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Title: A variational H (div) finite-element discretization approach for perfect incompressible fluids
Authors: Natale, A
Cotter, CJ
Item Type: Journal Article
Abstract: We propose a finite-element discretization approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite-element method that arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite- element H ( div ) vector fields are identified with advection operators; this is the first successful extension of the structure-preserving discretization of Pavlov et al. ( 2009 ) to the finite-element setting. The resulting algorithm coincides with the energy-conserving scheme proposed by Guzm ́ an et al. ( 2016 ). Through the variational derivation, we discover that it also satisfies a discrete analogous of Kelvin’s circulation theorem. Further, we propose an upwind-stabilized version of the scheme that dissipates enstrophy while preserving energy conservation and the discrete Kelvin’s theorem. We prove error estimates for this version of the scheme, and we study its behaviour through numerical tests.
Issue Date: 29-Jun-2017
Date of Acceptance: 16-May-2017
URI: http://hdl.handle.net/10044/1/50007
DOI: https://dx.doi.org/10.1093/imanum/drx033
ISSN: 0272-4979
Publisher: OXford University Press
Start Page: 1388
End Page: 1419
Journal / Book Title: IMA Journal of Numerical Analysis
Volume: 38
Issue: 3
Copyright Statement: © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Sponsor/Funder: Natural Environment Research Council (NERC)
Natural Environment Research Council (NERC)
Natural Environment Research Council (NERC)
Natural Environment Research Council (NERC)
Engineering & Physical Science Research Council (EPSRC)
Natural Environment Research Council (NERC)
Funder's Grant Number: NE/I016007/1
NE/I000747/1
NE/I02013X/1
NE/K006789/1
EP/L000407/1
NE/M013634/1
Keywords: math.NA
math-ph
math.MP
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
Numerical & Computational Mathematics
Publication Status: Published
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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