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Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 545-563. doi: 10.3934/nhm.2010.5.545
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Remarks on discretizations of convection terms in Hybrid mimetic mixed methods

  • 1.
    Université Montpellier 2, Institut de Mathématiques et de Modélisation de Montpellier, CC 051, Place Eugène Bataillon, 34095 Montpellier cedex 5
  • Received: 01 January 2010 Revised: 01 April 2010

  • Primary: 65N12, 65N30, 65M12, 65M60.

  • We present different ways, coming from Finite Volume or Mixed Finite Element frameworks, to discretize convection terms in Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods for elliptic equations. We compare them through several numerical tests, deducing some generic principles, depending on the situation, on the choice of an apropriate method and its parameters. We also present an adaptation to the Navier-Stokes equations, with a numerical tests in the case of the lid-driven cavity.

    Citation: Jérôme Droniou. Remarks on discretizations of convection terms in Hybrid mimetic mixed methods[J]. Networks and Heterogeneous Media, 2010, 5(3): 545-563. doi: 10.3934/nhm.2010.5.545

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  • We present different ways, coming from Finite Volume or Mixed Finite Element frameworks, to discretize convection terms in Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods for elliptic equations. We compare them through several numerical tests, deducing some generic principles, depending on the situation, on the choice of an apropriate method and its parameters. We also present an adaptation to the Navier-Stokes equations, with a numerical tests in the case of the lid-driven cavity.


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    1. Lorenzo Botti, Daniele A. Di Pietro, Jérôme Droniou, A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device, 2019, 376, 00219991, 786, 10.1016/j.jcp.2018.10.014
    2. Vikendra Singh, A Local Meshless Method for Steady State Convection Dominated Flows, 2017, 14, 0219-8762, 1750067, 10.1142/S0219876217500670
    3. Jerome Droniou, Finite volume schemes for diffusion equations: Introduction to and review of modern methods, 2014, 24, 0218-2025, 1575, 10.1142/S0218202514400041
    4. Claire Chainais-Hillairet, Maxime Herda, Simon Lemaire, Julien Moatti, Long-time behaviour of hybrid finite volume schemes for advection–diffusion equations: linear and nonlinear approaches, 2022, 151, 0029-599X, 963, 10.1007/s00211-022-01289-w
    5. Ophélie Angelini, Konstantin Brenner, Danielle Hilhorst, A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation, 2013, 123, 0029-599X, 219, 10.1007/s00211-012-0485-5
    6. Daniele A. Di Pietro, Jérôme Droniou, Alexandre Ern, A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes, 2015, 53, 0036-1429, 2135, 10.1137/140993971
    7. Daniele Antonio Di Pietro, Jérôme Droniou, 2020, Chapter 9, 978-3-030-37202-6, 421, 10.1007/978-3-030-37203-3_9
    8. A. Abushaikha, K. Terekhov, Adaptive Dynamic Grids and Mimetic Finite Difference Method for Miscible Displacement Problem, 2024, 45, 1995-0802, 143, 10.1134/S1995080224010025
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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