A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

Lorenzo Botti, Daniele A. Di Pietro, Jérôme Droniou

Research output: Contribution to journalArticleResearchpeer-review

Abstract

In this work we propose a novel Hybrid High-Order method for the incompressible Navier–Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in hk+1 (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problems).

LanguageEnglish
Pages786-816
Number of pages31
JournalJournal of Computational Physics
Volume376
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • A priori error estimate
  • Hybrid High-Order methods
  • Incompressible Navier–Stokes equations
  • Polyhedral element methods

Cite this

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title = "A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device",
abstract = "In this work we propose a novel Hybrid High-Order method for the incompressible Navier–Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in hk+1 (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problems).",
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A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device. / Botti, Lorenzo; Di Pietro, Daniele A.; Droniou, Jérôme.

In: Journal of Computational Physics, Vol. 376, 01.01.2019, p. 786-816.

Research output: Contribution to journalArticleResearchpeer-review

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