Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes

Daniele A. Di Pietro, Jérôme Droniou, Gianmarco Manzini

Research output: Contribution to journalArticleResearchpeer-review

15 Citations (Scopus)

Abstract

In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

Original languageEnglish
Pages (from-to)397-425
Number of pages29
JournalJournal of Computational Physics
Volume355
DOIs
Publication statusPublished - 15 Feb 2018

Keywords

  • Gradient discretisation methods
  • Gradient schemes
  • High-order Mimetic Finite Difference methods
  • Hybrid High-Order methods
  • Non-linear problems
  • Virtual Element methods

Cite this

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abstract = "In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–N{\'e}d{\'e}lec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.",
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Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes. / Di Pietro, Daniele A.; Droniou, Jérôme; Manzini, Gianmarco.

In: Journal of Computational Physics, Vol. 355, 15.02.2018, p. 397-425.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Droniou, Jérôme

AU - Manzini, Gianmarco

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N2 - In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

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