Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes
Abstract
Here, in this work we develop arbitraryorder 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 highorder 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 highorder residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limitconformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Lastly, links with Hybrid HighOrder, nonconforming Mimetic Finite Difference and nonconforming Virtual Element methods are also studied. Numerical examples complete the exposition.
 Authors:

 Univ. Montpellier (France)
 Monash University, Melbourne (Australia)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA); Laboratory Directed Research and Development Program (LDRD)
 OSTI Identifier:
 1415391
 Report Number(s):
 LAUR1724418
Journal ID: ISSN 00219991; TRN: US1800792
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 355; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; Gradient discretisation methods; Gradient schemes; Highorder Mimetic Finite Difference methods; Hybrid HighOrder methods; Virtual Element methods; Nonlinear problems
Citation Formats
Di Pietro, Daniele A., Droniou, Jérôme, and Manzini, Gianmarco. Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.11.018.
Di Pietro, Daniele A., Droniou, Jérôme, & Manzini, Gianmarco. Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes. United States. doi:10.1016/j.jcp.2017.11.018.
Di Pietro, Daniele A., Droniou, Jérôme, and Manzini, Gianmarco. Tue .
"Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes". United States. doi:10.1016/j.jcp.2017.11.018. https://www.osti.gov/servlets/purl/1415391.
@article{osti_1415391,
title = {Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes},
author = {Di Pietro, Daniele A. and Droniou, Jérôme and Manzini, Gianmarco},
abstractNote = {Here, in this work we develop arbitraryorder 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 highorder 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 highorder residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limitconformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Lastly, links with Hybrid HighOrder, nonconforming Mimetic Finite Difference and nonconforming Virtual Element methods are also studied. Numerical examples complete the exposition.},
doi = {10.1016/j.jcp.2017.11.018},
journal = {Journal of Computational Physics},
number = C,
volume = 355,
place = {United States},
year = {2017},
month = {11}
}
Web of Science
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