Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes
Abstract
Here, 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. Lastly, links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.
- Authors:
-
[1];
[3]
- Univ. Montpellier (France)
- Monash University, Melbourne (Australia)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (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):
- LA-UR-17-24418
Journal ID: ISSN 0021-9991; TRN: US1800792
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 355; Journal Issue: C; Journal ID: ISSN 0021-9991
- 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; High-order Mimetic Finite Difference methods; Hybrid High-Order methods; Virtual Element methods; Non-linear 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. https://doi.org/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. https://doi.org/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 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. Lastly, links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming 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 = {Tue Nov 21 00:00:00 EST 2017},
month = {Tue Nov 21 00:00:00 EST 2017}
}
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Works referenced in this record:
An Interior Penalty Finite Element Method with Discontinuous Elements
journal, August 1982
- Arnold, Douglas N.
- SIAM Journal on Numerical Analysis, Vol. 19, Issue 4
The nonconforming virtual element method
journal, May 2016
- Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
Basic Principles of Virtual Element Methods
journal, November 2012
- BeirÃO Da Veiga, L.; Brezzi, F.; Cangiani, A.
- Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 01
Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes
journal, May 2015
- Bonelle, Jérôme; Di Pietro, Daniele A.; Ern, Alexandre
- Computer Aided Geometric Design, Vol. 35-36
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes
journal, January 2005
- Brezzi, Franco; Lipnikov, Konstantin; Shashkov, Mikhail
- SIAM Journal on Numerical Analysis, Vol. 43, Issue 5
A Family of Mimetic Finite Difference Methods on Polygonal and Polyhedral Meshes
journal, October 2005
- Brezzi, Franco; Lipnikov, Konstantin; Simoncini, Valeria
- Mathematical Models and Methods in Applied Sciences, Vol. 15, Issue 10
The NonConforming Virtual Element Method for the Stokes Equations
journal, January 2016
- Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco
- SIAM Journal on Numerical Analysis, Vol. 54, Issue 6
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
journal, January 2000
- Castillo, Paul; Cockburn, Bernardo; Perugia, Ilaria
- SIAM Journal on Numerical Analysis, Vol. 38, Issue 5
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods
journal, May 2016
- Cockburn, Bernardo; Di Pietro, Daniele A.; Ern, Alexandre
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009
- Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho
- SIAM Journal on Numerical Analysis, Vol. 47, Issue 2
A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes
journal, December 2016
- Di Pietro, Daniele A.; Droniou, Jérôme
- Mathematics of Computation, Vol. 86, Issue 307
Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems
journal, April 2017
- Di Pietro, Daniele A.; Droniou, Jérôme
- Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 05
A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes
journal, January 2015
- Di Pietro, Daniele A.; Droniou, Jérôme; Ern, Alexandre
- SIAM Journal on Numerical Analysis, Vol. 53, Issue 5
A hybrid high-order locking-free method for linear elasticity on general meshes
journal, January 2015
- Di Pietro, Daniele A.; Ern, Alexandre
- Computer Methods in Applied Mechanics and Engineering, Vol. 283
Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015
- Di Pietro, Daniele A.; Ern, Alexandre
- Comptes Rendus Mathematique, Vol. 353, Issue 1
An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators
journal, October 2014
- Di Pietro, Daniele A.; Ern, Alexandre; Lemaire, Simon
- Computational Methods in Applied Mathematics, Vol. 14, Issue 4
An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow
journal, August 2014
- Di Pietro, Daniele A.; Lemaire, Simon
- Mathematics of Computation, Vol. 84, Issue 291
A mixed finite volume scheme for anisotropic diffusion problems on any grid
journal, October 2006
- Droniou, Jérôme; Eymard, Robert
- Numerische Mathematik, Vol. 105, Issue 1
A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods
journal, February 2010
- Droniou, JÉRÔMe; Eymard, Robert; GallouËT, Thierry
- Mathematical Models and Methods in Applied Sciences, Vol. 20, Issue 02
Gradient Schemes: a Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations
journal, September 2013
- Droniou, Jerome; Eymard, Robert; Gallouet, Thierry
- Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 13
Gradient schemes: Generic tools for the numerical analysis of diffusion equations
journal, May 2016
- Droniou, Jérome; Eymard, Robert; Herbin, Raphaèle
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
Gradient schemes for linear and non-linear elasticity equations
journal, June 2014
- Droniou, Jérôme; Lamichhane, Bishnu P.
- Numerische Mathematik, Vol. 129, Issue 2
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces
journal, June 2009
- Eymard, R.; Gallouet, T.; Herbin, R.
- IMA Journal of Numerical Analysis, Vol. 30, Issue 4
The Stefan problem in several space variables
journal, January 1968
- Friedman, Avner
- Transactions of the American Mathematical Society, Vol. 133, Issue 1
Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology
journal, January 2003
- Glowinski, Roland; Rappaz, Jacques
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 37, Issue 1
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation
journal, September 2014
- Lipnikov, K.; Manzini, G.
- Journal of Computational Physics, Vol. 272
Mixed finite elements in ?3
journal, September 1980
- Nedelec, J. C.
- Numerische Mathematik, Vol. 35, Issue 3
Stable Discontinuous Galerkin FEM Without Penalty Parameters
book, January 2016
- John, Lorenz; Neilan, Michael; Smears, Iain
- Lecture Notes in Computational Science and Engineering
A mixed finite volume scheme for anisotropic diffusion problems on any grid
journal, October 2006
- Droniou, Jérôme; Eymard, Robert
- Numerische Mathematik, Vol. 105, Issue 1
Acute Scrotum Following Traumatic Spermatic Cord Hematoma: A Case Report and Review
journal, March 2015
- Pepe, Pietro; Bonaccorsi, Astrid; Candiano, Giuseppe
- Urology Case Reports, Vol. 3, Issue 2
On a Nonlinear Parabolic Problem Arising in Some Models Related to Turbulent Flows
journal, July 1994
- Diaz, Jesus Ildefonso; De Thelin, Francois
- SIAM Journal on Mathematical Analysis, Vol. 25, Issue 4
The non-conforming virtual element method for the Stokes equations
preprint, January 2016
- Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco
- arXiv
An introduction to Hybrid High-Order methods
preprint, January 2017
- Di Pietro, Daniele A.; Tittarelli, Roberta
- arXiv
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- International Journal for Numerical Methods in Engineering, Vol. 120, Issue 3
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- Calcolo, Vol. 55, Issue 3
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- Journal of Scientific Computing, Vol. 80, Issue 3
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text, January 2017
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An advection-robust Hybrid High-Order method for the Oseen problem
preprint, January 2017
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- arXiv
Hybrid High-Order methods for finite deformations of hyperelastic materials
text, January 2017
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- arXiv