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Title: Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes

Journal Article · · Journal of Computational Physics
ORCiD logo [1];  [2]; ORCiD logo [3]
  1. Univ. Montpellier (France)
  2. Monash University, Melbourne (Australia)
  3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
+ Show Author Affiliations

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.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); Laboratory Directed Research and Development Program (LDRD)
Grant/Contract Number:
AC52-06NA25396
OSTI ID:
1415391
Report Number(s):
LA-UR-17-24418; TRN: US1800792
Journal Information:
Journal of Computational Physics, Vol. 355, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 34 works
Citation information provided by
Web of Science

References (32)

An Interior Penalty Finite Element Method with Discontinuous Elements journal August 1982
The nonconforming virtual element method journal May 2016
Basic Principles of Virtual Element Methods journal November 2012
Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes journal May 2015
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes journal January 2005
A Family of Mimetic Finite Difference Methods on Polygonal and Polyhedral Meshes journal October 2005
The NonConforming Virtual Element Method for the Stokes Equations journal January 2016
An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems journal January 2000
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods journal May 2016
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems journal January 2009
A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes journal December 2016
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
A Discontinuous-Skeletal Method for Advection-Diffusion-Reaction on General Meshes journal January 2015
A hybrid high-order locking-free method for linear elasticity on general meshes journal January 2015
Hybrid high-order methods for variable-diffusion problems on general meshes journal January 2015
An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators journal October 2014
An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow journal August 2014
A mixed finite volume scheme for anisotropic diffusion problems on any grid journal October 2006
A Unified Approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume Methods journal February 2010
Gradient Schemes: a Generic Framework for the Discretisation of Linear, Nonlinear and Nonlocal Elliptic and Parabolic Equations journal September 2013
Gradient schemes: Generic tools for the numerical analysis of diffusion equations journal May 2016
Gradient schemes for linear and non-linear elasticity equations journal June 2014
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces journal June 2009
The Stefan problem in several space variables journal January 1968
Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology journal January 2003
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation journal September 2014
Mixed finite elements in ?3 journal September 1980
Stable Discontinuous Galerkin FEM Without Penalty Parameters book January 2016
Acute Scrotum Following Traumatic Spermatic Cord Hematoma: A Case Report and Review journal March 2015
On a Nonlinear Parabolic Problem Arising in Some Models Related to Turbulent Flows journal July 1994
The non-conforming virtual element method for the Stokes equations preprint January 2016
An introduction to Hybrid High-Order methods preprint January 2017

Cited By (8)

An Advection-Robust Hybrid High-Order Method for the Oseen Problem journal March 2018
A Hybrid High‐Order method for finite elastoplastic deformations within a logarithmic strain framework
  • Abbas, Mickaël; Ern, Alexandre; Pignet, Nicolas
  • International Journal for Numerical Methods in Engineering, Vol. 120, Issue 3 https://doi.org/10.1002/nme.6137
journal July 2019
A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation journal September 2018
The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems journal July 2019
Hybrid High-Order methods for finite deformations of hyperelastic materials journal January 2018
A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity journal April 2020
Hybrid High-Order methods for finite deformations of hyperelastic materials text January 2017
An advection-robust Hybrid High-Order method for the Oseen problem preprint January 2017

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Related Subjects

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