# 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:

- 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):
- 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. 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 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 = {2017},

month = {11}

}

*Citation information provided by*

Web of Science

Web of Science

Works referencing / citing this record:

##
A Hybrid High‐Order method for finite elastoplastic deformations within a logarithmic strain framework

journal, July 2019

- Abbas, Mickaël; Ern, Alexandre; Pignet, Nicolas
- International Journal for Numerical Methods in Engineering, Vol. 120, Issue 3