# Conforming and nonconforming virtual element methods for elliptic problems

## Abstract

Here we present, in a unified framework, new conforming and nonconforming virtual element methods for general second-order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and nonsymmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal H ^{1}- and L ^{2}-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.

- Authors:

- Univ. of Leicester, Leicester (United Kingdom)
- 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 Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1331260

- Report Number(s):
- LA-UR-15-23951

Journal ID: ISSN 0272-4979

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- IMA Journal of Numerical Analysis

- Additional Journal Information:
- Journal Name: IMA Journal of Numerical Analysis; Journal ID: ISSN 0272-4979

- Publisher:
- Oxford University Press/Institute of Mathematics and its Applications

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; mathematics; elliptic problems; virtual element methods; polygonal and polyhedral meshes; convection-diffusion-reaction equations

### Citation Formats

```
Cangiani, Andrea, Manzini, Gianmarco, and Sutton, Oliver J.
```*Conforming and nonconforming virtual element methods for elliptic problems*. United States: N. p., 2016.
Web. doi:10.1093/imanum/drw036.

```
Cangiani, Andrea, Manzini, Gianmarco, & Sutton, Oliver J.
```*Conforming and nonconforming virtual element methods for elliptic problems*. United States. doi:10.1093/imanum/drw036.

```
Cangiani, Andrea, Manzini, Gianmarco, and Sutton, Oliver J. Wed .
"Conforming and nonconforming virtual element methods for elliptic problems". United States.
doi:10.1093/imanum/drw036. https://www.osti.gov/servlets/purl/1331260.
```

```
@article{osti_1331260,
```

title = {Conforming and nonconforming virtual element methods for elliptic problems},

author = {Cangiani, Andrea and Manzini, Gianmarco and Sutton, Oliver J.},

abstractNote = {Here we present, in a unified framework, new conforming and nonconforming virtual element methods for general second-order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and nonsymmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal H1- and L2-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.},

doi = {10.1093/imanum/drw036},

journal = {IMA Journal of Numerical Analysis},

number = ,

volume = ,

place = {United States},

year = {Wed Aug 03 00:00:00 EDT 2016},

month = {Wed Aug 03 00:00:00 EDT 2016}

}

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Cited by: 6 works

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