The nonconforming virtual element method

Blanca Ayuso de Dios; Konstantin Lipnikov; Gianmarco Manzini

doi:10.1051/m2an/2015090

The nonconforming virtual element method

Blanca Ayuso de Dios, Konstantin Lipnikov, Gianmarco Manzini

Research output: Contribution to journal › Article › peer-review

218 Scopus citations

Abstract

We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.

Original language	English (US)
Pages (from-to)	879-904
Number of pages	26
Journal	ESAIM: Mathematical Modelling and Numerical Analysis
Volume	50
Issue number	3
DOIs	https://doi.org/10.1051/m2an/2015090
State	Published - May 23 2016
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-02
Acknowledged KAUST grant number(s): BAS/1/1636-01-01
Acknowledgements: The first author is in-debt with Proff. F. Brezzi and D. Marini from Pavia, for the multiple and fruitful discussions and specially for the encouragement to carry out this work. The work of the first author was partially supported by KAUST grants BAS/1/1636-01-01 and Pocket ID 1000000193. She thanks KAUST for the support and hospitality, where part of the work was completed while she was Research Scientist with Peter Markowich. The work of the second and third authors was partially supported by the Laboratory Directed Research and Development Program (LDRD), U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under Contract DE-AC52-06NA25396.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Modeling and Simulation
Analysis
Applied Mathematics
General Mathematics
Numerical Analysis

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title = "The nonconforming virtual element method",

abstract = "We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.",

author = "{Ayuso de Dios}, Blanca and Konstantin Lipnikov and Gianmarco Manzini",

note = "KAUST Repository Item: Exported on 2021-04-02 Acknowledged KAUST grant number(s): BAS/1/1636-01-01 Acknowledgements: The first author is in-debt with Proff. F. Brezzi and D. Marini from Pavia, for the multiple and fruitful discussions and specially for the encouragement to carry out this work. The work of the first author was partially supported by KAUST grants BAS/1/1636-01-01 and Pocket ID 1000000193. She thanks KAUST for the support and hospitality, where part of the work was completed while she was Research Scientist with Peter Markowich. The work of the second and third authors was partially supported by the Laboratory Directed Research and Development Program (LDRD), U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences, and the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research, under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under Contract DE-AC52-06NA25396. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AB - We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.

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The nonconforming virtual element method

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