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Title: A posteriori error estimation and adaptivity in hp virtual elements

Abstract

An explicit and computable error estimator for the hp version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide, following the approach of Melenk and Wohlmuth (Adv Comput Math 15(1–4):311–331, 2001), hp adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM hp Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.

Authors:
 [1]; ORCiD logo [2];  [3]
+ Show Author Affiliations
  1. Univ. degli Studi di Milano-Bicocca, Milan (Italy)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Univ. of Vienna, Vienna (Austria)
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:
1542818
Report Number(s):
LA-UR-18-23445
Journal ID: ISSN 0029-599X
Grant/Contract Number:  
89233218CNA000001
Resource Type:
Accepted Manuscript
Journal Name:
Numerische Mathematik
Additional Journal Information:
Journal Name: Numerische Mathematik; Journal ID: ISSN 0029-599X
Publisher:
Springer Berlin Heidelberg
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics

Citation Formats

Beirão da Veiga, Lourenco, Manzini, Gianmarco, and Mascotto, Lorenzo. A posteriori error estimation and adaptivity in hp virtual elements. United States: N. p., 2019. Web. doi:10.1007/s00211-019-01054-6.
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Beirão da Veiga, Lourenco, Manzini, Gianmarco, & Mascotto, Lorenzo. A posteriori error estimation and adaptivity in hp virtual elements. United States. doi:10.1007/s00211-019-01054-6.
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Beirão da Veiga, Lourenco, Manzini, Gianmarco, and Mascotto, Lorenzo. Sat . "A posteriori error estimation and adaptivity in hp virtual elements". United States. doi:10.1007/s00211-019-01054-6. https://www.osti.gov/servlets/purl/1542818.
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@article{osti_1542818,
title = {A posteriori error estimation and adaptivity in hp virtual elements},
author = {Beirão da Veiga, Lourenco and Manzini, Gianmarco and Mascotto, Lorenzo},
abstractNote = {An explicit and computable error estimator for the hp version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide, following the approach of Melenk and Wohlmuth (Adv Comput Math 15(1–4):311–331, 2001), hp adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM hp Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.},
doi = {10.1007/s00211-019-01054-6},
journal = {Numerische Mathematik},
number = ,
volume = ,
place = {United States},
year = {2019},
month = {6}
}
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DOI:  10.1007/s00211-019-01054-6
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Works referenced in this record:

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journal, June 2016

  • Beir ao da Veiga, L.; Chernov, A.; Mascotto, L.
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Conforming polygonal finite elements
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A posteriori error estimates for the virtual element method
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  • Cangiani, Andrea; Georgoulis, Emmanuil H.; Pryer, Tristan
  • Numerische Mathematik, Vol. 137, Issue 4
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Some Estimates for Virtual Element Methods
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On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes
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hp -Interpolation of Nonsmooth Functions and an Application to hp -A posteriori Error Estimation
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High-order Virtual Element Method on polyhedral meshes
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$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems
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An H1-conforming virtual element for Darcy and Brinkman equations
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Ill-conditioning in the virtual element method: Stabilizations and bases
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Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems
journal, January 2009

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  • DOI: 10.1137/070706616

Residual a posteriori error estimation for the Virtual Element Method for elliptic problems
journal, March 2015

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  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 2
  • DOI: 10.1051/m2an/2014047

A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem
journal, April 2016

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  • DOI: 10.1093/imanum/drw002

A multigrid algorithm for the p -version of the virtual element method
journal, January 2018

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  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 52, Issue 1
  • DOI: 10.1051/m2an/2018007

Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions
journal, August 2018

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  • DOI: 10.1007/s10915-018-0797-4

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
  • DOI: 10.1051/m2an/2015090

A note on the design of hp-adaptive finite element methods for elliptic partial differential equations
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  • Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 2-5
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The Hitchhiker's Guide to the Virtual Element Method
journal, May 2014

  • Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.
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h p -adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems
journal, November 2019

  • Cangiani, A.; Georgoulis, E. H.; Giani, S.
  • Computers & Mathematics with Applications, Vol. 78, Issue 9
  • DOI: 10.1016/j.camwa.2019.04.002

Equivalent projectors for virtual element methods
journal, September 2013

Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015

A virtual element method for the Steklov eigenvalue problem
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  • DOI: 10.1142/S0218202515500372

Exploring high-order three dimensional virtual elements: Bases and stabilizations
journal, May 2018

Exponential convergence of the hp virtual element method in presence of corner singularities
journal, October 2017

A residual a posteriori error estimate for the Virtual Element Method
journal, April 2017

  • Berrone, Stefano; Borio, Andrea
  • Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 08
  • DOI: 10.1142/S0218202517500233

A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem
journal, November 2017

  • Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo
  • Computers & Mathematics with Applications, Vol. 74, Issue 9
  • DOI: 10.1016/j.camwa.2017.05.016

The fully nonconforming virtual element method for biharmonic problems
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  • Antonietti, P. F.; Manzini, G.; Verani, M.
  • Mathematical Models and Methods in Applied Sciences, Vol. 28, Issue 02
  • DOI: 10.1142/S0218202518500100

Mimetic finite difference method
journal, January 2014

  • Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
  • Journal of Computational Physics, Vol. 257
  • DOI: 10.1016/j.jcp.2013.07.031

Stability analysis for the virtual element method
journal, October 2017

  • Beirão da Veiga, Lourenço; Lovadina, Carlo; Russo, Alessandro
  • Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 13
  • DOI: 10.1142/S021820251750052X

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
  • DOI: 10.1142/S0218202512500492

A plane wave virtual element method for the Helmholtz problem
journal, May 2016

  • Perugia, Ilaria; Pietra, Paola; Russo, Alessandro
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
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An error indicator for mortar element solutions to the Stokes problem
journal, October 2001

Convergence and optimality of $${\mathbf {hp}}$$ hp -AFEM
journal, July 2016

  • Canuto, Claudio; Nochetto, Ricardo H.; Stevenson, Rob
  • Numerische Mathematik, Vol. 135, Issue 4
  • DOI: 10.1007/s00211-016-0826-x

The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains
journal, June 2018

  • Chernov, Alexey; Mascotto, Lorenzo
  • IMA Journal of Numerical Analysis, Vol. 39, Issue 4
  • DOI: 10.1093/imanum/dry038

A hybrid mortar virtual element method for discrete fracture network simulations
journal, February 2016

  • Benedetto, Matías Fernando; Berrone, Stefano; Borio, Andrea
  • Journal of Computational Physics, Vol. 306
  • DOI: 10.1016/j.jcp.2015.11.034

Virtual Elements for Linear Elasticity Problems
journal, January 2013

  • da Veiga, L. Beira͂o; Brezzi, F.; Marini, L. D.
  • SIAM Journal on Numerical Analysis, Vol. 51, Issue 2
  • DOI: 10.1137/120874746

Divergence free virtual elements for the stokes problem on polygonal meshes
journal, February 2017

  • da Veiga, Lourenco Beirão; Lovadina, Carlo; Vacca, Giuseppe
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 2
  • DOI: 10.1051/m2an/2016032

BDDC and FETI-DP for the virtual element method
journal, November 2017

A virtual element method for contact
journal, September 2016

    [ × clear filter / sort ]

    Works referencing / citing this record:

    Conforming polygonal finite elements
    journal, January 2004

    • Sukumar, N.; Tabarraei, A.
    • International Journal for Numerical Methods in Engineering, Vol. 61, Issue 12
    • DOI: 10.1002/nme.1141

    Ill-conditioning in the virtual element method: Stabilizations and bases
    journal, March 2018

    • Mascotto, Lorenzo
    • Numerical Methods for Partial Differential Equations, Vol. 34, Issue 4
    • DOI: 10.1002/num.22257

    Convergence and optimality of $${\mathbf {hp}}$$ hp -AFEM
    journal, July 2016

    • Canuto, Claudio; Nochetto, Ricardo H.; Stevenson, Rob
    • Numerische Mathematik, Vol. 135, Issue 4
    • DOI: 10.1007/s00211-016-0826-x

    A posteriori error estimates for the virtual element method
    journal, May 2017

    • Cangiani, Andrea; Georgoulis, Emmanuil H.; Pryer, Tristan
    • Numerische Mathematik, Vol. 137, Issue 4
    • DOI: 10.1007/s00211-017-0891-9

    Exponential convergence of the hp virtual element method in presence of corner singularities
    journal, October 2017

    A virtual element method for contact
    journal, September 2016

    BDDC and FETI-DP for the virtual element method
    journal, November 2017

    Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions
    journal, August 2018

    • Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander
    • Journal of Scientific Computing, Vol. 77, Issue 3
    • DOI: 10.1007/s10915-018-0797-4

    Equivalent projectors for virtual element methods
    journal, September 2013

    High-order Virtual Element Method on polyhedral meshes
    journal, September 2017

    • Beirão da Veiga, L.; Dassi, F.; Russo, A.
    • Computers & Mathematics with Applications, Vol. 74, Issue 5
    • DOI: 10.1016/j.camwa.2017.03.021

    A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem
    journal, November 2017

    • Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo
    • Computers & Mathematics with Applications, Vol. 74, Issue 9
    • DOI: 10.1016/j.camwa.2017.05.016

    Exploring high-order three dimensional virtual elements: Bases and stabilizations
    journal, May 2018

    h p -adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems
    journal, November 2019

    • Cangiani, A.; Georgoulis, E. H.; Giani, S.
    • Computers & Mathematics with Applications, Vol. 78, Issue 9
    • DOI: 10.1016/j.camwa.2019.04.002

    A note on the design of hp-adaptive finite element methods for elliptic partial differential equations
    journal, February 2005

    • Houston, Paul; Süli, Endre
    • Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 2-5
    • DOI: 10.1016/j.cma.2004.04.009

    On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes
    journal, December 2014

    • Gain, Arun L.; Talischi, Cameron; Paulino, Glaucio H.
    • Computer Methods in Applied Mechanics and Engineering, Vol. 282
    • DOI: 10.1016/j.cma.2014.05.005

    Hybrid high-order methods for variable-diffusion problems on general meshes
    journal, January 2015

    Mimetic finite difference method
    journal, January 2014

    • Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
    • Journal of Computational Physics, Vol. 257
    • DOI: 10.1016/j.jcp.2013.07.031

    A hybrid mortar virtual element method for discrete fracture network simulations
    journal, February 2016

    • Benedetto, Matías Fernando; Berrone, Stefano; Borio, Andrea
    • Journal of Computational Physics, Vol. 306
    • DOI: 10.1016/j.jcp.2015.11.034

    Residual a posteriori error estimation for the Virtual Element Method for elliptic problems
    journal, March 2015

    • Beirão da Veiga, L.; Manzini, G.
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 2
    • DOI: 10.1051/m2an/2014047

    A plane wave virtual element method for the Helmholtz problem
    journal, May 2016

    • Perugia, Ilaria; Pietra, Paola; Russo, Alessandro
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3
    • DOI: 10.1051/m2an/2015066

    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
    • DOI: 10.1051/m2an/2015090

    Divergence free virtual elements for the stokes problem on polygonal meshes
    journal, February 2017

    • da Veiga, Lourenco Beirão; Lovadina, Carlo; Vacca, Giuseppe
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 2
    • DOI: 10.1051/m2an/2016032

    A multigrid algorithm for the p -version of the virtual element method
    journal, January 2018

    • Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco
    • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 52, Issue 1
    • DOI: 10.1051/m2an/2018007

    An error indicator for mortar element solutions to the Stokes problem
    journal, October 2001

    A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem
    journal, April 2016

    • Cáceres, Ernesto; Gatica, Gabriel N.
    • IMA Journal of Numerical Analysis, Vol. 37, Issue 1
    • DOI: 10.1093/imanum/drw002

    The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains
    journal, June 2018

    • Chernov, Alexey; Mascotto, Lorenzo
    • IMA Journal of Numerical Analysis, Vol. 39, Issue 4
    • DOI: 10.1093/imanum/dry038

    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
    • DOI: 10.1137/070706616

    Higher Order BEM-Based FEM on Polygonal Meshes
    journal, January 2012

    • Rjasanow, Sergej; Weißer, Steffen
    • SIAM Journal on Numerical Analysis, Vol. 50, Issue 5
    • DOI: 10.1137/110849481

    Virtual Elements for Linear Elasticity Problems
    journal, January 2013

    • da Veiga, L. Beira͂o; Brezzi, F.; Marini, L. D.
    • SIAM Journal on Numerical Analysis, Vol. 51, Issue 2
    • DOI: 10.1137/120874746

    Some Estimates for Virtual Element Methods
    journal, January 2017

    • Brenner, Susanne C.; Guan, Qingguang; Sung, Li-Yeng
    • Computational Methods in Applied Mathematics, Vol. 17, Issue 4
    • DOI: 10.1515/cmam-2017-0008
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