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:
-
[2];
- Univ. degli Studi di Milano-Bicocca, Milan (Italy)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- 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 Volume: 143; Journal Issue: 1; 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.
Beirão da Veiga, Lourenco, Manzini, Gianmarco, & Mascotto, Lorenzo. A posteriori error estimation and adaptivity in hp virtual elements. United States. doi:https://doi.org/10.1007/s00211-019-01054-6
Beirão da Veiga, Lourenco, Manzini, Gianmarco, and Mascotto, Lorenzo. Sat .
"A posteriori error estimation and adaptivity in hp virtual elements". United States. doi:https://doi.org/10.1007/s00211-019-01054-6. https://www.osti.gov/servlets/purl/1542818.
@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 = 1,
volume = 143,
place = {United States},
year = {2019},
month = {6}
}
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Works referenced in this record:
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journal, January 2012
- Rjasanow, Sergej; Weißer, Steffen
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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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High-order Virtual Element Method on polyhedral meshes
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Residual a posteriori error estimation for the Virtual Element Method for elliptic problems
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A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem
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The nonconforming virtual element method
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A note on the design of hp-adaptive finite element methods for elliptic partial differential equations
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Equivalent projectors for virtual element methods
journal, September 2013
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- Computers & Mathematics with Applications, Vol. 66, Issue 3
A virtual element method for the Steklov eigenvalue problem
journal, April 2015
- Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo
- Mathematical Models and Methods in Applied Sciences, Vol. 25, Issue 08
Exploring high-order three dimensional virtual elements: Bases and stabilizations
journal, May 2018
- Dassi, F.; Mascotto, L.
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A residual a posteriori error estimate for the Virtual Element Method
journal, April 2017
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A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem
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The fully nonconforming virtual element method for biharmonic problems
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An error indicator for mortar element solutions to the Stokes problem
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Convergence and optimality of $${\mathbf {hp}}$$ hp -AFEM
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Divergence free virtual elements for the stokes problem on polygonal meshes
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BDDC and FETI-DP for the virtual element method
journal, November 2017
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A virtual element method for contact
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Basic principles of hp virtual elements on quasiuniform meshes
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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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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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A procedure for a posteriori error estimation for h-p finite element methods
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Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions
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- Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander
- Journal of Scientific Computing, Vol. 77, Issue 3
The Hitchhiker's Guide to the Virtual Element Method
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-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems
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Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015
- Di Pietro, Daniele A.; Ern, Alexandre
- Comptes Rendus Mathematique, Vol. 353, Issue 1
Exponential convergence of the hp virtual element method in presence of corner singularities
journal, October 2017
- da Veiga, L. Beirão; Chernov, A.; Mascotto, L.
- Numerische Mathematik, Vol. 138, Issue 3
Mimetic finite difference method
journal, January 2014
- Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
- Journal of Computational Physics, Vol. 257
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
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
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
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
Works referencing / citing this record:
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
BDDC and FETI-DP for the virtual element method
journal, November 2017
- Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele
- Calcolo, Vol. 54, Issue 4
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
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
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
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
-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
Mimetic finite difference method
journal, January 2014
- Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
- Journal of Computational Physics, Vol. 257
Hybrid high-order methods for variable-diffusion problems on general meshes
journal, January 2015
- Di Pietro, Daniele A.; Ern, Alexandre
- Comptes Rendus Mathematique, Vol. 353, Issue 1
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
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
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
Exponential convergence of the hp virtual element method in presence of corner singularities
journal, October 2017
- da Veiga, L. Beirão; Chernov, A.; Mascotto, L.
- Numerische Mathematik, Vol. 138, Issue 3
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
Exploring high-order three dimensional virtual elements: Bases and stabilizations
journal, May 2018
- Dassi, F.; Mascotto, L.
- Computers & Mathematics with Applications, Vol. 75, Issue 9
Convergence and optimality of $${\mathbf {hp}}$$ hp -AFEM
journal, July 2016
- Canuto, Claudio; Nochetto, Ricardo H.; Stevenson, Rob
- Numerische Mathematik, Vol. 135, Issue 4
An error indicator for mortar element solutions to the Stokes problem
journal, October 2001
- Bernardi, C.
- IMA Journal of Numerical Analysis, Vol. 21, Issue 4
Higher Order BEM-Based FEM on Polygonal Meshes
journal, January 2012
- Rjasanow, Sergej; Weißer, Steffen
- SIAM Journal on Numerical Analysis, Vol. 50, Issue 5
Conforming polygonal finite elements
journal, January 2004
- Sukumar, N.; Tabarraei, A.
- International Journal for Numerical Methods in Engineering, Vol. 61, Issue 12
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
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
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
A virtual element method for contact
journal, September 2016
- Wriggers, P.; Rust, W. T.; Reddy, B. D.
- Computational Mechanics, Vol. 58, Issue 6
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
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
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
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
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
Equivalent projectors for virtual element methods
journal, September 2013
- Ahmad, B.; Alsaedi, A.; Brezzi, F.
- Computers & Mathematics with Applications, Vol. 66, Issue 3
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