A posteriori error estimation and adaptivity in hp virtual elements

Beirão da Veiga, Lourenco; Manzini, Gianmarco; Mascotto, Lorenzo

doi:10.1007/s00211-019-01054-6

A posteriori error estimation and adaptivity in hp virtual elements

Journal Article · Sat Jun 15 00:00:00 EDT 2019 · Numerische Mathematik

DOI:https://doi.org/10.1007/s00211-019-01054-6· OSTI ID:1542818

Beirão da Veiga, Lourenco ^[1]; ^[2]; Mascotto, Lorenzo ^[3]

Univ. degli Studi di Milano-Bicocca, Milan (Italy)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Vienna, Vienna (Austria)

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.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: LDRD; USDOE

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1542818

Report Number(s):: LA-UR--18-23445

Journal Information:: Numerische Mathematik, Journal Name: Numerische Mathematik Journal Issue: 1 Vol. 143; ISSN 0029-599X

Publisher:: Springer Berlin HeidelbergCopyright Statement

Country of Publication:: United States

Language:: English

References (54)

A procedure for a posteriori error estimation for h-p finite element methods Ainsworth, Mark; Oden, J. Tinsley Computer Methods in Applied Mechanics and Engineering, Vol. 101, Issue 1-3 https://doi.org/10.1016/0045-7825(92)90016-d	journal	December 1992
The nonconforming virtual element method Ayuso de Dios, Blanca; Lipnikov, Konstantin; Manzini, Gianmarco ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3 https://doi.org/10.1051/m2an/2015090	journal	May 2016
The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains Chernov, Alexey; Mascotto, Lorenzo IMA Journal of Numerical Analysis, Vol. 39, Issue 4 https://doi.org/10.1093/imanum/dry038	journal	June 2018
$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems Dolejší, Vít; Ern, Alexandre; Vohralík, Martin SIAM Journal on Scientific Computing, Vol. 38, Issue 5 https://doi.org/10.1137/15m1026687	journal	January 2016
A residual a posteriori error estimate for the Virtual Element Method Berrone, Stefano; Borio, Andrea Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 08 https://doi.org/10.1142/s0218202517500233	journal	April 2017
Convergence and Optimality of hp-AFEM Canuto, Claudio; Nochetto, Ricardo H.; Stevenson, Rob arXiv https://doi.org/10.48550/arxiv.1503.03996	preprint	January 2015
A multigrid algorithm for the $p$-version of the Virtual Element Method Antonietti, P. F.; Mascotto, L.; Verani, M. arXiv https://doi.org/10.48550/arxiv.1703.02285	preprint	January 2017
Ill-conditioning in the Virtual Element Method: stabilizations and bases Mascotto, Lorenzo arXiv https://doi.org/10.48550/arxiv.1705.10581	preprint	January 2017
Exploring High-order three dimensional Virtual Elements: bases and stabilizations Mascotto, Lorenzo; Dassi, Franco arXiv https://doi.org/10.48550/arxiv.1709.04371	preprint	January 2017
Conforming polygonal finite elements Sukumar, N.; Tabarraei, A. International Journal for Numerical Methods in Engineering, Vol. 61, Issue 12 https://doi.org/10.1002/nme.1141	journal	January 2004
Polygonal finite elements for topology optimization: A unifying paradigm: POLYGONAL FINITE ELEMENTS FOR TOPOLOGY OPTIMIZATION Talischi, Cameron; Paulino, Glaucio H.; Pereira, Anderson International Journal for Numerical Methods in Engineering, Vol. 82, Issue 6 https://doi.org/10.1002/nme.2763	journal	December 2009
Ill-conditioning in the virtual element method: Stabilizations and bases Mascotto, Lorenzo Numerical Methods for Partial Differential Equations, Vol. 34, Issue 4 https://doi.org/10.1002/num.22257	journal	March 2018
The Mathematical Theory of Finite Element Methods Brenner, Susanne C.; Scott, L. Ridgway Texts in Applied Mathematics https://doi.org/10.1007/978-0-387-75934-0	book	January 2008
Convergence and optimality of $${\mathbf {hp}}$$ hp -AFEM Canuto, Claudio; Nochetto, Ricardo H.; Stevenson, Rob Numerische Mathematik, Vol. 135, Issue 4 https://doi.org/10.1007/s00211-016-0826-x	journal	July 2016
A posteriori error estimates for the virtual element method Cangiani, Andrea; Georgoulis, Emmanuil H.; Pryer, Tristan Numerische Mathematik, Vol. 137, Issue 4 https://doi.org/10.1007/s00211-017-0891-9	journal	May 2017
Exponential convergence of the hp virtual element method in presence of corner singularities da Veiga, L. Beirão; Chernov, A.; Mascotto, L. Numerische Mathematik, Vol. 138, Issue 3 https://doi.org/10.1007/s00211-017-0921-7	journal	October 2017
A virtual element method for contact Wriggers, P.; Rust, W. T.; Reddy, B. D. Computational Mechanics, Vol. 58, Issue 6 https://doi.org/10.1007/s00466-016-1331-x	journal	September 2016
BDDC and FETI-DP for the virtual element method Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele Calcolo, Vol. 54, Issue 4 https://doi.org/10.1007/s10092-017-0242-3	journal	November 2017
Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander Journal of Scientific Computing, Vol. 77, Issue 3 https://doi.org/10.1007/s10915-018-0797-4	journal	August 2018
A procedure for a posteriori error estimation for h-p finite element methods Ainsworth, Mark; Oden, J. Tinsley Computer Methods in Applied Mechanics and Engineering, Vol. 101, Issue 1-3 https://doi.org/10.1016/0045-7825(92)90016-D	journal	December 1992
Equivalent projectors for virtual element methods Ahmad, B.; Alsaedi, A.; Brezzi, F. Computers & Mathematics with Applications, Vol. 66, Issue 3 https://doi.org/10.1016/j.camwa.2013.05.015	journal	September 2013
High-order Virtual Element Method on polyhedral meshes Beirão da Veiga, L.; Dassi, F.; Russo, A. Computers & Mathematics with Applications, Vol. 74, Issue 5 https://doi.org/10.1016/j.camwa.2017.03.021	journal	September 2017
A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo Computers & Mathematics with Applications, Vol. 74, Issue 9 https://doi.org/10.1016/j.camwa.2017.05.016	journal	November 2017
Exploring high-order three dimensional virtual elements: Bases and stabilizations Dassi, F.; Mascotto, L. Computers & Mathematics with Applications, Vol. 75, Issue 9 https://doi.org/10.1016/j.camwa.2018.02.005	journal	May 2018
h p -adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems Cangiani, A.; Georgoulis, E. H.; Giani, S. Computers & Mathematics with Applications, Vol. 78, Issue 9 https://doi.org/10.1016/j.camwa.2019.04.002	journal	November 2019
A note on the design of hp-adaptive finite element methods for elliptic partial differential equations Houston, Paul; Süli, Endre Computer Methods in Applied Mechanics and Engineering, Vol. 194, Issue 2-5 https://doi.org/10.1016/j.cma.2004.04.009	journal	February 2005
On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes Gain, Arun L.; Talischi, Cameron; Paulino, Glaucio H. Computer Methods in Applied Mechanics and Engineering, Vol. 282 https://doi.org/10.1016/j.cma.2014.05.005	journal	December 2014
Hybrid high-order methods for variable-diffusion problems on general meshes Di Pietro, Daniele A.; Ern, Alexandre Comptes Rendus Mathematique, Vol. 353, Issue 1 https://doi.org/10.1016/j.crma.2014.10.013	journal	January 2015
Mimetic finite difference method Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail Journal of Computational Physics, Vol. 257 https://doi.org/10.1016/j.jcp.2013.07.031	journal	January 2014
A hybrid mortar virtual element method for discrete fracture network simulations Benedetto, Matías Fernando; Berrone, Stefano; Borio, Andrea Journal of Computational Physics, Vol. 306 https://doi.org/10.1016/j.jcp.2015.11.034	journal	February 2016
Residual a posteriori error estimation for the Virtual Element Method for elliptic problems Beirão da Veiga, L.; Manzini, G. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 2 https://doi.org/10.1051/m2an/2014047	journal	March 2015
A plane wave virtual element method for the Helmholtz problem Perugia, Ilaria; Pietra, Paola; Russo, Alessandro ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3 https://doi.org/10.1051/m2an/2015066	journal	May 2016
Divergence free virtual elements for the stokes problem on polygonal meshes da Veiga, Lourenco Beirão; Lovadina, Carlo; Vacca, Giuseppe ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 2 https://doi.org/10.1051/m2an/2016032	journal	February 2017
A multigrid algorithm for the p -version of the virtual element method Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 52, Issue 1 https://doi.org/10.1051/m2an/2018007	journal	January 2018
An error indicator for mortar element solutions to the Stokes problem Bernardi, C. IMA Journal of Numerical Analysis, Vol. 21, Issue 4 https://doi.org/10.1093/imanum/21.4.857	journal	October 2001
A mixed virtual element method for the pseudostress–velocity formulation of the Stokes problem Cáceres, Ernesto; Gatica, Gabriel N. IMA Journal of Numerical Analysis, Vol. 37, Issue 1 https://doi.org/10.1093/imanum/drw002	journal	April 2016
Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems Cockburn, Bernardo; Gopalakrishnan, Jayadeep; Lazarov, Raytcho SIAM Journal on Numerical Analysis, Vol. 47, Issue 2 https://doi.org/10.1137/070706616	journal	January 2009
Higher Order BEM-Based FEM on Polygonal Meshes Rjasanow, Sergej; Weißer, Steffen SIAM Journal on Numerical Analysis, Vol. 50, Issue 5 https://doi.org/10.1137/110849481	journal	January 2012
Virtual Elements for Linear Elasticity Problems da Veiga, L. Beira͂o; Brezzi, F.; Marini, L. D. SIAM Journal on Numerical Analysis, Vol. 51, Issue 2 https://doi.org/10.1137/120874746	journal	January 2013
$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems Dolejší, Vít; Ern, Alexandre; Vohralík, Martin SIAM Journal on Scientific Computing, Vol. 38, Issue 5 https://doi.org/10.1137/15M1026687	journal	January 2016
hp -Interpolation of Nonsmooth Functions and an Application to hp -A posteriori Error Estimation Melenk, J. M. SIAM Journal on Numerical Analysis, Vol. 43, Issue 1 https://doi.org/10.1137/S0036142903432930	journal	January 2005
Basic Principles of Virtual Element Methods BeirÃO Da Veiga, L.; Brezzi, F.; Cangiani, A. Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 01 https://doi.org/10.1142/S0218202512500492	journal	November 2012
The Hitchhiker's Guide to the Virtual Element Method Beirão da Veiga, L.; Brezzi, F.; Marini, L. D. Mathematical Models and Methods in Applied Sciences, Vol. 24, Issue 08 https://doi.org/10.1142/S021820251440003X	journal	May 2014
A virtual element method for the Steklov eigenvalue problem Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo Mathematical Models and Methods in Applied Sciences, Vol. 25, Issue 08 https://doi.org/10.1142/S0218202515500372	journal	April 2015
Basic principles of hp virtual elements on quasiuniform meshes Beir ao da Veiga, L.; Chernov, A.; Mascotto, L. Mathematical Models and Methods in Applied Sciences, Vol. 26, Issue 08 https://doi.org/10.1142/S021820251650038X	journal	June 2016
A residual a posteriori error estimate for the Virtual Element Method Berrone, Stefano; Borio, Andrea Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 08 https://doi.org/10.1142/S0218202517500233	journal	April 2017
Stability analysis for the virtual element method Beirão da Veiga, Lourenço; Lovadina, Carlo; Russo, Alessandro Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 13 https://doi.org/10.1142/S021820251750052X	journal	October 2017
An H1-conforming virtual element for Darcy and Brinkman equations Vacca, Giuseppe Mathematical Models and Methods in Applied Sciences, Vol. 28, Issue 01 https://doi.org/10.1142/S0218202518500057	journal	December 2017
The fully nonconforming virtual element method for biharmonic problems Antonietti, P. F.; Manzini, G.; Verani, M. Mathematical Models and Methods in Applied Sciences, Vol. 28, Issue 02 https://doi.org/10.1142/S0218202518500100	journal	December 2017
Some Estimates for Virtual Element Methods Brenner, Susanne C.; Guan, Qingguang; Sung, Li-Yeng Computational Methods in Applied Mathematics, Vol. 17, Issue 4 https://doi.org/10.1515/cmam-2017-0008	journal	January 2017
On the Virtual Element Method for Three-Dimensional Elasticity Problems on Arbitrary Polyhedral Meshes Gain, Arun L.; Talischi, Cameron; Paulino, Glaucio H. arXiv https://doi.org/10.48550/arxiv.1311.0932	text	January 2013
The nonconforming virtual element method de Dios, B. Ayuso; Lipnikov, K.; Manzini, G. arXiv https://doi.org/10.48550/arxiv.1405.3741	preprint	January 2014
A posteriori error estimates for the virtual element method Cangiani, Andrea; Georgoulis, Emmanuil H.; Pryer, Tristan arXiv https://doi.org/10.48550/arxiv.1603.05855	preprint	January 2016
High-order Virtual Element Method on polyhedral meshes da Veiga, L. Beirão; Dassi, F.; Russo, A. arXiv https://doi.org/10.48550/arxiv.1703.02882	preprint	January 2017

Similar Records

Residual-based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations

Journal Article · Tue Dec 17 23:00:00 EST 2019 · Journal of Numerical Mathematics · OSTI ID:1649541

Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates

Journal Article · Wed Dec 31 23:00:00 EST 2014 · Journal of Computational Physics · OSTI ID:22382155

Related Subjects

97 MATHEMATICS AND COMPUTING
Mathematics

A posteriori error estimation and adaptivity in hp virtual elements

Citation Formats

References (54)

Similar Records

Related Subjects