The nonconforming Virtual Element Method for eigenvalue problems

Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe

doi:10.1051/m2an/2018074

Title: The nonconforming Virtual Element Method for eigenvalue problems

Journal Article · 05 June 2019 · Mathematical Modelling and Numerical Analysis

DOI: https://doi.org/10.1051/m2an/2018074 · OSTI ID:1844151

Gardini, Francesca ^[1];

^[2]; Vacca, Giuseppe ^[3]

Università di Pavia (Italy)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. degli Studi di Milano Bicocca (Italy)

In this work, we analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the $L^2$-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

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

Sponsoring Organization:: European Research Council (ERC); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: 89233218CNA000001; AC52-06NA25396

OSTI ID:: 1844151

Report Number(s):: LA-UR-20-27582

Journal Information:: Mathematical Modelling and Numerical Analysis, Journal Name: Mathematical Modelling and Numerical Analysis Journal Issue: 3 Vol. 53; ISSN 0764-583X

Publisher:: EDP SciencesCopyright Statement

Country of Publication:: United States

Language:: English

References (49)

Singularities in Boundary Value Problems and Exact Controllability of Hyperbolic Systems Grisvard, Pierre Optimization, Optimal Control and Partial Differential Equations ISNM 107 https://doi.org/10.1007/978-3-0348-8625-3_8	book	January 1992
Convergence of the mimetic finite difference method for eigenvalue problems in mixed form Cangiani, Andrea; Gardini, Francesca; Manzini, Gianmarco Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 9-12 https://doi.org/10.1016/j.cma.2010.06.011	journal	February 2011
A stress/displacement Virtual Element method for plane elasticity problems Artioli, E.; de Miranda, S.; Lovadina, C. Computer Methods in Applied Mechanics and Engineering, Vol. 325 https://doi.org/10.1016/j.cma.2017.06.036	journal	October 2017
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 non-conforming virtual element method for the Stokes equations Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco arXiv https://doi.org/10.48550/arxiv.1608.01210	preprint	January 2016
Hourglass stabilization and the virtual element method: Hourglass stabilization and the virtual element method Cangiani, A.; Manzini, G.; Russo, A. International Journal for Numerical Methods in Engineering, Vol. 102, Issue 3-4 https://doi.org/10.1002/nme.4854	journal	January 2015
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
PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab Talischi, Cameron; Paulino, Glaucio H.; Pereira, Anderson Structural and Multidisciplinary Optimization, Vol. 45, Issue 3 https://doi.org/10.1007/s00158-011-0706-z	journal	January 2012
A virtual element method for the acoustic vibration problem Beirão da Veiga, Lourenço; Mora, David; Rivera, Gonzalo Numerische Mathematik, Vol. 136, Issue 3 https://doi.org/10.1007/s00211-016-0855-5	journal	December 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
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
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
Serendipity Virtual Elements for General Elliptic Equations in Three Dimensions Beirão Da Veiga, Lourenço; Brezzi, Franco; Dassi, Franco Chinese Annals of Mathematics, Series B, Vol. 39, Issue 2 https://doi.org/10.1007/s11401-018-1066-4	journal	February 2018
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
Virtual Element Methods for hyperbolic problems on polygonal meshes Vacca, Giuseppe Computers & Mathematics with Applications, Vol. 74, Issue 5 https://doi.org/10.1016/j.camwa.2016.04.029	journal	September 2017
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
Convergence of the mimetic finite difference method for eigenvalue problems in mixed form Cangiani, Andrea; Gardini, Francesca; Manzini, Gianmarco Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 9-12 https://doi.org/10.1016/j.cma.2010.06.011	journal	February 2011
Virtual Element Methods for plate bending problems Brezzi, Franco; Marini, L. Donatella Computer Methods in Applied Mechanics and Engineering, Vol. 253 https://doi.org/10.1016/j.cma.2012.09.012	journal	January 2013
Some basic formulations of the virtual element method (VEM) for finite deformations Chi, H.; da Veiga, L. Beirão; Paulino, G. H. Computer Methods in Applied Mechanics and Engineering, Vol. 318 https://doi.org/10.1016/j.cma.2016.12.020	journal	May 2017
A stress/displacement Virtual Element method for plane elasticity problems Artioli, E.; de Miranda, S.; Lovadina, C. Computer Methods in Applied Mechanics and Engineering, Vol. 325 https://doi.org/10.1016/j.cma.2017.06.036	journal	October 2017
Serendipity Nodal VEM spaces Beirão da Veiga, L.; Brezzi, F.; Marini, L. D. Computers & Fluids, Vol. 141 https://doi.org/10.1016/j.compfluid.2016.02.015	journal	December 2016
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
A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation Lipnikov, K.; Manzini, G. Journal of Computational Physics, Vol. 272 https://doi.org/10.1016/j.jcp.2014.04.021	journal	September 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
Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes Di Pietro, Daniele A.; Droniou, Jérôme; Manzini, Gianmarco Journal of Computational Physics, Vol. 355 https://doi.org/10.1016/j.jcp.2017.11.018	journal	February 2018
Finite element approximation of eigenvalue problems Boffi, Daniele Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000012	journal	May 2010
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
Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods Cockburn, Bernardo; Di Pietro, Daniele A.; Ern, Alexandre ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 3 https://doi.org/10.1051/m2an/2015051	journal	May 2016
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
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
Finite element quasi-interpolation and best approximation Ern, Alexandre; Guermond, Jean-Luc ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 4 https://doi.org/10.1051/m2an/2016066	journal	July 2017
A virtual element method for the vibration problem of Kirchhoff plates Mora, David; Rivera, Gonzalo; Velásquez, Iván ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1051/m2an/2017041	journal	July 2018
Mimetic finite differences for elliptic problems Brezzi, Franco; Buffa, Annalisa; Lipnikov, Konstantin ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 43, Issue 2 https://doi.org/10.1051/m2an:2008046	journal	December 2008
Spectral approximation of elliptic operators by the Hybrid High-Order method Calo, Victor; Cicuttin, Matteo; Deng, Quanling Mathematics of Computation, Vol. 88, Issue 318 https://doi.org/10.1090/mcom/3405	journal	December 2018
A virtual element method with arbitrary regularity da Veiga, L. B.; Manzini, G. IMA Journal of Numerical Analysis, Vol. 34, Issue 2 https://doi.org/10.1093/imanum/drt018	journal	July 2013
Virtual element method for second-order elliptic eigenvalue problems Gardini, Francesca; Vacca, Giuseppe IMA Journal of Numerical Analysis, Vol. 38, Issue 4 https://doi.org/10.1093/imanum/drx063	journal	November 2017
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes da Veiga, L. Beirão; Lipnikov, K.; Manzini, G. SIAM Journal on Numerical Analysis, Vol. 49, Issue 5 https://doi.org/10.1137/100807764	journal	January 2011
A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes Antonietti, P. F.; da Veiga, L. Beira͂o; Scacchi, S. SIAM Journal on Numerical Analysis, Vol. 54, Issue 1 https://doi.org/10.1137/15M1008117	journal	January 2016
The NonConforming Virtual Element Method for the Stokes Equations Cangiani, Andrea; Gyrya, Vitaliy; Manzini, Gianmarco SIAM Journal on Numerical Analysis, Vol. 54, Issue 6 https://doi.org/10.1137/15M1049531	journal	January 2016
Poincaré--Friedrichs Inequalities for Piecewise H ¹ Functions Brenner, Susanne C. SIAM Journal on Numerical Analysis, Vol. 41, Issue 1 https://doi.org/10.1137/S0036142902401311	journal	January 2003
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
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
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
An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators Di Pietro, Daniele A.; Ern, Alexandre; Lemaire, Simon Computational Methods in Applied Mathematics, Vol. 14, Issue 4 https://doi.org/10.1515/cmam-2014-0018	journal	October 2014
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

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A priori error analysis for a mixed VEM discretization of the spectral problem for the Laplacian operator Lepe, Felipe; Rivera, Gonzalo Calcolo, Vol. 58, Issue 2 https://doi.org/10.1007/s10092-021-00412-x	journal	April 2021
A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem Rivera, Gonzalo; Lepe, Felipe Unpublished https://doi.org/10.13140/rg.2.2.32055.34725	text	January 2020
Sharper error estimates for Virtual Elements and a bubble-enriched version da Veiga, L. Beirao; Vacca, G. arXiv https://doi.org/10.48550/arxiv.2005.12009	preprint	January 2020
A virtual element method for the Steklov eigenvalue problem allowing small edges Lepe, Felipe; Mora, David; Rivera, Gonzalo arXiv https://doi.org/10.48550/arxiv.2006.09573	preprint	January 2020
Virtual element approximation of eigenvalue problems Boffi, Daniele; Gardini, Francesca; Gastaldi, Lucia arXiv https://doi.org/10.48550/arxiv.2012.15660	preprint	January 2020
The conforming virtual element method for polyharmonic and elastodynamics problems: a review Antonietti, Paola Francesca; Manzini, Gianmarco; Mazzieri, Ilario arXiv https://doi.org/10.48550/arxiv.2102.01652	preprint	January 2021
Stabilization of the nonconforming virtual element method Bertoluzza, Silvia; Manzini, Gianmarco; Pennacchio, Micol arXiv https://doi.org/10.48550/arxiv.2103.03742	preprint	January 2021
A nonconforming Trefftz virtual element method for the Helmholtz problem Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander Mathematical Models and Methods in Applied Sciences, Vol. 29, Issue 09 https://doi.org/10.1142/s0218202519500301	journal	August 2019
A nonconforming Trefftz virtual element method for the Helmholtz problem Mascotto, L.; Perugia, I.; Pichler, A. arXiv https://doi.org/10.48550/arxiv.1805.05634	preprint	January 2018

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