A plane wave virtual element method for the Helmholtz problem
Perugia, Ilaria1, 2 ; Pietra, Paola3 ; Russo, Alessandro4
1 Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
2 Department of Mathematics, University of Pavia, 27100 Pavia, Italy
3 Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR, 27100 Pavia, Italy
4 University of Milano Bicocca, 20126 Milano, Italy
ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 783-808.

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: (i) a low order VEM space whose basis functions, which are associated to the mesh vertices, are not explicitly computed in the element interiors; (ii) a proper local projection operator onto the plane wave space; (iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.

Reçu le :
DOI : 10.1051/m2an/2015066
Classification : 65N30, 65N12, 65N15, 35J05
Mots-clés : Helmholtz equation, virtual element method, plane wave basis functions, error analysis, duality estimates
Perugia, Ilaria 1, 2 ; Pietra, Paola 3 ; Russo, Alessandro 4

1 Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria
2 Department of Mathematics, University of Pavia, 27100 Pavia, Italy
3 Istituto di Matematica Applicata e Tecnologie Informatiche “Enrico Magenes”, CNR, 27100 Pavia, Italy
4 University of Milano Bicocca, 20126 Milano, Italy 
@article{M2AN_2016__50_3_783_0,
     author = {Perugia, Ilaria and Pietra, Paola and Russo, Alessandro},
     title = {A plane wave virtual element method for the {Helmholtz} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {783--808},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {3},
     year = {2016},
     doi = {10.1051/m2an/2015066},
     zbl = {1343.65137},
     mrnumber = {3507273},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2015066/}
}
TY  - JOUR
AU  - Perugia, Ilaria
AU  - Pietra, Paola
AU  - Russo, Alessandro
TI  - A plane wave virtual element method for the Helmholtz problem
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 783
EP  - 808
VL  - 50
IS  - 3
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2015066/
DO  - 10.1051/m2an/2015066
LA  - en
ID  - M2AN_2016__50_3_783_0
ER  - 
%0 Journal Article
%A Perugia, Ilaria
%A Pietra, Paola
%A Russo, Alessandro
%T A plane wave virtual element method for the Helmholtz problem
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 783-808
%V 50
%N 3
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2015066/
%R 10.1051/m2an/2015066
%G en
%F M2AN_2016__50_3_783_0
Perugia, Ilaria; Pietra, Paola; Russo, Alessandro. A plane wave virtual element method for the Helmholtz problem. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue – Polyhedral discretization for PDE, Tome 50 (2016) no. 3, pp. 783-808. doi : 10.1051/m2an/2015066. https://www.numdam.org/articles/10.1051/m2an/2015066/

P.F. Antonietti, L. Beirão Da Veiga, D. Mora and M. Verani, A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52 (2014) 386–404. | DOI | MR | Zbl

B. Ayuso De Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. To appear in Special issue – Polyhedral discretization for PDE. ESAIM: M2AN 50 (2016). DOI: | DOI | Numdam | MR

I.M. Babuška and S.A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation? SIAM Rev. 42 (2000) 451–484. | MR | Zbl

L. Beirão Da Veiga, and G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34 (2014) 759–781. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl

L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming virtual element method. To appear in Numer. Math. (2015). DOI: | DOI | MR

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems. To appear in Special issue – Polyhedral discretization for PDE. ESAIM M2AN 50 (2016). DOI: | DOI

L Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016) 729–750. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi, L.D. Marini and A. Russo, The Hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24 (2014) 1541–1573. | DOI | MR | Zbl

L. Beirão Da Veiga, C. Lovadina and D. Mora, A virtual element method for elastic and inelastic problems on polytope meshes. Comput. Methods Appl. Mech. Eng. 295 (2015) 327–346. | DOI | MR | Zbl

M.F. Benedetto, S. Berrone, S. Pieraccini and S. Scialò, The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280 (2014) 135–156. | DOI | MR | Zbl

J.H. Bramble and L.E. Payne, Bounds in the Neumann problem for second order uniformly elliptic operators. Pacific J. Math 12 (1962) 823–833. | DOI | MR | Zbl

F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253 (2013) 455–462. | DOI | MR | Zbl

F. Brezzi and L.D. Marini, Virtual Element and Discontinuous Galerkin Methods. In Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. Springer (2014) 209–221. | MR | Zbl

F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN 48 (2014) 1227–1240. | DOI | Numdam | MR | Zbl

A. Buffa and P. Monk, Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation. ESAIM: M2AN 42 (2008) 925–940. | DOI | Numdam | MR | Zbl

O. Cessenat, Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques, Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine (1996).

O. Cessenat and B. Després, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz equation. SIAM J. Numer. Anal. 35 (1998) 255–299. | DOI | MR | Zbl

E. Deckers, O. Atak, L. Coox, R. D’Amico, H. Devriendt, S. Jonckheere, K. Koo, B. Pluymers, D. Vandepitte and W. Desmet, The wave based method: An overview of 15 years of research. Innovations in Wave Modelling. Wave Motion 51 (2014) 550–565. | DOI | MR | Zbl

W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis. Ph.D. thesis, KU Leuven, Belgium, 1998.

C. Farhat, I. Harari and L. Franca, The discontinuous enrichment method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6455–6479. | DOI | MR | Zbl

C. Farhat, I. Harari and U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput. Methods Appl. Mech. Eng. 192 (2003) 1389–1419. | DOI | MR | Zbl

G. Gabard, Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225 (2007) 1961–1984. | DOI | MR | Zbl

G. Gabard, Exact integration of polynomial-exponential products with application to wave-based numerical methods. Comm. Numer. Methods Eng. 25 (2009) 237–246. | DOI | MR | Zbl

A.L. Gain, C. Talischi and G.H. Paulino, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282 (2014) 132–160. | DOI | MR | Zbl

C.J. Gittelson, Plane wave discontinuous Galerkin methods. Master’s thesis, SAM-ETH Zürich, Switzerland (2008).

C.J. Gittelson, R. Hiptmair and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM: M2AN 43 (2009) 297–332. | DOI | Numdam | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Approximation by plane waves. Technical report 2009-27, SAM-ETH Zürich, Switzerland (2009). Available at http://www.sam.math.ethz.ch/reports/2009/27.

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–284. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79 (2014) 79–91. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, A Survey of Trefftz Methods for the Helmholtz Equation. “Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations”. Edited by G.R. Barrenechea, A. Cangiani, E.H. Geogoulis. In Lect. Notes Comput. Sci. Eng. Springer. Preprint [math.NA] (2015). | arXiv | MR

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin Methods: Exponential convergence of the hp-version. To appear in Found. Comput. Math. (2015). DOI: | DOI | MR

F. Ihlenburg and I. Babuska, Solution of Helmholtz problems by knowledge-based fem. Comp. Ass. Mech. Eng. Sci. 4 (1997) 397–416. | Zbl

J. Ladevèze and P. Ladevèze, Bounds of the Poincaré constant with respect to the problem of star-shaped membrane regions. Z. Angew. Math. Phys. 29 (1978) 670–683. | MR | Zbl

P. Ladevèze and H. Riou, On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 278 (2014) 729–743. | DOI | MR | Zbl

G. Manzini, A. Russo and N. Sukumar. New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24 (2014) 1665–1699. | DOI | MR | Zbl

J.M. Melenk, On Generalized Finite Element Methods. Ph.D. thesis, University of Maryland, 1995. | MR

J.M. Melenk and I. Babuška, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139 (1996) 289–314. | DOI | MR | Zbl

J.M. Melenk and I Babuska, Approximation with harmonic and generalized harmonic polynomials in the partition of unity method. Comp. Ass. Mech. Eng. Sci. 4 (1997) 607–632. | Zbl

J.M. Melenk and S. Sauter, Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal. 49 (2011) 1210–1243. | DOI | MR | Zbl

A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonic wave problems. Ph.D. thesis, Seminar for applied mathematics, ETH Zürich (2011). Available at http://e-collection.library.ethz.ch/view/eth:4515.

A. Moiola, R. Hiptmair and I. Perugia, Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62 (2011) 809–837. | DOI | MR | Zbl

P. Monk and D.Q. Wang, A least squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 175 (1999) 121–136. | DOI | MR | Zbl

L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl

H. Riou, P. Ladevéze and B. Sourcis, The multiscale VTCR approach applied to acoustics problems. J. Comput. Acoust. 16 (2008) 487–505. | DOI | Zbl

M. Stojek, Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Methods Eng. 41 (1998) 831–849. | DOI | MR | Zbl

N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61 (2004) 2045–2066. | DOI | MR | Zbl

R. Tezaur and C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems. Int. J. Numer. Methods Eng. 66 (2006) 796–815. | DOI | MR | Zbl

  • Hooshyarfarzin, Baharak; Abbaszadeh, Mostafa; Dehghan, Mehdi Numerical simulation and error estimation of the Davey-Stewartson equations with virtual element method, Applied Mathematics and Computation, Volume 489 (2025), p. 129146 | DOI:10.1016/j.amc.2024.129146
  • Wu, Bangmin; Wang, Fei; Han, Weimin The virtual element method for a contact problem with wear and unilateral constraint, Applied Numerical Mathematics, Volume 206 (2024), p. 29 | DOI:10.1016/j.apnum.2024.08.004
  • Trezzi, Manuel; Zerbinati, Umberto When rational functions meet virtual elements: the lightning virtual element method, Calcolo, Volume 61 (2024) no. 3 | DOI:10.1007/s10092-024-00585-1
  • Feng, Haoqiang; Huang, Jianping; Li, Caipin; Liu, Kang; Wang, Yi-Yao; Sha, Wei E. I.; Zhan, Qiwei Phase-Informed Discontinuous Galerkin Method for Extremely High-Frequency Wave Modeling, IEEE Transactions on Antennas and Propagation, Volume 72 (2024) no. 8, p. 6614 | DOI:10.1109/tap.2024.3427389
  • Yang, Qian; Shen, Xiaoqin; Zhao, Jikun; Cheng, Yumin An enriched virtual element method for 2D‐3C generalized membrane shell model on surface, International Journal for Numerical Methods in Engineering, Volume 125 (2024) no. 8 | DOI:10.1002/nme.7432
  • Bertoluzza, S.; Montardini, M.; Pennacchio, M.; Prada, D. The virtual element method on polygonal pixel–based tessellations, Journal of Computational Physics, Volume 518 (2024), p. 113334 | DOI:10.1016/j.jcp.2024.113334
  • Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele Interior estimates for the virtual element method, Numerische Mathematik, Volume 156 (2024) no. 3, p. 1163 | DOI:10.1007/s00211-024-01408-9
  • Duan, Huoyuan; Li, Ziliang A mixed virtual element method for nearly incompressible linear elasticity equations, Advances in Computational Mathematics, Volume 49 (2023) no. 1 | DOI:10.1007/s10444-023-10013-7
  • Shen, Xiaoqin; Wang, Chen; Yang, Qian; Zhao, Jikun; Gao, Zhiming Virtual element method for linear elastic clamped plate model, Applied Numerical Mathematics, Volume 191 (2023), p. 1 | DOI:10.1016/j.apnum.2023.04.020
  • Qiu, Jiali; Wang, Fei; Ling, Min; Zhao, Jikun The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality, Communications in Nonlinear Science and Numerical Simulation, Volume 127 (2023), p. 107547 | DOI:10.1016/j.cnsns.2023.107547
  • Antonietti, Paola F.; Bonizzoni, Francesca; Verani, Marco A cVEM-DG space-time method for the dissipative wave equation, Computers Mathematics with Applications, Volume 152 (2023), p. 341 | DOI:10.1016/j.camwa.2023.10.022
  • Lehrenfeld, Christoph; Stocker, Paul Embedded Trefftz discontinuous Galerkin methods, International Journal for Numerical Methods in Engineering, Volume 124 (2023) no. 17, p. 3637 | DOI:10.1002/nme.7258
  • Sreekumar, Abhilash; Triantafyllou, Savvas P.; Chevillotte, Fabien; Bécot, François‐Xavier Resolving vibro‐acoustics in poroelastic media via a multiscale virtual element method, International Journal for Numerical Methods in Engineering, Volume 124 (2023) no. 7, p. 1510 | DOI:10.1002/nme.7173
  • Antonietti, P. F.; Manzini, G.; Scacchi, S.; Verani, M. On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1, Volume 137 (2023), p. 3 | DOI:10.1007/978-3-031-20432-6_1
  • Dassi, F.; Gedicke, J.; Mascotto, L. Adaptive virtual element methods with equilibrated fluxes, Applied Numerical Mathematics, Volume 173 (2022), p. 249 | DOI:10.1016/j.apnum.2021.11.015
  • Manzini, Gianmarco; Mazzia, Annamaria Conforming virtual element approximations of the two-dimensional Stokes problem, Applied Numerical Mathematics, Volume 181 (2022), p. 176 | DOI:10.1016/j.apnum.2022.06.002
  • Wu, Bangmin; Wang, Fei; Han, Weimin Virtual element method for a frictional contact problem with normal compliance, Communications in Nonlinear Science and Numerical Simulation, Volume 107 (2022), p. 106125 | DOI:10.1016/j.cnsns.2021.106125
  • Sreekumar, Abhilash; Triantafyllou, Savvas P.; Chevillotte, Fabien Virtual elements for sound propagation in complex poroelastic media, Computational Mechanics, Volume 69 (2022) no. 1, p. 347 | DOI:10.1007/s00466-021-02078-2
  • Benvenuti, E.; Chiozzi, A.; Manzini, G.; Sukumar, N. Extended virtual element method for two-dimensional linear elastic fracture, Computer Methods in Applied Mechanics and Engineering, Volume 390 (2022), p. 114352 | DOI:10.1016/j.cma.2021.114352
  • Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements, Computer Methods in Applied Mechanics and Engineering, Volume 400 (2022), p. 115454 | DOI:10.1016/j.cma.2022.115454
  • Sorgente, T.; Biasotti, S.; Manzini, G.; Spagnuolo, M. Polyhedral mesh quality indicator for the Virtual Element Method, Computers Mathematics with Applications, Volume 114 (2022), p. 151 | DOI:10.1016/j.camwa.2022.03.042
  • Bertoluzza, S.; Manzini, G.; Pennacchio, M.; Prada, D. Stabilization of the nonconforming virtual element method, Computers Mathematics with Applications, Volume 116 (2022), p. 25 | DOI:10.1016/j.camwa.2021.10.009
  • Kaya, Adem; Freitag, Melina A. Conditioning analysis for discrete Helmholtz problems, Computers Mathematics with Applications, Volume 118 (2022), p. 171 | DOI:10.1016/j.camwa.2022.05.016
  • Rupp, Andreas; Gahn, Markus; Kanschat, Guido Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 56 (2022) no. 2, p. 505 | DOI:10.1051/m2an/2022011
  • Li, Lulu; Su, Haiyan; He, Yinnian Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh, Entropy, Volume 24 (2022) no. 8, p. 1129 | DOI:10.3390/e24081129
  • Wang, Fei; Reddy, B. Daya A priori error analysis of virtual element method for contact problem, Fixed Point Theory and Algorithms for Sciences and Engineering, Volume 2022 (2022) no. 1 | DOI:10.1186/s13663-022-00720-z
  • Manzini, Gianmarco; Mazzia, Annamaria A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem, Journal of Computational Dynamics, Volume 9 (2022) no. 2, p. 207 | DOI:10.3934/jcd.2021020
  • Adak, D.; Mora, D.; Natarajan, S. Convergence Analysis of Virtual Element Method for Nonlinear Nonlocal Dynamic Plate Equation, Journal of Scientific Computing, Volume 91 (2022) no. 1 | DOI:10.1007/s10915-022-01794-y
  • Sorgente, Tommaso; Prada, Daniele; Cabiddu, Daniela; Biasotti, Silvia; Patanè, Giuseppe; Pennacchio, Micol; Bertoluzza, Silvia; Manzini, Gianmarco; Spagnuolo, Michela VEM and the Mesh, The Virtual Element Method and its Applications, Volume 31 (2022), p. 1 | DOI:10.1007/978-3-030-95319-5_1
  • Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander The Nonconforming Trefftz Virtual Element Method: General Setting, Applications, and Dispersion Analysis for the Helmholtz Equation, The Virtual Element Method and its Applications, Volume 31 (2022), p. 363 | DOI:10.1007/978-3-030-95319-5_9
  • Antonietti, Paola F.; Manzini, Gianmarco; Mazzieri, Ilario; Scacchi, Simone; Verani, Marco The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review, The Virtual Element Method and its Applications, Volume 31 (2022), p. 411 | DOI:10.1007/978-3-030-95319-5_10
  • Alvarez, Sebastian Naranjo; Bokil, Vrushali A.; Gyrya, Vitaliy; Manzini, Gianmarco The Virtual Element Method for the Coupled System of Magneto-Hydrodynamics, The Virtual Element Method and its Applications, Volume 31 (2022), p. 499 | DOI:10.1007/978-3-030-95319-5_12
  • Bürger, Raimund; Kumar, Sarvesh; Mora, David; Ruiz-Baier, Ricardo; Verma, Nitesh Virtual element methods for the three-field formulation of time-dependent linear poroelasticity, Advances in Computational Mathematics, Volume 47 (2021) no. 1 | DOI:10.1007/s10444-020-09826-7
  • Bachini, Elena; Manzini, Gianmarco; Putti, Mario Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces, Calcolo, Volume 58 (2021) no. 3 | DOI:10.1007/s10092-021-00418-5
  • Arrutselvi, M.; Natarajan, E. Virtual Element Method for Nonlinear Time-Dependent Convection-Diffusion-Reaction Equation, Computational Mathematics and Modeling, Volume 32 (2021) no. 3, p. 376 | DOI:10.1007/s10598-021-09537-8
  • Artioli, E.; Mascotto, L. Enrichment of the nonconforming virtual element method with singular functions, Computer Methods in Applied Mechanics and Engineering, Volume 385 (2021), p. 114024 | DOI:10.1016/j.cma.2021.114024
  • Desiderio, L.; Falletta, S.; Scuderi, L. A Virtual Element Method coupled with a Boundary Integral Non Reflecting condition for 2D exterior Helmholtz problems, Computers Mathematics with Applications, Volume 84 (2021), p. 296 | DOI:10.1016/j.camwa.2021.01.002
  • Lovadina, Carlo; Mora, David; Velásquez, Iván A virtual element method for the von Kármán equations, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 55 (2021) no. 2, p. 533 | DOI:10.1051/m2an/2020085
  • Wang, Fei; Zhao, Jikun Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem, IMA Journal of Numerical Analysis, Volume 41 (2021) no. 2, p. 1496 | DOI:10.1093/imanum/draa005
  • Hessari, Pedram; Chegeni, Farhad THE IMPACT OF ENVIRONMENTAL CONSTRUCTION ON THE SPATIAL CONFIGURATION OF TRADITIONAL IRANIAN HOUSING (CASE STUDY: COMPARISON OF DEZFUL AND BOROUJERD TRADITIONAL HOUSING), JOURNAL OF ARCHITECTURE AND URBANISM, Volume 45 (2021) no. 1, p. 50 | DOI:10.3846/jau.2021.14230
  • Lepe, Felipe; Mora, David; Rivera, Gonzalo; Velásquez, Iván A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges, Journal of Scientific Computing, Volume 88 (2021) no. 2 | DOI:10.1007/s10915-021-01555-3
  • Antonietti, Paola F.; Manzini, Gianmarco; Scacchi, Simone; Verani, Marco A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations, Mathematical Models and Methods in Applied Sciences, Volume 31 (2021) no. 14, p. 2825 | DOI:10.1142/s0218202521500627
  • Adak, Dibyendu; Manzini, Gianmarco; Natarajan, Sundararajan The Mixed Virtual Element Method for the Richards Equation, Polyhedral Methods in Geosciences, Volume 27 (2021), p. 259 | DOI:10.1007/978-3-030-69363-3_7
  • Mora, David; Velásquez, Iván Virtual Elements for the Transmission Eigenvalue Problem on Polytopal Meshes, SIAM Journal on Scientific Computing, Volume 43 (2021) no. 4, p. A2425 | DOI:10.1137/20m1347887
  • Mascotto, Lorenzo; Pichler, Alexander Extension of the nonconforming Trefftz virtual element method to the Helmholtz problem with piecewise constant wave number, Applied Numerical Mathematics, Volume 155 (2020), p. 160 | DOI:10.1016/j.apnum.2019.04.005
  • Mora, David; Velásquez, Iván Virtual element for the buckling problem of Kirchhoff–Love plates, Computer Methods in Applied Mechanics and Engineering, Volume 360 (2020), p. 112687 | DOI:10.1016/j.cma.2019.112687
  • Liu, Xin; He, Zhengkang; Chen, Zhangxin A fully discrete virtual element scheme for the Cahn–Hilliard equation in mixed form, Computer Physics Communications, Volume 246 (2020), p. 106870 | DOI:10.1016/j.cpc.2019.106870
  • Dong, Qiannan; Wu, Jiming; Su, Shuai Relationship between the vertex-centered linearity-preserving scheme and the lowest-order virtual element method for diffusion problems on star-shaped polygons, Computers Mathematics with Applications, Volume 79 (2020) no. 11, p. 3117 | DOI:10.1016/j.camwa.2020.01.009
  • Huang, Jianguo; Lin, Sen A C0P2 time-stepping virtual element method for linear wave equations on polygonal meshes, Electronic Research Archive, Volume 28 (2020) no. 2, p. 911 | DOI:10.3934/era.2020048
  • Mora, David; Rivera, Gonzalo A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations, IMA Journal of Numerical Analysis, Volume 40 (2020) no. 1, p. 322 | DOI:10.1093/imanum/dry063
  • Wang, Fei; Wei, Huayi Virtual element methods for the obstacle problem, IMA Journal of Numerical Analysis, Volume 40 (2020) no. 1, p. 708 | DOI:10.1093/imanum/dry055
  • Anaya, Verónica; Bendahmane, Mostafa; Mora, David; Sepúlveda, Mauricio A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology, IMA Journal of Numerical Analysis, Volume 40 (2020) no. 2, p. 1544 | DOI:10.1093/imanum/drz001
  • D'Altri, Antonio Maria; de Miranda, Stefano; Patruno, Luca; Artioli, Edoardo; Lovadina, Carlo Error estimation and mesh adaptivity for the virtual element method based on recovery by compatibility in patches, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 19, p. 4374 | DOI:10.1002/nme.6438
  • Deng, Yanling; Wang, Fei; Wei, Huayi A Posteriori Error Estimates of Virtual Element Method for a Simplified Friction Problem, Journal of Scientific Computing, Volume 83 (2020) no. 3 | DOI:10.1007/s10915-020-01242-9
  • Ling, Min; Wang, Fei; Han, Weimin The Nonconforming Virtual Element Method for a Stationary Stokes Hemivariational Inequality with Slip Boundary Condition, Journal of Scientific Computing, Volume 85 (2020) no. 3 | DOI:10.1007/s10915-020-01333-7
  • Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A. Polynomial preserving virtual elements with curved edges, Mathematical Models and Methods in Applied Sciences, Volume 30 (2020) no. 08, p. 1555 | DOI:10.1142/s0218202520500311
  • Peterseim, Daniel; Verfürth, Barbara Computational high frequency scattering from high-contrast heterogeneous media, Mathematics of Computation, Volume 89 (2020) no. 326, p. 2649 | DOI:10.1090/mcom/3529
  • Chiozzi, Andrea; Benvenuti, Elena Extended virtual element method for the torsion problem of cracked prismatic beams, Meccanica, Volume 55 (2020) no. 4, p. 637 | DOI:10.1007/s11012-019-01073-5
  • Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele FETI-DP for the Three Dimensional Virtual Element Method, SIAM Journal on Numerical Analysis, Volume 58 (2020) no. 3, p. 1556 | DOI:10.1137/18m1233303
  • Weißer, Steffen Introduction, BEM-based Finite Element Approaches on Polytopal Meshes, Volume 130 (2019), p. 1 | DOI:10.1007/978-3-030-20961-2_1
  • Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander A nonconforming Trefftz virtual element method for the Helmholtz problem: Numerical aspects, Computer Methods in Applied Mechanics and Engineering, Volume 347 (2019), p. 445 | DOI:10.1016/j.cma.2018.12.039
  • Benvenuti, E.; Chiozzi, A.; Manzini, G.; Sukumar, N. Extended virtual element method for the Laplace problem with singularities and discontinuities, Computer Methods in Applied Mechanics and Engineering, Volume 356 (2019), p. 571 | DOI:10.1016/j.cma.2019.07.028
  • Beirão da Veiga, L.; Russo, A.; Vacca, G. The Virtual Element Method with curved edges, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 53 (2019) no. 2, p. 375 | DOI:10.1051/m2an/2018052
  • Zepeda-Núñez, Leonardo; Scheuer, Adrien; Hewett, Russell J.; Demanet, Laurent The method of polarized traces for the 3D Helmholtz equation, GEOPHYSICS, Volume 84 (2019) no. 4, p. T313 | DOI:10.1190/geo2018-0153.1
  • Zhang, Bei; Zhao, Jikun; Yang, Yongqin; Chen, Shaochun The nonconforming virtual element method for elasticity problems, Journal of Computational Physics, Volume 378 (2019), p. 394 | DOI:10.1016/j.jcp.2018.11.004
  • Li, Qiuqi; Jiang, Lijian A multiscale virtual element method for elliptic problems in heterogeneous porous media, Journal of Computational Physics, Volume 388 (2019), p. 394 | DOI:10.1016/j.jcp.2019.03.031
  • Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander A nonconforming Trefftz virtual element method for the Helmholtz problem, Mathematical Models and Methods in Applied Sciences, Volume 29 (2019) no. 09, p. 1619 | DOI:10.1142/s0218202519500301
  • Beirão da Veiga, L.; Manzini, G.; Mascotto, L. A posteriori error estimation and adaptivity in hp virtual elements, Numerische Mathematik, Volume 143 (2019) no. 1, p. 139 | DOI:10.1007/s00211-019-01054-6
  • Zhao, Jikun; Zhang, Bei; Mao, Shipeng; Chen, Shaochun The Divergence-Free Nonconforming Virtual Element for the Stokes Problem, SIAM Journal on Numerical Analysis, Volume 57 (2019) no. 6, p. 2730 | DOI:10.1137/18m1200762
  • Taylor, R. L.; Artioli, E. VEM for Inelastic Solids, Advances in Computational Plasticity, Volume 46 (2018), p. 381 | DOI:10.1007/978-3-319-60885-3_18
  • Wang, Fei; Wei, Huayi Virtual element method for simplified friction problem, Applied Mathematics Letters, Volume 85 (2018), p. 125 | DOI:10.1016/j.aml.2018.06.002
  • Zhang, Baiju; Feng, Minfu Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation, Applied Mathematics and Computation, Volume 328 (2018), p. 1 | DOI:10.1016/j.amc.2018.01.023
  • Dassi, F.; Mascotto, L. Exploring high-order three dimensional virtual elements: Bases and stabilizations, Computers Mathematics with Applications, Volume 75 (2018) no. 9, p. 3379 | DOI:10.1016/j.camwa.2018.02.005
  • Antonietti, Paola F.; Mascotto, Lorenzo; Verani, Marco A multigrid algorithm for the p-version of the virtual element method, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 52 (2018) no. 1, p. 337 | DOI:10.1051/m2an/2018007
  • Gardini, Francesca; Vacca, Giuseppe Virtual element method for second-order elliptic eigenvalue problems, IMA Journal of Numerical Analysis, Volume 38 (2018) no. 4, p. 2026 | DOI:10.1093/imanum/drx063
  • Mascotto, Lorenzo; Perugia, Ilaria; Pichler, Alexander Non-conforming Harmonic Virtual Element Method: h h - and p p -Versions, Journal of Scientific Computing, Volume 77 (2018) no. 3, p. 1874 | DOI:10.1007/s10915-018-0797-4
  • Antonietti, P. F.; Manzini, G.; Verani, M. The fully nonconforming virtual element method for biharmonic problems, Mathematical Models and Methods in Applied Sciences, Volume 28 (2018) no. 02, p. 387 | DOI:10.1142/s0218202518500100
  • Mora, David; Velásquez, Iván A virtual element method for the transmission eigenvalue problem, Mathematical Models and Methods in Applied Sciences, Volume 28 (2018) no. 14, p. 2803 | DOI:10.1142/s0218202518500616
  • Ohlberger, Mario; Verfurth, Barbara A New Heterogeneous Multiscale Method for the Helmholtz Equation with High Contrast, Multiscale Modeling Simulation, Volume 16 (2018) no. 1, p. 385 | DOI:10.1137/16m1108820
  • Chinosi, Claudia Virtual Elements for the Reissner‐Mindlin plate problem, Numerical Methods for Partial Differential Equations, Volume 34 (2018) no. 4, p. 1117 | DOI:10.1002/num.22248
  • Mascotto, Lorenzo Ill‐conditioning in the virtual element method: Stabilizations and bases, Numerical Methods for Partial Differential Equations, Volume 34 (2018) no. 4, p. 1258 | DOI:10.1002/num.22257
  • da Veiga, L. Beirão; Chernov, A.; Mascotto, L.; Russo, A. Exponential convergence of the hp virtual element method in presence of corner singularities, Numerische Mathematik, Volume 138 (2018) no. 3, p. 581 | DOI:10.1007/s00211-017-0921-7
  • Rizzuti, Gabrio, SEG Technical Program Expanded Abstracts 2018 (2018), p. 3898 | DOI:10.1190/segam2018-2996158.1
  • da Veiga, L. Beira͂o; Lovadina, C.; Vacca, G. Virtual Elements for the Navier–Stokes Problem on Polygonal Meshes, SIAM Journal on Numerical Analysis, Volume 56 (2018) no. 3, p. 1210 | DOI:10.1137/17m1132811
  • da Veiga, L. Beira͂o; Brezzi, F.; Dassi, F.; Marini, L. D.; Russo, A. A Family of Three-Dimensional Virtual Elements with Applications to Magnetostatics, SIAM Journal on Numerical Analysis, Volume 56 (2018) no. 5, p. 2940 | DOI:10.1137/18m1169886
  • Ohlberger, Mario; Verfürth, Barbara Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems, AIMS Mathematics, Volume 2 (2017) no. 3, p. 458 | DOI:10.3934/math.2017.2.458
  • Bertoluzza, Silvia; Pennacchio, Micol; Prada, Daniele BDDC and FETI-DP for the virtual element method, Calcolo, Volume 54 (2017) no. 4, p. 1565 | DOI:10.1007/s10092-017-0242-3
  • Banerjee, Subhajit; Sukumar, N. Exact integration scheme for planewave-enriched partition of unity finite element method to solve the Helmholtz problem, Computer Methods in Applied Mechanics and Engineering, Volume 317 (2017), p. 619 | DOI:10.1016/j.cma.2017.01.001
  • Beirão da Veiga, L.; Brezzi, F.; Dassi, F.; Marini, L.D.; Russo, A. Virtual Element approximation of 2D magnetostatic problems, Computer Methods in Applied Mechanics and Engineering, Volume 327 (2017), p. 173 | DOI:10.1016/j.cma.2017.08.013
  • Antonietti, P.F.; Bruggi, M.; Scacchi, S.; Verani, M. On the virtual element method for topology optimization on polygonal meshes: A numerical study, Computers Mathematics with Applications, Volume 74 (2017) no. 5, p. 1091 | DOI:10.1016/j.camwa.2017.05.025
  • Beirão da Veiga, L.; Lopez, L.; Vacca, G. Mimetic finite difference methods for Hamiltonian wave equations in 2D, Computers Mathematics with Applications, Volume 74 (2017) no. 5, p. 1123 | DOI:10.1016/j.camwa.2017.05.022
  • Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo A posteriori error estimates for a Virtual Element Method for the Steklov eigenvalue problem, Computers Mathematics with Applications, Volume 74 (2017) no. 9, p. 2172 | DOI:10.1016/j.camwa.2017.05.016
  • Beirão da Veiga, Lourenço; Lovadina, Carlo; Russo, Alessandro Stability analysis for the virtual element method, Mathematical Models and Methods in Applied Sciences, Volume 27 (2017) no. 13, p. 2557 | DOI:10.1142/s021820251750052x
  • Sutton, Oliver J. The virtual element method in 50 lines of MATLAB, Numerical Algorithms, Volume 75 (2017) no. 4, p. 1141 | DOI:10.1007/s11075-016-0235-3
  • Beirão da Veiga, Lourenço; Mora, David; Rivera, Gonzalo; Rodríguez, Rodolfo A virtual element method for the acoustic vibration problem, Numerische Mathematik, Volume 136 (2017) no. 3, p. 725 | DOI:10.1007/s00211-016-0855-5
  • Fang, Jun; Qian, Jianliang; Zepeda-Núñez, Leonardo; Zhao, Hongkai Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations, Research in the Mathematical Sciences, Volume 4 (2017) no. 1 | DOI:10.1186/s40687-017-0098-9
  • Chinosi, Claudia; Marini, L. Donatella Virtual Element Method for fourth order problems: L2-estimates, Computers Mathematics with Applications, Volume 72 (2016) no. 8, p. 1959 | DOI:10.1016/j.camwa.2016.02.001
  • Beirão da Veiga, L.; Ern, A. Preface, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 50 (2016) no. 3, p. 633 | DOI:10.1051/m2an/2016034
  • Zhao, Jikun; Chen, Shaochun; Zhang, Bei The nonconforming virtual element method for plate bending problems, Mathematical Models and Methods in Applied Sciences, Volume 26 (2016) no. 09, p. 1671 | DOI:10.1142/s021820251650041x

Cité par 99 documents. Sources : Crossref