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The virtual element method

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  11 May 2023

Lourenço Beirão Da Veiga ,
Franco Brezzi ,
L. Donatella Marini  and
Alessandro Russo Open the ORCID record for Alessandro Russo [Opens in a new window]
Lourenço Beirão Da Veiga
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Italy and IMATI-CNR, Pavia, Italy E-mail: lourenco.beirao@unimib.it
Franco Brezzi
Affiliation:
IUSS, Istituto Universitario di Studi Superiori, Pavia, Italy and IMATI-CNR, Pavia, Italy E-mail: brezzi@imati.cnr.it
L. Donatella Marini
Affiliation:
Dipartimento di Matematica, Università di Pavia, Italy, and IMATI-CNR, Pavia, Italy E-mail: marini@imati.cnr.it
Alessandro Russo
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano–Bicocca, Italy and IMATI-CNR, Pavia, Italy E-mail: alessandro.russo@unimib.it
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Abstract

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The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.

Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.

MSC classification

Primary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Information

Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 123 - 202
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

References

Adams, R. A. (1975), Sobolev Spaces, Vol. 65 of Pure and Applied Mathematics, Academic Press.Google Scholar
Aghili, J. and Di Pietro, D. A. (2018), An advection-robust hybrid high-order method for the Oseen problem, J. Sci. Comput. 77, 1310–1338.CrossRefGoogle Scholar
Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L. D. and Russo, A. (2013), Equivalent projectors for virtual element methods, Comput. Math. Appl. 66, 376–391.CrossRefGoogle Scholar
Antonietti, P. F., Beirão da Veiga, L., Mora, D. and Verani, M. (2014), A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 52, 386–404.CrossRefGoogle Scholar
Antonietti, P. F., Beirão da Veiga, L., Scacchi, S. and Verani, M. (2016), A C 1 virtual element method for the Cahn–Hilliard equation with polygonal meshes, SIAM J. Numer. Anal. 54, 34–57.CrossRefGoogle Scholar
Antonietti, P. F., Manzini, G. and Verani, M. (2018), The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci. 28, 387–407.CrossRefGoogle Scholar
Antonietti, P. F., Mascotto, L., Verani, M. and Zonca, S. (2022), Stability analysis of polytopic discontinuous Galerkin approximations of the Stokes problem with applications to fluid–structure interaction problems, J. Sci. Comput. 90, 23.CrossRefGoogle Scholar
Antonietti, P. F., Verani, M., Vergara, C. and Zonca, S. (2019), Numerical solution of fluid–structure interaction problems by means of a high order discontinuous Galerkin method on polygonal grids, Finite Elem. Anal. Des. 159, 1–14.CrossRefGoogle Scholar
Arnold, D. N. and Awanou, G. (2011), The serendipity family of finite elements, Found . Comput. Math. 11, 337–344.Google Scholar
Arnold, D. N., Boffi, D. and Falk, R. S. (2002), Approximation by quadrilateral finite elements, Math. Comp. 71, 909–922.CrossRefGoogle Scholar
Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D. (2001), Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39, 1749–1779.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006a), Differential complexes and stability of finite element methods, I: The de Rham complex, in Compatible Spatial Discretizations, Vol. 142 of IMA Volumes in Mathematics and its Applications, Springer, pp. 24–46.CrossRefGoogle Scholar
Arnold, D. N., Falk, R. S. and Winther, R. (2006b), Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15, 1–155.CrossRefGoogle Scholar
Artioli, E., Beirão da Veiga, L., Lovadina, C. and Sacco, E. (2017a), Arbitrary order 2D virtual elements for polygonal meshes, I: Elastic problem, Comput. Mech. 60, 355–377.CrossRefGoogle Scholar
Artioli, E., Beirão da Veiga, L., Lovadina, C. and Sacco, E. (2017b), Arbitrary order 2D virtual elements for polygonal meshes, II: Inelastic problem, Comput. Mech. 60, 643–657.CrossRefGoogle Scholar
Artioli, E., de Miranda, S., Lovadina, C. and Patruno, L. (2018), A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle, Comput . Methods Appl. Mech. Engrg 340, 978–999.CrossRefGoogle Scholar
Artioli, E., Marfia, S. and Sacco, E. (2020), VEM-based tracking algorithm for cohesive/ frictional 2D fracture, Comp . Meth. Appl. Mech. Engrg 365, 112956.CrossRefGoogle Scholar
Auchmuty, G. and Alexander, J. C. (2001), ${L}^2$ well-posedness of planar div-curl systems, Arch. Ration. Mech. Anal. 160, 91–134.CrossRefGoogle Scholar
Auchmuty, G. and Alexander, J. C. (2005), ${L}^2$ -well-posedness of 3D div-curl boundary value problems, Quart. Appl. Math. 63, 479–508.CrossRefGoogle Scholar
de Dios, B. Ayuso, Lipnikov, K. and Manzini, G. (2016), The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50, 879–904.CrossRefGoogle Scholar
Beirão da Veiga, L. and Lipnikov, K. (2010), A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput. 32, 875–893.CrossRefGoogle Scholar
Beirão da Veiga, L. and Mascotto, L. (2022), Interpolation and stability properties of low-order face and edge virtual element spaces, IMA J. Numer. Anal. 2022, drac008.Google Scholar
Beirão da Veiga, L., Brezzi, F. and Marini, L. D. (2013a), Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51, 794–812.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D. and Russo, A. (2013b), Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23, 199–214.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Dassi, F., Marini, L. D. and Russo, A. (2017a), Virtual element approximation of 2D magnetostatic problems, Comput. Methods Appl. Mech. Engrg 327, 173–195.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Dassi, F., Marini, L. D. and Russo, A. (2018a), A family of three-dimensional virtual elements with applications to magnetostatics, SIAM J. Numer. Anal. 56, 2940–2962.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Dassi, F., Marini, L. D. and Russo, A. (2018b), Lowest order virtual element approximation of magnetostatic problems, Comput. Methods Appl. Mech. Engrg 332, 343–362.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2014), The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci. 24, 1541–1573.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2016a), $H\left(\operatorname{div}\right)$ and $H\left(\operatorname{curl}\right)$ -conforming VEM, Numer. Math. 133, 303–332.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2016b), Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal. 50, 727–747.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2016c), Serendipity nodal VEM spaces, Comp . Fluids 141, 2–12.Google Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2016d), Virtual element methods for general second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci. 26, 729–750.CrossRefGoogle Scholar
Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. (2020), Polynomial preserving virtual elements with curved edges, Math. Models Methods Appl. Sci. 30, 1555–1590.CrossRefGoogle Scholar
Beirão da Veiga, L., Canuto, C., Nochetto, R. H. and Vacca, G. (2021), Equilibrium analysis of an immersed rigid leaflet by the virtual element method, Math. Models Methods Appl. Sci. 31, 1323–1372.CrossRefGoogle Scholar
Beirão da Veiga, L., Chernov, A., Mascotto, L. and Russo, A. (2016e), Basic principles of hp virtual elements on quasiuniform meshes, Math. Models Methods Appl. Sci. 26, 1567–1598.CrossRefGoogle Scholar
Beirão da Veiga, L., Dassi, F. and Vacca, G. (2018c), The Stokes complex for virtual elements in three dimensions, Math. Models Methods Appl. Sci. 30, 477–512.CrossRefGoogle Scholar
Beirão da Veiga, L., Dassi, F., Manzini, G. and Mascotto, L. (2022a), The virtual element method for the 3D resistive magnetohydrodynamic model. Submitted for publication. Available at arXiv:2201.04417v1.CrossRefGoogle Scholar
Beirão da Veiga, L., Dassi, F., Manzini, G. and Mascotto, L. (2022b), Virtual elements for Maxwell’s equations, Comput. Math. Appl. 116, 82–99.CrossRefGoogle Scholar
Beirão da Veiga, L., Lovadina, C. and Mora, D. (2015), A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mech. Engrg 295, 327–346.CrossRefGoogle Scholar
Beirão da Veiga, L., Lovadina, C. and Russo, A. (2017b), Stability analysis for the virtual element method, Math. Models Methods Appl. Sci. 27, 2557–2594.CrossRefGoogle Scholar
Beirão da Veiga, L., Lovadina, C. and Vacca, G. (2017c), Divergence free virtual elements for the Stokes problem on polygonal meshes, ESAIM Math. Model. Numer. Anal. 51, 509–535.CrossRefGoogle Scholar
Beirão da Veiga, L., Lovadina, C. and Vacca, G. (2018d), Virtual elements for the Navier–Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 56, 1210–1242.CrossRefGoogle Scholar
Beirão da Veiga, L., Manzini, G. and Mascotto, L. (2019a), A posteriori error estimation and adaptivity in hp virtual elements, Numer. Math. 143, 139–175.CrossRefGoogle Scholar
Beirão da Veiga, L., Mascotto, L. and Meng, J. (2022c), Interpolation and stability estimates for edge and face virtual elements of general order, Math. Models Methods Appl. Sci. 32, 1589–1631.CrossRefGoogle Scholar
Beirão da Veiga, L., Mora, D. and Vacca, G. (2019b), The Stokes complex for virtual elements with application to Navier–Stokes flows, J. Sci. Comput. 81, 990–1018.CrossRefGoogle Scholar
Beirão da Veiga, L., Russo, A. and Vacca, G. (2019c), The virtual element method with curved edges, ESAIM Math. Model. Numer. Anal. 53, 375–404.CrossRefGoogle Scholar
Benedetto, M. F., Berrone, S., Borio, A., Pieraccini, S. and Scialò, S. (2016), A hybrid mortar virtual element method for discrete fracture network simulations, J. Comput. Phys. 306, 148–166.CrossRefGoogle Scholar
Benedetto, M. F., Borio, A. and Scialò, S. (2017), Mixed virtual elements for discrete fracture network simulations, Finite Elem. Anal. Des. 134, 55–67.CrossRefGoogle Scholar
Benedetto, M. F., Borio, A., Kyburg, F., Mollica, J. and Scialò, S. (2022), An arbitrary order mixed virtual element formulation for coupled multi-dimensional flow problems, Comput. Methods Appl. Mech. Engrg 391, 114204.CrossRefGoogle Scholar
Berrone, S. and Raeli, A. (2022), Efficient partitioning of conforming virtual element discretizations for large scale discrete fracture network flow parallel solvers, Eng . Geol. 306, 106747.Google Scholar
Bertoluzza, S., Pennacchio, M. and Prada, D. (2019), High order VEMs on curved domains, Rend . Lincei Mat. Appl. 30, 391–412.Google Scholar
Boffi, D., Brezzi, F. and Fortin, M. (2013), Mixed Finite Element Methods and Applications, Vol. 44 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Bossavit, A. (1988), Mixed finite elements and the complex of Whitney forms, in The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987), Academic Press, pp. 137–144.Google Scholar
Botti, L., Di Pietro, D. A. and Droniou, J. (2018), A hybrid high-order discretisation of the Brinkman problem robust in the Darcy and Stokes limits, Comput. Methods Appl. Mech. Engrg 341, 278–310.CrossRefGoogle Scholar
Brenner, S. C. and Scott, L. R. (2008), The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics, third edition, Springer.CrossRefGoogle Scholar
Brenner, S. C. and Sung, L.-Y. (2018), Virtual element methods on meshes with small edges or faces, Math. Models Methods Appl. Sci. 28, 1291–1336.CrossRefGoogle Scholar
Brenner, S. C., Guan, Q. and Sung, L.-Y. (2017), Some estimates for virtual element methods, Comput . Methods Appl. Math. 17, 553–574.Google Scholar
Brenner, S. C., Sung, L.-Y. and Tan, Z. (2021), A C 1 virtual element method for an elliptic distributed optimal control problem with pointwise state constraints, Math. Models Methods Appl. Sci. 31, 2887–2906.CrossRefGoogle Scholar
Brezzi, F. and Marini, L. D. (2013), Virtual element methods for plate bending problems, Comput . Methods Appl. Mech. Engrg 253, 455–462.CrossRefGoogle Scholar
Brezzi, F. and Marini, L. D. (2021), Finite elements and virtual elements on classical meshes, Vietnam J. Math. 49, 871–899.CrossRefGoogle Scholar
Brezzi, F., Falk, R. S. and Marini, L. D. (2014), Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal. 48, 1227–1240.CrossRefGoogle Scholar
Cáceres, E. and Gatica, G. N. (2016), A mixed Virtual Element Method for the pseudostress-velocity formulation of the Stokes problem, IMA J. Numer. Anal. 37, 296–331.CrossRefGoogle Scholar
Cáceres, E., Gatica, G. N. and Sequeira, F. A. (2017), A mixed virtual element method for the Brinkman problem, Math. Models Methods Appl. Sci. 27, 707–743.CrossRefGoogle Scholar
Cáceres, E., Gatica, G. N. and Sequeira, F. A. (2018), A mixed virtual element method for quasi-Newtonian Stokes flows, SIAM J. Numer. Anal. 56, 317–343.CrossRefGoogle Scholar
Cangiani, A., Georgoulis, E. H., Pryer, T. and Sutton, O. J. (2017), A posteriori error estimates for the virtual element method, Numer . Math. 137, 857–893.Google Scholar
Cangiani, A., Gyrya, V. and Manzini, G. (2016), The nonconforming virtual element method for the Stokes equations, SIAM J. Numer. Anal. 54, 3411–3435.CrossRefGoogle Scholar
Cao, S. and Chen, L. (2018), Anisotropic error estimates of the linear virtual element method on polygonal meshes, SIAM J. Numer. Anal. 56, 2913–2939.CrossRefGoogle Scholar
Cao, S., Chen, L. and Guo, R. (2022a), Immersed virtual element methods for Maxwell interface problems in three dimensions. Available at arXiv:2202.09987.Google Scholar
Cao, S., Chen, L., Guo, R. and Lin, F. (2022b), Immersed virtual element methods for elliptic interface problems in two dimensions, J. Sci. Comput. 93, 12.CrossRefGoogle Scholar
Quiroz, D. Castañón and Di Pietro, D. A. (2020), A hybrid high-order method for the incompressible Navier–Stokes problem robust for large irrotational body forces, Comput. Math. Appl. 79, 2655–2677.CrossRefGoogle Scholar
Chen, L. and Huang, J. (2018), Some error analysis on virtual element methods, Calcolo 55, 5.CrossRefGoogle Scholar
Chen, L. and Wang, F. (2019), A divergence free weak virtual element method for the Stokes problem on polytopal meshes, J. Sci. Comput. 78, 864–886.CrossRefGoogle Scholar
Chernov, A., Marcati, C. and Mascotto, L. (2021), $p$ and $hp$ - virtual elements for the Stokes problem, Adv. Comput. Math. 47, 24.CrossRefGoogle Scholar
Chi, H., Beirão da Veiga, L. and Paulino, G. (2017), Some basic formulations of the virtual element method (VEM) for finite deformations, Comput. Methods Appl. Mech. Engrg 318, 148–192.CrossRefGoogle Scholar
Chi, H., Pereira, A., Menezes, I. F. M. and Paulino, G. H. (2020), Virtual element method (VEM)-based topology optimization: An integrated framework, Struct . Mult. Optim. 62, 1089–1114.CrossRefGoogle Scholar
Chin, E. B. and Sukumar, N. (2021), Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries, Comput. Methods Appl. Mech. Engrg 380, 113796.CrossRefGoogle Scholar
Chin, E. B., Lasserre, J. B. and Sukumar, N. (2015), Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra, Comput. Mech. 56, 967–981.CrossRefGoogle Scholar
Chinosi, C. and Marini, L. D. (2016), Virtual element method for fourth order problems: L 2-estimates, .Comput. Math. Appl 72, 1959–1967.CrossRefGoogle Scholar
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, Vol. 4 of Studies in Mathematics and its Applications, North-Holland.CrossRefGoogle Scholar
Cihan, M., Hudobivnik, B., Korelc, J. and Wriggers, P. (2022), A virtual element method for 3D contact problems with non-conforming meshes, Comput. Methods Appl. Mech. Engrg 402, 115385.CrossRefGoogle Scholar
Cockburn, B., Fu, G. and Qiu, W. (2017), A note on the devising of superconvergent HDG methods for Stokes flow by M-decompositions, IMA J. Numer. Anal. 37, 730–749.Google Scholar
Dassi, F. and Scacchi, S. (2020), Parallel block preconditioners for three-dimensional virtual element discretizations of saddle-point problems, Comput . Methods Appl. Mech. Engrg 372, 113424.CrossRefGoogle Scholar
Dassi, F. and Vacca, G. (2020), Bricks for mixed high-order virtual element method: Projectors and differential operators, Appl. Numer. Math. 155, 140–159.CrossRefGoogle Scholar
Dassi, F., Di Barba, P. and Russo, A. (2020a), Virtual element method and permanent magnet simulations: Potential and mixed formulations, IET Sci . Meas. Technol. 14, 1098–1104.Google Scholar
Dassi, F., Fumagalli, A., Scotti, A. and Vacca, G. (2022), Bend 3d mixed virtual element method for Darcy problems, Comput. Math. Appl. 119, 1–12.CrossRefGoogle Scholar
Dassi, F., Lovadina, C. and Visinoni, M. (2020b), A three-dimensional Hellinger–Reissner virtual element method for linear elasticity problems, Comput . Methods Appl. Mech. Engrg 364, 112910.CrossRefGoogle Scholar
Di Pietro, D. A. and Krell, S. (2018), A hybrid high-order method for the steady incompressible Navier–Stokes problem, J. Sci. Comput. 74, 1677–1705.CrossRefGoogle Scholar
Falk, R. S. and Neilan, M. (2013), Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal. 51, 1308–1326.CrossRefGoogle Scholar
Frerichs, D. and Merdon, C. (2022), Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem, IMA J. Numer. Anal. 42, 597–619.CrossRefGoogle Scholar
Gain, A. L., Paulino, G. H., Duarte, L. S. and Menezes, I. F. M. (2015), Topology optimization using polytopes, Comput . Methods Appl. Mech. Engrg 293, 411–430.CrossRefGoogle Scholar
Gardini, F., Manzini, G. and Vacca, G. (2019), The nonconforming virtual element method for eigenvalue problems, ESAIM Math. Model. Numer. Anal. 53, 749–774.CrossRefGoogle Scholar
Gatica, G. N., Munar, M. and Sequeira, F. A. (2018a), A mixed virtual element method for a nonlinear Brinkman model of porous media flow, Calcolo 55, 21.CrossRefGoogle Scholar
Gatica, G. N., Munar, M. and Sequeira, F. A. (2018b), A mixed virtual element method for the Navier–Stokes equations, Math. Models Methods Appl. Sci. 28, 2719–2762.CrossRefGoogle Scholar
Gauger, N., Linke, A. and Schroeder, P. (2019), On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond, SMAI J. Comput. Math. 5, 88–129.CrossRefGoogle Scholar
Girault, V. and Raviart, P.-A. (1979), Finite Element Approximation of the Navier–Stokes Equations, Vol. 749 of Lecture Notes in Mathematics, Springer.CrossRefGoogle Scholar
Guzmán, J. and Neilan, M. (2014), Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp. 83, 15–36.CrossRefGoogle Scholar
Guzmán, J. and Neilan, M. (2018), Inf-sup stable finite elements on barycentric refinements producing divergence-free approximations in arbitrary dimensions, SIAM J. Numer. Anal. 56, 2826–2844.CrossRefGoogle Scholar
Guzmán, J. and Scott, L. R. (2019), The Scott–Vogelius finite elements revisited, Math. Comp. 88, 515–529.CrossRefGoogle Scholar
Hiptmair, R. (2001), Discrete Hodge operators, Numer . Math. 90, 265–289.Google Scholar
Hussein, A., Aldakheel, F., Hudobivnik, B., Wriggers, P., Guidault, P.-A. and Allix, O. (2019), A computational framework for brittle crack-propagation based on efficient virtual element method, Finite Elem. Anal. Des. 159, 15–32.CrossRefGoogle Scholar
John, V., Linke, A., Merdon, C., Neilan, M. and Rebholz, L. G. (2017), On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev. 59, 492–544.CrossRefGoogle Scholar
Linke, A. and Merdon, C. (2016a), On velocity errors due to irrotational forces in the Navier–Stokes momentum balance, J. Comput. Phys. 313, 654–661.CrossRefGoogle Scholar
Linke, A. and Merdon, C. (2016b), Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg 311, 304–326.CrossRefGoogle Scholar
Lipnikov, K., Vassilev, D. and Yotov, I. (2014), Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes–Darcy flows on polygonal and polyhedral grids, Numer . Math. 126, 321–360.Google Scholar
Liu, X. and Chen, Z. (2019), The nonconforming virtual element method for the Navier–Stokes equations, Adv. Comput. Math. 45, 51–74.CrossRefGoogle Scholar
Liu, X., Li, J. and Chen, Z. (2017), A nonconforming virtual element method for the Stokes problem on general meshes, Comput . Methods Appl. Mech. Engrg 320, 694–711.CrossRefGoogle Scholar
Liu, X., Li, R. and Chen, Z. (2019), A virtual element method for the coupled Stokes–Darcy problem with the Beaver–Joseph–Saffman interface condition, Calcolo 56, 48.CrossRefGoogle Scholar
Liu, X., Li, R. and Nie, Y. (2020), A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem, Comput . Methods Appl. Mech. Engrg 372, 113351.CrossRefGoogle Scholar
Mascotto, L., Perugia, I. and Pichler, A. (2019), A nonconforming Trefftz virtual element method for the Helmholtz problem, Math. Models Methods Appl. Sci. 29, 1619–1656.CrossRefGoogle Scholar
Mattiussi, C. (1997), An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comput. Phys. 133, 289–309.CrossRefGoogle Scholar
Mora, D., Rivera, G. and Rodrguez, R. (2015), A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci. 25, 1421–1445.CrossRefGoogle Scholar
Mora, D., Rivera, G. and Rodriguez, R. (2017), A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem, Comput. Math. Appl. 74, 2172–2190.CrossRefGoogle Scholar
Munar, M. and Sequeira, F. A. (2020), A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow, Comput. Math. Appl. 80, 1240–1259.CrossRefGoogle Scholar
Natarajan, S. (2020), On the application of polygonal finite element method for Stokes flow: A comparison between equal order and different order approximation, Comput . Aided Geom. Design 77, 101813.CrossRefGoogle Scholar
Park, K., Chi, H. and Paulino, G. H. (2020), Numerical recipes for elastodynamic virtual element methods with explicit time integration, Int. J. Num. Meth. Engrg 121, 1–31.CrossRefGoogle Scholar
Park, K., Chi, H. and Paulino, G. H. (2021), B-bar virtual element method for nearly incompressible and compressible materials, Meccanica 56, 1423–1439.CrossRefGoogle Scholar
Simo, J. C. and Hughes, T. J. R. (2006), Computational Inelasticity, Vol. 7, Springer Science & Business Media.Google Scholar
Vacca, G. (2018), An H 1-conforming virtual element for Darcy and Brinkman equations, Math. Models Methods Appl. Sci. 28, 159–194.CrossRefGoogle Scholar
Wang, F. and Zhao, J. (2021), Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem, IMA J. Numer. Anal. 41, 1496–1521.CrossRefGoogle Scholar
Wang, G., Wang, Y. and He, Y. (2020), A posteriori error estimates for the virtual element method for the Stokes problem, J. Sci. Comput. 84, 37.CrossRefGoogle Scholar
Wriggers, P. and Hudobivnik, B. (2017), A low order virtual element formulation for finite elasto-plastic deformations, Comput . Methods Appl. Mech. Engrg 327, 459–477.CrossRefGoogle Scholar
Wriggers, P., Reddy, B. D., Rust, W. and Hudobivnik, B. (2017), Efficient virtual element formulations for compressible and incompressible finite deformations, Comput. Mech. 60, 253–268.CrossRefGoogle Scholar
Wriggers, P., Rust, W. T. and Reddy, B. D. (2016), A virtual element method for contact, Comput. Mech. 58, 1039–1050.CrossRefGoogle Scholar
Zhao, J., Chen, S. and Zhang, B. (2016), The nonconforming virtual element method for plate bending problems, Math. Models Methods Appl. Sci. 26, 1671–1687.CrossRefGoogle Scholar
Zhao, J., Zhang, B., Mao, S. and Chen, S. (2020), The nonconforming virtual element method for the Darcy–Stokes problem, Comput . Methods Appl. Mech. Engrg 370, 113251.CrossRefGoogle Scholar