Skip to main content
SpringerLink
Log in

The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review

  • Chapter
  • First Online:
The Virtual Element Method and its Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 31))

  • 918 Accesses

Abstract

In this chapter we review recent results on the conforming virtual element approximation of polyharmonic and eleastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the chapter. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R.A. Adams, J.J.F. Fournier, in Sobolev spaces, 2 edn. Pure and Applied Mathematics (Academic Press, New York, 2003)

    Google Scholar 

  2. B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Aldakheel, B. Hudobivnik, A. Hussein, P. Wriggers, Phase-field modeling of brittle fracture using an efficient virtual element scheme. Comput. Methods Appl. Mech. Eng. 341, 443–466 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. P.F. Antonietti, I. Mazzieri, A. Quarteroni, F. Rapetti. Non-conforming high order approximations of the elastodynamics equation. Comput. Methods Appl. Mech. Eng. 209/212, 212–238 (2012)

    Google Scholar 

  5. P.F. Antonietti, B. Ayuso de Dios, I. Mazzieri, A. Quarteroni, Stability analysis of discontinuous Galerkin approximations to the elastodynamics problem. J. Sci. Comput. 68(1), 143–170 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. P.F. Antonietti, L. Beirão da Veiga, S. Scacchi, M. Verani, A C1 virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34–56 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. P.F. Antonietti, G. Manzini, M. Verani, The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28(2), 387–407 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.F. Antonietti, I. Mazzieri, High-order discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes. Comput. Methods Appl. Mech. Eng. 342, 414–437 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. P.F. Antonietti, G. Manzini, M. Verani, The conforming virtual element method for polyharmonic problems. Comput. Math. Appl. 79(7), 2021–2034 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. P.F. Antonietti, F. Bonaldi, I. Mazzieri, A high-order discontinuous Galerkin approach to the elasto-acoustic problem. Comput. Methods Appl. Mech. Eng. 358, 112634, 29 (2020)

    Google Scholar 

  11. P.F. Antonietti, G. Manzini, I. Mazzieri, H.M. Mourad, M. Verani, The arbitrary-order virtual element method for linear elastodynamics models. convergence, stability and dispersion-dissipation analysis. Int. J. Numer. Methods Eng. 122(4), 934–971 (2021)

    Google Scholar 

  12. P.F. Antonietti, G. Manzini, S. Scacchi, M. Verani, A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations. Math. Models Methods Appl. Sci. 31(14), 2825–2853 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Ayuso de Dios, K. Lipnikov, G. Manzini, The non-conforming virtual element method. ESAIM Math. Model. Numer. 50(3), 879–904 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.W. Barrett, S. Langdon, R. Nürnberg, Finite element approximation of a sixth order nonlinear degenerate parabolic equation. Numer. Math. 96(3), 401–434 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Beirão da Veiga, K. Lipnikov, G. Manzini, Arbitrary order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49(5), 1737–1760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Beirão da Veiga, F. Brezzi, L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Beirão da Veiga, K. Lipnikov, G. Manzini, in The Mimetic Finite Difference Method, ed. MS&A. Modeling, Simulations and Applications, vol. 11, I edn. (Springer, Berlin, 2014)

    Google Scholar 

  19. L. Beirão da Veiga, G. Manzini, A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34(2), 782–799 (2014).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. L. Beirão da Veiga, G. Manzini, Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM Math. Model. Numer. Anal. 49(2), 577–599 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM. Math. Model. Numer. Anal. 50(3), 727–747 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Beirão da Veiga, F. Brezzi, L.D. Marini, A. Russo, Virtual element methods for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Beirão da Veiga, A. Chernov, L. Mascotto, A. Russo, Basic principles of hp virtual elements on quasiuniform meshes. Math. Models Methods Appl. Sci. 26(8), 1567–1598 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Beirão da Veiga, A. Chernov, L. Mascotto, A. Russo, Exponential convergence of the hp virtual element method in presence of corner singularities. Numer. Math. 138(3), 581–613 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Beirão da Veiga, F. Dassi, A. Russo, A C1 virtual element method on polyhedral meshes. Comput. Math. Appl. 79(7), 1936–1955 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Beirão da Veiga, F. Dassi, G. Manzini, L. Mascotto, Virtual elements for Maxwell’s equations. Comput. Math. Appl. 116, 82–99 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Benvenuti, A. Chiozzi, G. Manzini, N. Sukumar, Extended virtual element method for the Laplace problem with singularities and discontinuities. Comput. Methods Appl. Mech. Eng. 356, 571–597 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Bernardi, M. Dauge, Y. Maday, Polynomials in the Sobolev world. Technical report, HAL (2007). hal-00153795

    Google Scholar 

  30. S. Berrone, S. Pieraccini, S. Scialò, F. Vicini, A parallel solver for large scale DFN flow simulations. SIAM J. Sci. Comput. 37(3), C285–C306 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Berrone, A. Borio, Orthogonal polynomials in badly shaped polygonal elements for the virtual element method. Finite Elem. Anal. Des. 129, 14–31 (2017)

    Article  MathSciNet  Google Scholar 

  32. S. Berrone, A. Borio, G. Manzini, SUPG stabilization for the nonconforming virtual element method for advection–diffusion–reaction equations. Comput. Methods Appl. Mech. Eng. 340, 500–529 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. M.J. Borden, T.J.R. Hughes, C.M. Landis, C.V. Verhoosel, A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework. Comput. Methods Appl. Mech. Eng. 273, 100–118 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. J.H. Bramble, R.S. Falk, A mixed-Lagrange multiplier finite element method for the polyharmonic equation. RAIRO Modél. Math. Anal. Numér. 19(4), 519–557 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. S.C. Brenner, R. Scott, The mathematical theory of finite element methods, vol. 15 (Springer, Berlin, 2008)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. F. Brezzi, A. Buffa, K. Lipnikov, Mimetic finite differences for elliptic problems. M2AN Math. Model. Numer. Anal. 43, 277–295 (2009)

    Google Scholar 

  38. F. Brezzi, A. Buffa, G. Manzini, Mimetic scalar products for discrete differential forms. J. Comput. Phys. 257(Part B), 1228–1259 (2014)

    Google Scholar 

  39. F. Brezzi, R.S. Falk, L.D. Marini, Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227–1240 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. J.W. Cahn, On spinodal decomposition. Acta Metall. 9, 795–801 (1961)

    Article  Google Scholar 

  41. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    MATH  Google Scholar 

  42. J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)

    Google Scholar 

  43. A. Cangiani, G. Manzini, A. Russo, N. Sukumar, Hourglass stabilization of the virtual element method. Internat. J. Numer. Methods Eng. 102(3–4), 404–436 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Cangiani, V. Gyrya, G. Manzini, The non-conforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411–3435 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Cangiani, E.H. Georgoulis, T. Pryer, O.J. Sutton, A posteriori error estimates for the virtual element method. Numer. Math. 137, 857–893 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Cangiani, V. Gyya, G. Manzini, O. Sutton, Chapter 14: virtual element methods for elliptic problems on polygonal meshes, in K. Hormann, N. Sukumar, eds. Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics (CRC Press, Taylor & Francis Group, Boca Raton, 2017), pp. 1–20

    Google Scholar 

  47. A. Cangiani, G. Manzini, O. Sutton, Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37, 1317–1354 (2017). (online August 2016)

    Google Scholar 

  48. O. Certik, F. Gardini, G. Manzini, G. Vacca, The virtual element method for eigenvalue problems with potential terms on polytopic meshes. Appl. Math. 63(3), 333–365 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  49. O. Certik, F. Gardini, G. Manzini, L. Mascotto, G. Vacca, The p- and hp-versions of the virtual element method for elliptic eigenvalue problems. Comput. Math. Appl. 79(7), 2035–2056 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. F. Chave, D.A. Di Pietro, F. Marche, F. Pigeonneau, A hybrid high-order method for the Cahn-Hilliard problem in mixed form. SIAM J. Numer. Anal. 54(3), 1873–1898 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. F. Chen, J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Commun. Comput. Phys. 13(5), 1189–1208 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. L. Chen, X. Huang, Nonconforming virtual element method for 2mth order partial differential equations in \(\mathbb {R}^n\). Math. Comp. 89(324), 1711–1744 (2020)

    Google Scholar 

  53. C. Chinosi, L.D. Marini, Virtual element method for fourth order problems: L2-estimates. Comput. Math. Appl. 72(8), 1959–1967 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. F. Dassi, L. Mascotto, Exploring high-order three dimensional virtual elements: bases and stabilizations. Comput. Math. Appl. 75(9), 3379–3401 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. D.A. Di Pietro, J. Droniou, G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes. J. Comput. Phys. 355, 397–425 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  56. C.M. Elliott, S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Math. Comp. 58(198), 603–630, S33–S36 (1992)

    Google Scholar 

  57. C.M. Elliott, Z. Songmu, On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96(4), 339–357 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  58. C.M. Elliott, D.A. French, Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math. 38(2), 97–128 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  59. C.M. Elliott, D.A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26(4), 884–903 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. C.M. Elliott, D.A. French, F.A. Milner, A second-order splitting method for the Cahn-Hilliard equation. Numer. Math. 54(5), 575–590 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  61. E. Faccioli, F. Maggio, A. Quarteroni, A. Taghan, Spectral-domain decomposition methods for the solution of acoustic and elastic wave equations. The Leading Edge 61, 1160–1174 (1996). Faccioli1996

    Google Scholar 

  62. D. Gallistl, Stable splitting of polyharmonic operators by generalized Stokes systems. Math. Comp. 86(308), 2555–2577 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  63. F. Gardini, G. Manzini, G. Vacca, The nonconforming virtual element method for eigenvalue problems. ESAIM Math. Model. Numer. 53, 749–774 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  64. F. Gazzola, H.-C. Grunau, G. Sweers, in Polyharmonic Boundary Value Problems. Lecture Notes in Mathematics, vol. 1991. (Springer, Berlin, 2010). Positivity preserving and nonlinear higher order elliptic equations in bounded domains.

    Google Scholar 

  65. H. Gómez, V.M. Calo, Y. Bazilevs, T.J.R. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197(49–50), 4333–4352 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  66. P. Grisvard, Elliptic problems in nonsmooth domains, in Monographs and Studies in Mathematics, vol. 24. (Pitman (Advanced Publishing Program), Boston, 1985)

    MATH  Google Scholar 

  67. T. Gudi, M. Neilan, An interior penalty method for a sixth-order elliptic equation. IMA J. Numer. Anal. 31(4), 1734–1753 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. D. Kay, V. Styles, E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection. SIAM J. Numer. Anal. 47(4), 2660–2685 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. D. Komatitsch, J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999)

    Article  Google Scholar 

  70. D.J. Korteweg, Sur la forme que prenent les équations du mouvements des fluides si l’on tient compte des forces capilaires causées par des variations de densité considérables mains continues et sur la théorie de la capillarité dans l’hypothése d’une varation continue de la densité. Arch. Néerl Sci. Exactes Nat. Ser. II (1901)

    Google Scholar 

  71. L.D. Landau, On the theory of superconductivity, in D. ter Haar, (ed.) Collected papers of L. D. Landau, pp. 546–568 (Pergamon, Oxford, 1965)

    Google Scholar 

  72. K. Lipnikov, G. Manzini, A high-order mimetic method for unstructured polyhedral meshes. J. Comput. Phys. 272, 360–385 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  73. K. Lipnikov, G. Manzini, F. Brezzi, A. Buffa, The mimetic finite difference method for 3D magnetostatics fields problems. J. Comp. Phys. 230(2), 305–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  74. K. Lipnikov, G. Manzini, M. Shashkov, Mimetic finite difference method. J. Comput. Phys. 257, Part B:1163–1227 (2014)

    Google Scholar 

  75. X. Liu, Z. Chen, A virtual element method for the Cahn-Hilliard problem in mixed form. Appl. Math. Lett. 87, 115–124 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  76. C. Lovadina, D. Mora, I. Velásquez, A virtual element method for the von Kármán equations. Technical report, Preprint CI2MA:2019-36 (2019)

    Google Scholar 

  77. G. Manzini, A. Russo, N. Sukumar, New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24(8), 1621–1663 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  78. G. Manzini, K. Lipnikov, J.D. Moulton, M. Shashkov, Convergence analysis of the mimetic finite difference method for elliptic problems with staggered discretizations of diffusion coefficients. SIAM J. Numer. Anal. 55(6), 2956–2981 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  79. L. Mascotto, Ill-conditioning in the virtual element method: stabilizations and bases. Numer. Methods Partial Differential Equations 34(4), 1258–1281 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  80. D. Mora, I. Velásquez, A virtual element method for the transmission eigenvalue problem. Math. Models Methods Appl. Sci. 28(14), 2803–2831 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  81. D. Mora, I. Velásquez, Virtual element for the buckling problem of Kirchhoff-Love plates. Comput. Methods Appl. Mech. Eng. 360, 112687, 22 (2020)

    Google Scholar 

  82. D. Mora, G. Rivera, R. Rodríguez, A virtual element method for the Steklov eigenvalue problem. Math. Methods Appl. Sci. 25(08), 1421–1445 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  83. D. Mora, G. Rivera, I. Velásquez, A virtual element method for the vibration problem of Kirchhoff plates. ESAIM Math. Model. Numer. Anal. 52(4), 1437–1456 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  84. S. Naranjo-Alvarez, V. Bokil, V. Gyrya, G. Manzini, The virtual element method for resistive magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 381, 113815 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  85. K. Park, H. Chi, G.H. Paulino, On nonconvex meshes for elastodynamics using virtual element methods with explicit time integration. Comput. Methods Appl. Mech. Eng. 356, 669–684 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  86. K. Park, H. Chi, G.H. Paulino, Numerical recipes for elastodynamic virtual element methods with explicit time integration. Internat. J. Numer. Methods Eng. 121(1), 1–31 (2020)

    Article  MathSciNet  Google Scholar 

  87. G.H. Paulino, A.L. Gain, Bridging art and engineering using Escher-based virtual elements. Struct. Multidisciplinary Optim. 51(4), 867–883 (2015)

    Article  MathSciNet  Google Scholar 

  88. I. Perugia, P. Pietra, A. Russo, A plane wave virtual element method for the Helmholtz problem. ESAIM Math. Model. Num. 50(3), 783–808 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  89. A. Quarteroni, R. Sacco, F. Saleri, Numerical mathematics, in Texts in Applied Mathematics, vol. 37 (Springer, Berlin, 2007)

    Book  MATH  Google Scholar 

  90. P.-A. Raviart, J.-M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree] (Masson, Paris, 1983)

    Google Scholar 

  91. B. Rivière, M.F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, in Current trends in scientific computing (Xi’an, 2002). Contemporary Mathematical, vol. 329, pp. 271–282 (American Mathematical Society, Providence, RI, 2003)

    Google Scholar 

  92. J.S. Rowlinson, Translation of J. D. van der Waals’ “The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density”. J. Statist. Phys. 20(2), 197–244 (1979)

    Google Scholar 

  93. M. Schedensack, A new discretization for mth-Laplace equations with arbitrary polynomial degrees. SIAM J. Numer. Anal. 54(4), 2138–2162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  94. T. Sorgente, S. Biasotti, G. Manzini, M. Spagnuolo, The role of mesh quality and mesh quality indicators in the virtual element method. Adv. Comput. Math. 48(3), 1–34 (2022)

    MathSciNet  MATH  Google Scholar 

  95. S. Torabi, J. Lowengrub, A. Voigt, S. Wise, A new phase-field model for strongly anisotropic systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465(2105), 1337–1359 (2009). With supplementary material available online

    Google Scholar 

  96. G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes. Comput. Math. Appl. 74(5), 882–898 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  97. M. Wang, J. Xu, Minimal finite element spaces for 2m-th-order partial differential equations in Rn. Math. Comp. 82(281), 25–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  98. G.N. Wells, E. Kuhl, K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 218(2), 860–877 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  99. P. Wriggers, W.T. Rust, B.D. Reddy, A virtual element method for contact. Comput. Mech. 58(6), 1039–1050 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  100. J. Zhao, S. Chen, B. Zhang, The nonconforming virtual element method for plate bending problems. Math. Models Methods Appl. Sci. 26(9), 1671–1687 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  101. J. Zhao, B. Zhang, S. Chen, S. Mao, The Morley-type virtual element for plate bending problems. J. Sci. Comput. 76(1), 610–629 (2018)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

PFA and MV acknowledge the financial support of PRIN research grant number 201744KLJL “Virtual Element Methods: Analysis and Applications” funded by MIUR. PFA, IM, and MV, annd SS acknowledges the financial support of INdAM-GNCS. GM acknowledges the financial support of the ERC Project CHANGE, which has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 694515).

Author information

Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    Paola F. Antonietti, Ilario Mazzieri & Marco Verani

  2. IMATI-CNR, Pavia, Italy

    Gianmarco Manzini

  3. Dipartimento di Matematica, Università di Milano, Milan, Italy

    Simone Scacchi

Authors
  1. Paola F. Antonietti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianmarco Manzini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ilario Mazzieri
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Simone Scacchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marco Verani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Verani .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, MOX - Politecnico di Milano, Milano, Italy

    Paola F. Antonietti

  2. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italy

    Lourenço Beirão da Veiga

  3. Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Gianmarco Manzini

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Antonietti, P.F., Manzini, G., Mazzieri, I., Scacchi, S., Verani, M. (2022). The Conforming Virtual Element Method for Polyharmonic and Elastodynamics Problems: A Review. In: Antonietti, P.F., Beirão da Veiga, L., Manzini, G. (eds) The Virtual Element Method and its Applications. SEMA SIMAI Springer Series, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-030-95319-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation