A Modified Interior Penalty Virtual Element Method for Fourth-Order Singular Perturbation Problems (original) (raw)

References

  1. Adak, D., Natarajan, S.: Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique. Math. Comput. Simul. 172, 224–243 (2020)
    MathSciNet Google Scholar
  2. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)
    MathSciNet Google Scholar
  3. Alvarez, S.N., Beirão Da Veiga, L., Dassi, F., Gyrya, V., Manzini, G.: The virtual element method for a 2D incompressible MHD system. Math. Comput. Simul. 211, 301–328 (2023)
    MathSciNet Google Scholar
  4. Antonietti, P.F., Beirão da Veiga, L., Manzini, G.: The Virtual Element Method and Its Applications. Springer, Cham (2022)
    Google Scholar
  5. Antonietti, P.F., Bruggi, M., Scacchi, S., Verani, M.: On the virtual element method for topology optimization on polygonal meshes: a numerical study. Comput. Math. Appl. 74(5), 1091–1109 (2017)
    MathSciNet Google Scholar
  6. Antonietti, P.F., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28(2), 387–407 (2018)
    MathSciNet Google Scholar
  7. Beirão Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)
    MathSciNet Google Scholar
  8. Beirão Da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The Hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541–1573 (2014)
    MathSciNet Google Scholar
  9. Beirão Da Veiga, L., Dassi, F., Manzini, G., Mascotto, L.: The virtual element method for the 3D resistive magnetohydrodynamic model. Math. Models Methods Appl. Sci. 33(3), 643–686 (2023)
    MathSciNet Google Scholar
  10. Beirão Da Veiga, L., Lovadina, C., Vacca, G.: Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56(3), 1210–1242 (2018)
    MathSciNet Google Scholar
  11. Beirão Da Veiga, L., Mora, D., Vacca, G.: The Stokes complex for virtual elements with application to Navier–Stokes flows. J. Sci. Comput. 81, 990–1018 (2019)
    MathSciNet Google Scholar
  12. Brenner, S.C.: Poincaré–Friedrichs inequalities for piecewise \(H^1\) functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003)
    MathSciNet Google Scholar
  13. Brenner, S.C., Neilan, M.: A \(C^0\) interior penalty method for a fourth order elliptic singular perturbation problem. SIAM J. Numer. Anal. 49, 869–892 (2011)
    MathSciNet Google Scholar
  14. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2008)
    Google Scholar
  15. Brenner, S.C., Sung, L.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22(23), 83–118 (2005)
    MathSciNet Google Scholar
  16. Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. M2AN Math. Model. Numer. Anal. 43(2), 277–295 (2009)
    MathSciNet Google Scholar
  17. Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)
    MathSciNet Google Scholar
  18. Bringmann, P., Carstensen, C., Streitberger, J.: Local parameter selection in the \(C^0\) interior penalty method for the biharmonic equation. J. Numer. Math. 6, 66 (2023)
    Google Scholar
  19. Cáceres, E., Gatica, G.N.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37, 296–331 (2017)
    MathSciNet Google Scholar
  20. Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37(3), 1317–1354 (2016)
    MathSciNet Google Scholar
  21. Chen, L., Huang, J.: Some error analysis on virtual element methods. Calcolo 55(1), 5 (2018)
    MathSciNet Google Scholar
  22. Chen, L., Huang, X.: Nonconforming virtual element method for \(2m\)-th order partial differential equations in \(R^n\). Math. Comput. 89(324), 1711–1744 (2020)
    Google Scholar
  23. Chi, H., Pereira, A., Menezes, I.F.M., Paulino, G.H.: Virtual element method (VEM)-based topology optimization: an integrated framework. Struct. Multidiscip. Optim. 62(3), 1089–1114 (2020)
    MathSciNet Google Scholar
  24. Chinosi, C., Marini, L.D.: Virtual element method for fourth order problems: \(L^2\)-estimates. Comput. Math. Appl. 72(8), 1959–1967 (2016)
    MathSciNet Google Scholar
  25. Ciarlet, P.G.: The Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978)
    Google Scholar
  26. De Dios, B.A., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879–904 (2016)
    MathSciNet Google Scholar
  27. Feng, F., Han, W., Huang, J.: Virtual element method for an elliptic hemivariational inequality with applications to contact mechanics. J. Sci. Comput. 81(3), 2388–2412 (2019)
    MathSciNet Google Scholar
  28. Feng, F., Han, W., Huang, J.: A nonconforming virtual element method for a fourth-order hemivariational inequality in Kirchhoff plate problem. J. Sci. Comput. 90(3), 89 (2022)
    MathSciNet Google Scholar
  29. Gatica, G.N., Munar, M.: A mixed virtual element method for the Navier–Stokes equations. Math. Models Methods Appl. Sci. 28(14), 2719–2762 (2018)
    MathSciNet Google Scholar
  30. Huang, J., Yu, Y.: A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations. J. Comput. Appl. Math. 386, 113229 (2021)
    MathSciNet Google Scholar
  31. Ling, M., Wang, F., Han, W.: The nonconforming virtual element method for a stationary Stokes hemivariational inequality with slip boundary condition. J. Sci. Comput. 85(3), Paper No. 56 (2020)
  32. Liu, X., Li, J., Chen, Z.: A nonconforming virtual element method for the Stokes problem on general meshes. Comput. Methods Appl. Mech. Eng. 320, 694–711 (2017)
    MathSciNet Google Scholar
  33. Nilssen, T.K., Tai, X., Winther, R.: A robust nonconforming \(H^2\)-element. Math. Comput. 70(234), 489–505 (2001)
    Google Scholar
  34. Qiu, J., Wang, F., Ling, M., Zhao, J.: The interior penalty virtual element method for the fourth-order elliptic hemivariational inequality. Commun. Nonlinear Sci. Numer. Simul. 127(4644807), Paper No. 107547 (2023)
  35. Semper, B.: Conforming finite element approximations for a fourth-order singular perturbation problem. SIAM J. Numer. Anal. 29(4), 1043–1058 (1992)
    MathSciNet Google Scholar
  36. Talischi, C., Paulino, G.H., Pereira, A., Ivan Menezes, F.M.: Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45(3), 309–328 (2012)
    MathSciNet Google Scholar
  37. Wang, F., Wei, H.: Virtual element method for simplified friction problem. Appl. Math. Lett. 85(3820290), 125–131 (2018)
    MathSciNet Google Scholar
  38. Wang, F., Wu, B., Han, W.: The virtual element method for general elliptic hemivariational inequalities. J. Comput. Appl. Math. 389(4194398), Paper No. 113330 (2021)
  39. Wang, F., Zhao, J.: Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem. IMA J. Numer. Anal. 41(2), 1496–1521 (2021)
    MathSciNet Google Scholar
  40. Wang, M.: On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J. Numer. Anal. 39(2), 363–384 (2001)
    MathSciNet Google Scholar
  41. Wang, M., Xu, J., Hu, Y.: Modified Morley element method for a fourth order elliptic singular perturbation problem. J. Comput. Math. 24(2), 113–120 (2006)
    MathSciNet Google Scholar
  42. Warburton, T., Hesthaven, J.S.: On the constants in \(hp\)-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765–2773 (2003)
    MathSciNet Google Scholar
  43. Xiao, W., Ling, M.: Virtual element method for a history-dependent variational–hemivariational inequality in contact problems. J. Sci. Comput. 96(3), Paper No. 82 (2023)
  44. Zhang, B., Zhao, J.: The virtual element method with interior penalty for the fourth-order singular perturbation problem. Commun. Nonlinear Sci. Numer. Simul. 133, Paper No. 107964 (2024)
  45. Zhang, B., Zhao, J., Chen, S.: The nonconforming virtual element method for fourth-order singular perturbation problem. Adv. Comput. Math. 46(2), Paper No. 19 (2020)
  46. Zhang, X., Chi, H., Paulino, G.H.: Adaptive multi-material topology optimization with hyperelastic materials under large deformations: a virtual element approach. Comput. Methods Appl. Mech. Eng. 370(4129484), 112976 (2020)
  47. Zhao, J., Mao, S., Zhang, B., Wang, F.: The interior penalty virtual element method for the biharmonic problem. Math. Comp. 92(342), 1543–1574 (2023)
    MathSciNet Google Scholar
  48. Zhao, J., Zhang, B., Chen, S., Mao, S.: The Morley-type virtual element for plate bending problems. J. Sci. Comput. 76(1), 610–629 (2018)
    MathSciNet Google Scholar

Download references