Open Access
Issue
ESAIM: M2AN
Volume 58, Number 3, May-June 2024
Page(s) 881 - 926
DOI https://doi.org/10.1051/m2an/2024017
Published online 03 June 2024
  1. D. Adak, E. Natarajan and S. Kumar, Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35 (2019) 222–245. [Google Scholar]
  2. B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini and A. Russo, Equivalent projectors for virtual element methods. Comput. Math. Appl. 66 (2013) 376–391. [Google Scholar]
  3. V. Anaya, M. Bendahmane, D. Mora and M. Sepúlveda, A virtual element method for a nonlocal fitzhugh–nagumo model of cardiac electrophysiology. IMA J. Numer. Anal. 40 (2020) 1544–1576. [Google Scholar]
  4. M. Arrutselvi and E. Natarajan, Virtual element stabilization for the system of time-dependent nonlinear convection-diffusion-reaction equations. Comput. Math. Appl. 142 (2023) 121–139. [Google Scholar]
  5. L. Beirãao 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 (2012) 199–214. [Google Scholar]
  6. L. Beirãao 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. [Google Scholar]
  7. L. Beirãao da Veiga, F. Brezzi, L.D. Marini and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26 (2016) 729–750. [Google Scholar]
  8. L. Beirãao da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method. Math. Models Methods Appl. Sci. 27 (2017) 2557–2594. [Google Scholar]
  9. L. Beirãao da Veiga, A. Pichler and G. Vacca, A virtual element method for the miscible displacement of incompressible fluids in porous media. Comput. Methods Appl. Mech. Eng. 375 (2021) 113649. [Google Scholar]
  10. S.C. Brenner and L.-Y. Sung, Virtual element methods on meshes with small edges or faces. Math. Models Methods Appl. Sci. 28 (2018) 1291–1336. [Google Scholar]
  11. A. Cangiani, P. Chatzipantelidis, G. Diwan and E.H. Georgoulis, Virtual element method for quasilinear elliptic problems. IMA J. Numer. Anal. 40 (2019) 2450–2472. [Google Scholar]
  12. J.R. Cannon and y. Lin, Non-classical h1 projection and galerkin methods for non-linear parabolic integro-differential equations. Calcolo 25 (1988) 187–201. [Google Scholar]
  13. L. Chen and J. Huang, Some error analysis on virtual element methods. Calcolo 55 (2018) 1–23. [Google Scholar]
  14. Y. Chen, Y. Huang and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction–diffusion equations. Int. J. Numer. Methods Eng. 57 (2003) 193–209. [Google Scholar]
  15. C.N. Dawson, Q. Du and T.F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equation. Math. comput. 57 (1991) 63–71. [Google Scholar]
  16. L.C. Evans, Partial Differential Equations, Vol. 19. American Mathematical Society (2022). [Google Scholar]
  17. I. Farago, Finite element method for solving nonlinear parabolic equations. Comput. Math. Appl. 21 (1991) 59–69. [Google Scholar]
  18. S.M.F. Garcia, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: The continuous-time case. Numer. Methods Partial Differ. Equ. 10 (1994) 129–147. [Google Scholar]
  19. A.T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier–Stokes equations. IMA J. Numer. Anal. 20 (2000) 633–667. [Google Scholar]
  20. M. Li, Cut-off error splitting technique for conservative nonconforming vem for n-coupled nonlinear schrödinger–boussinesq equations. J. Sci. Comput. 93 (2022) 86. [Google Scholar]
  21. B. Li and W. Sun, Error analysis of linearized semi-implicit galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10 (2013) 622–633. [Google Scholar]
  22. B. Li and W. Sun, Unconditional convergence and optimal error estimates of a galerkin-mixed fem for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51 (2013) 1959–1977. [Google Scholar]
  23. B. Li and W. Sun, Unconditionally optimal error analysis of fully discrete galerkin methods for general nonlinear parabolic equations. Preprint: arXiv:1303.6410 (2013). [Google Scholar]
  24. D. Li and J. Wang, Unconditionally optimal error analysis of crank–nicolson galerkin fems for a strongly nonlinear parabolic system. J. Sci. Comput. 72 (2017) 892–915. [Google Scholar]
  25. B. Li, H. Gao and W. Sun, Unconditionally optimal error estimates of a crank–nicolson galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52 (2014) 933–954. [Google Scholar]
  26. M. Li, J. Zhao and Sh. Chen, Unconditional error analysis of vems for a generalized nonlinear schrodinger equation. J. Comput. Math. (2023). [Google Scholar]
  27. W. Liu, Y. Chen, Q. Gu and Y. Huang, Virtual element method for nonlinear sobolev equation on polygonal meshes. Numer. Algorithms (2023). [Google Scholar]
  28. M. Luskin, A galerkin method for nonlinear parabolic equations with nonlinear boundary conditions. SIAM J. Numer. Anal. 16 (1979) 284–299. [Google Scholar]
  29. H. Malchow, S.V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. CRC Press (2007). [Google Scholar]
  30. J. Marshall, A. Adcroft, C. Hill, L. Perelman and C. Heisey, A finite-volume, incompressible navier stokes model for studies of the ocean on parallel computers. J. Geophys. Res. Oceans 102 (1997) 5753–5766. [Google Scholar]
  31. A. Pinelli, I.Z. Naqavi, U. Piomelli and J. Favier, Immersed-boundary methods for general finite-difference and finite-volume navier–stokes solvers. J. Comput. Phys. 229 (2010) 9073–9091. [Google Scholar]
  32. H.H. Rachford Jr., Two-level discrete-time galerkin approximations for second order nonlinear parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 1010–1026. [Google Scholar]
  33. B. Rivière and M.F. Wheeler, A discontinuous galerkin method applied to nonlinear parabolic equations. In: Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer (2000) 231–244. [Google Scholar]
  34. C. Talischi, G.H. Paulino, A. Pereira and I.F. Menezes, PolyMesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. Optim. 45 (2012) 309–328. [Google Scholar]
  35. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems Second Edition, Vol. 1. Springer Science and Media (2006). [Google Scholar]
  36. G. Vacca and L. Beirãao da Veiga, Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 31 (2015) 2110–2134. [Google Scholar]
  37. N. Wang and M. Li, Unconditional error analysis of a linearized BDF2 virtual element method for nonlinear ginzburg–landau equation with variable time step. Commun. Nonlinear Sci. Numer. Simul. 116 (2023) 106889. [Google Scholar]
  38. Y. Wang, Y. Chen and Y. Huang, A two-grid eulerian–lagrangian localized adjoint method to miscible displacement problems with dispersion term. Comput. Math. Appl. 80 (2020) 54–68. [Google Scholar]
  39. H. Yi, Y. Chen, Y. Wang and Y. Huang, Optimal convergence analysis of a linearized second-order BDF-PPIFE method for semi-linear parabolic interface problems. Appl Math Comput. 438 (2023) 127581. [Google Scholar]
  40. Y. Yu, mVEM: MATLAB Programming for Virtual Element Methods (2019–2022). [Google Scholar]
  41. Y. Yu, mvem: A matlab software package for the virtual element methods. Preprint: arXiv:2204.01339 (2022). [Google Scholar]

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