References
- B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, and A. Russo, Equivalent projectors for Virtual element methods, Comput. Math. Appl. 66 (3) (2013), pp. 376–391.
- P.F. Antonietti, L. Beirao da Veiga, D. Mora, and M. Verani, A stream Virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 52(1) (2014), pp. 386–404.
- P.F. Antonietti, L. Beirao Da Veiga, S. Scacchi, and M. Verani, A C1 Virtual element method for the Cahn–Hilliard equation with polygonal meshes, SIAM J. Numer. Anal. 54(1) (2016), pp. 34–56.
- J. Argyris, M. Haase, and J.C. Heinrich, Finite element approximation to two-dimensional Sine-Gordon solitons, Comput. Methods in Appl. Mech. Eng. 86 (1) (1991), pp. 1–26.
- A. Ashyralyev and A. Sirma, A note on the numerical solution of the semilinear Schrodinger equation, Nonlinear Anal. Theory Methods Appl. 71(12) (2009), pp. 2507–2516.
- G.A. Baker, V.A Dougalis, and O. Karakashian, On multistep-galerkin discretizations of semilinear hyperbolic and parabolic equations, Nonlinear Anal. Theory Methods Appl. 4(3) (1980), pp. 579–597.
- L. Beirao Da Veiga and K. Lipnikov, A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput. 32 (2) (2010), pp. 875–893.
- L. Beirao da Veiga and G. Manzini, Residual a posteriori error estimation for the Virtual element method for elliptic problems, ESAIM Math. Model. Numer. Anal. 49(2) (2015), pp. 577–599.
- L. Beirao Da Veiga, K. Lipnikov, and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal. 48(4) (2010), pp. 1419–1443.
- L. Beirao Da Veiga, K. Lipnikov, and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes, SIAM J. Numer. Anal. 49 (5) (2011), pp. 1737–1760.
- L. Beirao 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(01) (2013), pp. 199–214.
- L. Beirao Da Veiga, F. Brezzi, and L.D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51 (2) (2013), pp. 794–812.
- L. Beirao da Veiga, K. Lipnikov, and G. Manzini, The mimetic finite difference method for elliptic problems, 11, Springer, 2014.
- L. Beirao 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 (08) (2014), pp. 1541–1573.
- L. Beirao Da Veiga, C. Lovadina, and D. Mora, A Virtual element method for elastic and inelastic problems on polytope meshes, Comput. Method Appl. Mech. Eng. 295 (2015), pp. 327–346.
- L. Beirao da Veiga, F. Brezzi, L.D. Marini, and A. Russo, Mixed Virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Math. 50(3) (2016), pp. 727–747.
- L. Beirao da Veiga, F. Brezzi, L.D. Marini, and A. Russo, Virtual element method for general second-order elliptic problems on polygonal meshes, Math. Model. Methods Appl. Sci. 26(04) (2016), pp. 729–750.
- L. Beirao da Veiga, F. Dassi, and A. Russo, High-order Virtual element method on polyhedral meshes, Comput. Math. Appl. 74(5) (2017), pp. 1110–1122.
- L. Beirao Da Veiga, C. Lovadina, and G. Vacca, Divergence free Virtual elements for the Stokes problem on polygonal meshes, ESIAM Math. Model. Numer. Anal. 51(2) (2017), pp. 509–535.
- A.G. Bratsos, The solution of the two-dimensional Sine Gordon equation using the method of lines, J. Comput. Appl. Math. 206(1) (2007), pp. 251–277.
- F. Brezzi and L.D. Marini, Virtual element methods for plate bending problems, Comput. Method Appl. Mech. Eng. 253 (2013), pp. 455–462.
- F. Brezzi, K. Lipnikov, and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal. 43(5) (2005), pp. 1872–1896.
- F. Brezzi, R.S. Falk, and L. D. Marini, Basic principles of mixed Virtual element methods, ESIAM Math. Model. Numer. Anal. 48 (4) (2014), pp. 1227–1240.
- A. Cangiani, E.H. Georgoulis, T. Pryer, and O.J. Sutton, A posteriori error estimates for the Virtual element method, Numer. Math. (2016), pp. 1–37.
- A. Cangiani, G. Manzini, and O.J. Sutton, Conforming and nonconforming Virtual element methods for elliptic problems, IMA J. Numer. Anal. 37(3) (2016), pp. 1317–1354.
- C.M. Chen, S. Larsson, and N.Y. Zhang, Error estimates of optimal order for finite element methods with interpolated coefficients for the nonlinear heat equation, IMA J. Numer. Anal. 9 (4) (1989), pp. 507–524.
- M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional Sine-Gordon equation using the radial basis functions, Math. Comput. Simul. 79(3) (2008), pp. 700–715.
- B.A. de Dios, K. Lipnikov, and G. Manzini, The nonconforming Virtual element method, ESAIM Math. Model. Numer. Anal. 50(3) (2016), pp. 879–904.
- T. Dupont, L2-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10(5) (1973), pp. 880–889.
- 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), pp. 132–160.
- F. Gardini and G. Vacca, Virtual element method for Second Order Elliptic Eigenvalue Problems, IMA J. Numer. Anal. (2017). Available at http://doi.org/10.1093/imanum/drx063
- D. Lupo, K.R. Paye, and N.I. Popivanov, On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type, Nonlinear Anal Theory Methods Appl. 108 (2014), pp. 29–56.
- L. Mascotto, A therapy for the ill-conditioning in the Virtual element method, preprint (2017). Available at arxiv:1705.10581
- S.E Mousavi and N. Sukumar, Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons, Comput. Mech. 47(5) (2011), pp. 535–554.
- S. Natarajan, S. Bordas, and D. Roy Mahapatra, Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping, Int. J. Numer. Methods Eng. 80(1) (2009), pp. 103–134.
- N.M. Newmark, A method of computation for structural dynamics, J. Eng. Mech. Div. 85(3) (1959), pp. 67–94.
- 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(3) (2012), pp. 309–328.
- V. Thomee, Galerkin finite element methods for parabolic problems, Springer, Berlin, 1984.
- G. Vacca, Virtual element methods for hyperbolic problems on polygonal meshes, Comput. Math. Appl. 74(1) (2017), pp. 882–898.
- G. Vacca and L. Beirao da Veiga, Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods Partial Differ. Equ. 31(6) (2015), pp. 2110–2134.
- M. Zlamal, A finite element solution of the nonlinear heat equation, RAIRO Anal. Numer. 14 (1980), pp. 203–216.