Abstract
Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for polygonal meshes, each element of which admits a virtual quasi-uniform triangulation. This sub-mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are presented by means of standard technical tools in finite element methods.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L., Russo, A.: Equivalent projectors for virtual element methods. Computers & Mathematics with Applications 66(3), 376–391 (2013)
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), 1–16 (2012)
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)
Beirão Da Veiga, L., Brezzi, F., Marini, L., Russo, A.: Serendipity nodal VEM spaces. Comput. Fluids 141, 2–12 (2016)
Beirão Da Veiga, L., Lovadina, C., Russo, A.: Stability analysis for the virtual element method. arXiv:1607.05988 (2016) (to appear on M3AS)
Beirão Da Veiga, L., Manzini, G.: Residual a posteriori error estimation for the virtual element method for elliptic problems. ESAIM Math. Model. Numer. Anal. (M2AN) 49, 577–599 (2015)
Beirão Da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Virtual element method for general second-order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)
Brenner, S.C.: Poincaré–Friedrichs inequalities for piecewise \(H^1\) functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)
Brenner, S.C., Guan, Q., Sung, L.: Some estimates for virtual element methods. Comput. Methods Appl. Math. (2017). https://doi.org/10.1515/cmam-2017-0008
Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. ESAIM Math. Model. Numer. Anal. 43, 277–295 (2009)
Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)
Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37, 1317–1354 (2017)
Chen, L., Wei, H.Y., Wen, M.: An interface-fitted mesh generator and virtual element methods for elliptic interface problems. J. Comput. Phys. 334, 327–348 (2017)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Cockburn, B., Guzm, J., Soon, S.-C., Stolarski, H.K.: An analysis of the embedded discontinuous galerkin method for second-order elliptic problems. SIAM J. Numer. Anal. 47(4), 2686–2707 (2009)
Dautray, R., Lions, J.-L.: Mathematicl Analysis and Numerical Methods for Science and Technology, Volume 1 Physical Origins and Classical Methods. Springer, Berlin (2000)
Dios, B.A.D., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. (M2AN) 50, 879–904 (2014)
Dupont, T., Scott, R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34(150), 441–463 (1980)
Guzey, S., Cockburn, B., Stolarski, H.K.: The embedded discontinuous Galerkin method: application to linear shell problems. Int. J. Numer. Methods Eng. 70, 757–790 (2007)
Mora, D., Rivera, G., Rodríguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25(08), 1421–1445 (2015)
Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element method with polynomial reduction. J. Comput. Appl. Math. 285, 45–58 (2015)
Shi, Z.C., Wang, M.: Finite Element Methods. Science Press, Beijing (2013)
Wriggers, P., Rust, W., Reddy, B.: A virtual element method for contact. Comput. Mech. 58, 1039–1050 (2016)
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We thank the referees for valuable suggestions and comments which improved an early version of the paper. The first author was supported by the National Science Foundation (NSF) DMS-1418934 and in part by the Sea Poly Project of Beijing Overseas Talents. The second author was partially supported by NSFC (Grant No. 11571237).
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Chen, L., Huang, J. Some error analysis on virtual element methods. Calcolo 55, 5 (2018). https://doi.org/10.1007/s10092-018-0249-4
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DOI: https://doi.org/10.1007/s10092-018-0249-4