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We consider a model Poisson problem in Rd (d = 2, 3) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges (d = 2) or small faces (d = 3). Our results extend the ones in [L. Beirão da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method, Math. Models Methods Appl. Sci. 27 (2017) 2557-2594] for the original two-dimensional virtual element method from [L. Beirão 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 (2013) 199-214] to the version of the virtual element method in [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] that can also be applied to problems in three dimensions.

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Mathematical Models and Methods in Applied Sciences

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