We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959-962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [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]), we use here, in a systematic way, the L2-projection operators as designed 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]. In particular, the present method does not reduce to the original Virtual Element Method of [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] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [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 case of variable coefficients produces, in general, sub-optimal results.
Beirao da Veiga, L., Brezzi, F., Marini, L.D., & Russo, A. (2016). Virtual Element Method for general second-order elliptic problems on polygonal meshes. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 26(4), 729-750.
Citazione: | Beirao da Veiga, L., Brezzi, F., Marini, L.D., & Russo, A. (2016). Virtual Element Method for general second-order elliptic problems on polygonal meshes. MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES, 26(4), 729-750. |
Tipo: | Articolo in rivista - Articolo scientifico |
Carattere della pubblicazione: | Scientifica |
Presenza di un coautore afferente ad Istituzioni straniere: | No |
Titolo: | Virtual Element Method for general second-order elliptic problems on polygonal meshes |
Autori: | Beirao da Veiga, L; Brezzi, F; Marini, LD; Russo, A |
Autori: | |
Data di pubblicazione: | 2016 |
Lingua: | English |
Rivista: | MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1142/S0218202516500160 |
Appare nelle tipologie: | 01 - Articolo su rivista |