Beirão da Veiga, Lourenco; Brezzi, F.; Dassi, Franco; Marini, L. D.; Russo, A. A family of three-dimensional virtual elements with applications to magnetostatics. (English) Zbl 1412.65201 SIAM J. Numer. Anal. 56, No. 5, 2940-2962 (2018). As a simple model problem, the application of virtual element methods (VEMs) to the linear magnetostatic three-dimensional problem in the formulation of Kikuchi is considered. New serendipity VEM spaces are also introduced. The paper is organized as follows. Section 1 is an introduction. In Section 2, some basic notations and well-known properties of polynomial spaces are given. Section 3 consists of the following six subsections: 3.1. The Kikuchi variational formulation; 3.2. The local spaces on faces; 3.3. The local serendipity spaces on faces; 3.4. The local spaces on polyhedra; 3.5. The global spaces and 3.6. Scalar products for VEM spaces in three dimensions. In Section 4, the discretized problem is introduced and the a priori error bounds for it are proved. Finally, some numerical results are reported and discussed in Section 5. Reviewer: Temur A. Jangveladze (Tbilisi) Cited in 38 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 78A30 Electro- and magnetostatics 78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory 35B45 A priori estimates in context of PDEs Keywords:virtual element methods; serendipity; magnetostatic problems Software:Voro++ PDFBibTeX XMLCite \textit{L. Beirão da Veiga} et al., SIAM J. Numer. Anal. 56, No. 5, 2940--2962 (2018; Zbl 1412.65201) Full Text: DOI arXiv References: [1] B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl., 66 (2013), pp. 376–391. · Zbl 1347.65172 [2] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. 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