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A family of three-dimensional virtual elements with applications to magnetostatics. (English) Zbl 1412.65201

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.

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

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References:

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