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The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. (English) Zbl 1207.78041

In the present paper it is proposed a mimetic finite difference method that extends the mimetic formulation to the numerical treatment of magnetostatic field problems. The approach developed in this paper uses degrees of freedom attached to the vertices and edges, and employs natural discrete operators that mimic the curl and the gradient operator of the differential setting.

Using the discrete curl and gradient operators and two suitable quadrature rules for the numerical discretization of volume integrals on the computational domain, the authors provide a numerical discretization of the div-curl variational formulation of magnetostatics. It is established the existence and uniqueness of the numerical solution by means of an argument that generalizes the concept of logically rectangular or cubic meshes by Hyman and Shashkov to the case of unstructured polyhedral meshes. In the final part of the present paper, the accuracy of the method is illustrated by numerically solving several problems.

MSC:
78M20Finite difference methods (optics)
78A30Electro- and magnetostatics
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