Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism

It has been recognised that numerical computations of magnetic fields by the finite-element method may require new types of elements, whose degrees of freedom are not field values at mesh nodes, but other field-related quantities like e.g. circulations along edges of the mesh. A rationale for the use of these special ‘mixed’ elements can be obtained if one cxpresses basic equations in terms of differential forms, instead of vector fields. The paper gives an elementary introduction to this point of view, presents Whitney forms (the mixed finite elements alluded to), and sketches two numerical methods (dual, in some sense), for eddy-current studies, based on these elements.