New vector finite elements for three-dimensional magnetic field computation

Finite-element vector potential solutions of three-dimensional magnetic field problems are usually obtained by approximating each component of the vector potential by a separate set of scalar finite-element approximation functions and by imposing continuity conditions between elements on all three components. This procedure is equivalent to imposing continuity of both the normal and the tangential components of the vector potential. We show in this paper that this procedure is too restrictive: While continuity of the tangential component of the vector potential is required, continuity of the normal components is not essential in the variational formulation. We introduce a new type of vector finite-element approximation function that has the property that it interpolates not to point values of each component of vector potential, but rather to the tangential projection of the vector potential on each edge of tetrahedral finite elements. With the new basis functions, continuity of the normal component of the vector potential is provided only approximately by means of the natural interface conditions inherent in the variational procedure. This results in a more efficient procedure for the solution of three-dimensional magnetostatic field problems than is obtained by enforcing normal component continuity exactly.