Vectorial solution to double curl equation with generalized coulomb gauge for magnetostatic problems

Abstract

In this paper, a solution to the double curl equation with generalized Coulomb gauge is proposed based on the vectorial representation of the magnetic vector potential. Traditional Coulomb gauge is applied to remove the null space of the curl operator and hence the uniqueness of the solution is guaranteed. However, as the divergence operator cannot act on edge elements (curl-conforming) directly, the magnetic vector potential is represented by nodal elements, which is too restrictive, since both the tangential continuity and the normal continuity are required. Inspired by the mapping of Whitney forms by mathematical operators and Hodge (star) operators, the divergence of the magnetic vector potential, as a whole, can be approximated by Whitney elements. Hence, the magnetic vector potential can be expanded by the edge elements, where its vectorial nature is retained and only the tangential continuity is required. Finally, the original equation can be rewritten in a generalized form and solved in a more natural and accurate way using finite-element method.