Contents I. Introduction
Given a steady current density and a spatial distribution of reluctivity , magnetostatics consists in finding fields and such that {\rm div}\ B=0,\quad H=\nu B,\quad{\rm rot}\ H=J\eqno{\hbox{(1)}}(plus boundary conditions that we shall ignore here). In terms of differential forms , and for which , and stand proxy, this becomes {\ssr d}b=0,\quad h=\nu b,\quad{\ssr d}h=j \eqno{\hbox{(2)}}where is a Hodge operator. The differential geometric framework in which (2) makes sense has a discrete counterpart, based on a cell-complex (i.e., a finite element mesh) with sets of nodes, edges, faces, volumes, and incidence matrices , in which (2) transposes as {\bf Db}=0,\quad{\bf h}={\mmb\nu}{\bf b},\quad{\bf R}^{\rm t}{\bf h}={\bf j} \eqno{\hbox{(3)}}where and are face-based arrays of degrees of freedom (DoF), and a matrix derived from in order to satisfy a consistency criterium that ensures the convergence, when is refined, of the field , as reconstructed from by using Whitney forms (“face elements”) , to the solution of (2). The array is obtained by integration of over faces of the complex dual to , and is thus edge-based.
