Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches

We propose an analysis (discretization techniques, convergence) of numerical schemes for Maxwell equations which use two meshes (not necessarily tetrahedral), dual to each other. Schemes of this class generalize Yee's "finite difference in time domain" method (FDTD). We distinguish network equations (the discrete equivalents of Faraday's law and Ampere's relation) which can be set up without any recourse to finite elements, and network constitutive laws, whose validity cannot be assessed without them. This establishes a complementarity between "finite integration techniques" (FIT) and the finite element method (FEM). As an example, a Yee-like method on a simplicial mesh and its so-called "orthogonal" dual, is described, and its convergence is proved.