On a Finite-Element Method for Solving the Three-Dimensional Maxwell Equations

A Simplified Implicit Maxwell Solver

“…The spurious solutions, in particular, do not satisfy the divergence laws [32]. In this context, it is shown that boundary conditions play a crucial role in satisfying the constraints on the solution of Maxwell's equations [1,32]. Jiang et al show that if one satisfies Gauss's law on a part of the boundary at all times, the solution of Faraday's and Ampère's laws will satisfy Gauss's law throughout the volume [32].…”

Resolution of the Maxwell equations in a domain with reentrant corners

“…As moreover div \ s = 0, we deduce that Av s = curl/? 5 . p s being regular outside corners, this is also true for \ s .…”

“…In the same way as in the previous section, we can find a basis of the space V s = curl" l N, for a given p s e N, by exhibitmg a non vamshing element of V s , which we shall call v 5 …”

“…which yields the value of the constant c. Then, the regular part u^ can be determined with the help of a usual finite element method ( [33], [5] or [14]). On the other hand, due to the steep gradients, we might need to refine the mesh considerably at the reentrant corner if the global method (43)- (45) …”

“…The resolution of the steady-state or time-dependent Maxwell équations in a bounded domain has become classical thanks to finite différence methods in rectangular domains or finite element methods conforming in //(curl) [33] or mixed, conforming in //(curl, div) ( [5], [14]) in more complicated geometries. However, when the boundary is not regular and when the domain is not convex, that is in présence of reentrant corners, the mesh needs to be refined drastically in the neighborhood of the reentrant corners in order to get an acceptable numerical solution (see [7], [18] among others for a study of this approach).…”