Nodal Finite Elements for Maxwell's equations: how to handle singularities in regions with corners
In this talk, we are concerned with the time-harmonic Maxwell equations which are given in a regularized form similar to the vector Helmholtz equation. In the case of a regular or convex domain, this problem can be discretized by means of nodal finite elements. Here, we are interested in the same problem in a non-convex polyhedron. The mathematical analysis shows that a nodal Finite Element Method does not approach in general the solution to Maxwell's equations, but actually the solution to a neighbouring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near reentrant edges and corners of the domain, singularities which cannot be approximated by Lagrange finite elements. The numerical method that we propose to overcome these difficulties is based on the decomposition of the electromagnetic field into a regular part that can be treated numerically by nodal finite elements, and a singular part that is determined and taken into account explicitly. Numerical results illustrate two variants of the method.
E-mail : lohrenge@math.unice.fr