On the solution of time-harmonic scattering problems for Maxwell's equations

This paper deals with the scattering of a monochromatic electromagnetic wave by a perfect conductor surrounded by a locally inhomogeneous medium. The direct numerical solution of this problem by a finite-element method requires special edge elements. The aim of the present paper is to give an equivalent formulation of the problem well suited for both easy theoretical investigation and numerical implementation. Following a well-known idea, this formulation is obtained by adding a regularizing term such as "grad div" in the time-harmonic Maxwell equations, which leads us to solve an elliptic problem similar to the vector Helmholtz equation instead of Maxwell's equation. The numerical treatment of this new formulation requires only standard Lagrange finite elements. A unified approach, which is valid for the equations satisfied by either the electric or the magnetic field, is presented. It applies for a conductor with a Lipschitz-continuous boundary surrounded by a dissipative or nondissipative medium whose electromagnetic coefficients (permittivity and permeability) may be irregular. A family of scattering problems is defined, that is, the classical problem (which follows from Maxwell's equations) and the so-called "regularized problem" obtained by adding a regularizing term in Maxwell's equations. These problems are shown to be well posed and to have the same solution. An integral representation technique is described.