A coercive bilinear form for Maxwell's equations

Abstract

When one wants to treat the time-harmonic Maxwell equations with variational methods, one has to face the problem that the natural bilinear form is not coercive on the whole Sobolev space H 1. One can, however, make it coercive by adding a certain bilinear form on the boundary of the domain. This addition causes a change in the natural boundary conditions. The additional bilinear form (see (2.7), (2.21), (3.3)) contains tangential derivatives of the normal and tangential components of the field on the boundary, and it vanishes on the subspaces of H 1 that consist of fields with either vanishing tangential components or vanishing normal components on the boundary. Thus the variational formulations of the “electric” or “magnetic” boundary value problems with homogeneous boundary conditions are not changed. A useful change is caused in the method of boundary integral equations for the boundary value problems and for transmission problems where one has to use nonzero boundary data. The idea of this change emerged from the desire to have strongly elliptic boundary integral equations for the “electric” boundary value problem that are suitable for numerical approximation. Subsequently, it was shown how to incorporate the “magnetic” boundary data and to apply the idea to transmission problems. In the present note we present this idea in full generality, also for the anisotropic case, and prove coercivity without using symbols of pseudo-differential operators on the boundary.