Mathematical tools for electromagnetic modelling under realistic assumptions

Orthogonal decompositions and related results provide the framework in which electromagnetics naturally fits but are proved under restrictive assumptions that often are not fulfilled in real applications. Hence, only in fairly simple situations, a sound theoretical basis can be given to the mathematical models of computational electromagnetics, even though they are used in general applications. In this paper, similar mathematical tools that hold under more realistic assumptions, tuned to cover very many cases of practical interest, are given. These tools provide a suitable framework for models involving different inhomogeneous anisotropic materials, boundary and interfaces with edges and vertices, mixed boundary conditions, and topologically nontrivial configurations. How easily the proposed tools can be applied to rigorously prove the correctness of some basic models of electromagnetics under realistic assumptions is also shown.