Existence and Uniqueness of a Solution of a Singular Volume Integral Equation in a Diffraction Problem

摘要

The present paper deals with external electromagnetic field diffraction by a locally inhomo-geneous body placed in an ideally conducting parallelepiped. The research is important in that the results can be used, for example, when solving diffraction problems for microwave ovens and biological objects. The problem can be solved numerically by finite-element methods. However, some difficulties are encountered if one attempts a straightforward application of these methods. First, the boundary value problem for the system of Maxwell equations is not elliptic, and therefore, the standard schemes cannot be used to prove the convergence of projection methods [1, p. 240]. Second, to achieve reasonable accuracy in computing the field in a body with permittivity e = 10-20 ?Q (the body mainly consists of water), one needs a very fine grid, and then one has to take a fine grid also in the volume outside the body. (The choice of different-scale grids inside and outside the body leads to wrong results.) In turn, this, together with the fact that we deal with a three-dimensional vector problem, results in sparse matrices of very large size in the finite-element method.