Existence and uniqueness of a solution of a singular volume integral equation in a diffraction problem

The present paper deals with external electromagnetic field diffraction by a locally inhomogeneous 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 ε ≅ 10-20 ε0 (the body mainly consists of water), one needs a very one 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. The method of volume integral equations [2, pp. 59-153] is free of these disadvantages. Here the operator proves to be elliptic, and the integral equation should be solved only inside the body (in the inhomogeneity domain). In contrast to [2], we study the integral equation mainly by using the results of analysis of the corresponding boundary value problem [3] and the theorem on the equivalence of the boundary value problem and the integral equation. More precisely, we prove the existence and uniqueness theorem for the integral equation in L2 and the convergence of the Galerkin numerical method and obtain some results on the smoothness of solutions. © 2005 Pleiades Publishing, Inc.