S. P. Kulikov

Problems of the numerical solution of two-dimensional (2D) and three-dimensional (3D) scattering and absorption of stationary electromagnetic waves Rayleigh and resonant ranges by locally-non-uniform non-magnetic dielectrics are considered. The integral operator which influences a vector of amplitude of an electric field is Fredholm (compact) operator in 2D case of E-polarization, but the singular operator in 2D case of H-polarization and in the general 3D case. The method of the final sums and digitization of the integral equation leads in these cases to system of the linear algebraic equations (SLAE) from N, 2N and 3N complex unknown values of an electric field in units of a settlement grid. Value N can be rather great for problems of resonant area of wavelengths. Therefore the numerical solution received SLAE can be based on iterative methods. In the given work the method of optimum simple iteration (MOSI) which essentially uses the information of a spectrum of the integral operator. Numerically and also analytically laws of a spectrum of integral transition operator and corresponding matrix of a problem are established. This spectrum consists of two parts in considered vector cases - a continuous part, which has the established dependence on values of permittivity of a body, and a discrete part, which depends only from geometry and scattering region size. The points of accumulation of discrete spectrum are on a continuous component. In a Rayleigh range of wavelengths the discrete component is in rather small area near with a continuous one and at definition of optimum parameter it can be neglected. For a homogeneous body, for example, a continuous spectrum of the transition operator is a piecewise from a point 0 up to points 1- , where is the relative value permittivity of a body and in this case optimum parameter is . For the non-uniform body without losses the formula is the same, where is the maximal value in the body. In a resonant case there is a discrete spectrum which for a body without losses fills some significant area in a low half of a complex plane and which forms an imaginary value component of optimum parameter. In cases of an extensive discrete spectrum the optimum parameter MOSI is defined by special algorithms for a spectrum of a triangle or a polygon with one straight line on a continuous real spectrum and with the remote tops near to a point of singularity and in low half of a complex plane. As the body has the losses and value of has negative imagine part, the spectral area extends due to the continuous part on upper half of plane and contains also piece-wises from 0 up to 1- . In this case, the optimal parameter proposed explicit algorithm for determining its. The formula of the first approximation MOSI is proposed on the basis of explicit formulas and algorithms to determine the optimal parameter. This formula allows to evaluate the field without solving the integral equation. It has been shown that the use of approximation based on the first member of the optimal number of iterations expands the area of the effective use of so called the Born approximation. The principle of an inva-riance of a spectral picture and optimum parameter at transition from a rare settlement grid to a grid with a good approximation of the field is used for the range of resonant scattering. Numerical solutions of some 2D and 3D scattering and absorption problems in Rayleigh and a resonant range on non-uniform dielectric structures are presented.