Super-algebraically convergent mathematical model of hollow waveguldes by Analytical Regularization Method

In the paper the definition of a non-saturated algorithm is given, which roughly says that for such an algorithm the smoothness of output data is proportional to smoothness of input data, and infinitely smooth input data correspond to infinitely smooth output data. The only input data for the problem considered is the contour of waveguide cross-section, and output data are eigen frequency and eigen current density (which is infinitely smooth with necessity for infinitely smooth contour see). We managed our algorithm to be of non-saturated kind. In spite of the fact that the kernel of corresponding integral equation has logarithmic singularity, by using a generalization of singular expansion of the kernel it is necessary to calculate numerically the coefficients of an infinitely smooth function only. The Fourier coefficients of the kernel can be obtained after that analytically. As the result, for the infinitely smooth contour, we obtain super-algebraic (i.e. faster than any algebraic) convergence in respect to the system size (in spite of the very slow linear convergence of the system determinant). The crucial point of this achievement is the ability of our algorithm to calculate fast all the necessary Fourier coefficients of the kernel with (almost) all the correct figures of the computer mantissa length. It is noteworthy that for non-infinitely smooth contour we loose immediately the super-algebraic convergence, but have algebraic convergence only, rate of which is dictated by the smoothness degree of the contour.