Application of a New Method for Approximate Solutions and Error Estimates to Waveguide Discontinuity and Phased Array Problems

Many electromagnetic boundary value problems can be put in the form y=Kx , where K is a scattering operator, y is a vector or function proportional to the incident field, and x represents the scattered field. A useful and common solution to these problems is obtained by approximating K by an N th order degenerate operator so that y = K N x N . A number of authors formulated upper bounds to the error ∥ x − x N ∥ that are essentially equivalent, but which we found to be unnecessarily restrictive. A reformulation of the problem leads to a solution in terms of a Neumann series involving the difference operator K − K N with the range of K N properly extended to the range of K . This approach is found to yield, with much less restrictive conditions, close error estimates as opposed to bounds. In fact, for some otherwise intractable scattering problems, this formulation converges so rapidly for N = 1 that an essentially closed form approximate solution may be written. Other salient features are also discussed together with examples and new physical results for a diverse class of boundary value problems including analysis of coaxial TE 11 discontinuities and dielectric covered phased arrays.