Convergence in the spectral domain formulation of waveguide and scattering problems

Many problems in numerical electromagnetics are formulated in the spectral domain, for example, microstrip and frequency-selective surfaces. In the formulation of a linear set of equations, infinite inner product integrals or summations need to be evaluated. The concepts of absolute and relative convergence using common basis functions are discussed using specific numerical examples. No evidence was found for the existence of a relative convergence phenomenon. It is shown that one should rely on absolute convergence where a sufficiently large number of satisfactory basis functions are used together with a large (approaching infinite) inner product truncation point