Solution to the Time-Harmonic Maxwell's Equations in a Waveguide; Use of Higher-Order Derivatives for Solving the Discrete Problem

The authors study time-harmonic Maxwell's equations in the junction of two rectangular waveguides. They prove existence and uniqueness of the solution, except for a discrete set of values of the frequency of the incident wave. Introducing an artificial boundary condition, they prove that the so-obtained approximate solution converges to the exact solution, and depends analytically on the frequency. They use next higher-order derivatives for solving the approximate problem for a family of shapes of the domain on a large frequency band. This leads to a very efficient method for the numerical simulation of the waveguide, which can be used, for instance, in optimal shape design.