A domain decomposition method for computing the scattering matrix of waveguide circuits

We analyze and develop numerical methods for time-harmonic wave scattering in metallic waveguide structures of infinite extent. We show that radiation boundary conditions formulated via projectors onto outgoing modes determine the coefficients of propagating modes uniquely, even when the structure supports trapped modes. Building on this, we introduce a fast divide-and-conquer solver that constructs solution operators on subdomains as impedance-to-impedance maps and couples them by enforcing continuity conditions across their interfaces. For Dirichlet waveguides, the computation of impedance-to-impedance maps requires the solution of mixed Dirichlet-Impedance boundary value problems. We construct a second-kind Fredholm integral equation that avoids near-hypersingular operators, requiring only integral operators whose kernels are at most weakly singular. Numerical experiments on large structures with many circuit elements demonstrate substantial efficiency gains: the proposed approach typically outperforms state-of-the-art fast iterative and fast direct solvers by one to two orders of magnitude.