Output Error Estimates in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations

Abstract

The Reduced Basis Method (RBM) (Rozza et al., Archiv Comput Methods Eng 15:229–275, 2008) generates low-order models for efficient evaluation of parametrized PDEs in many-query and real-time contexts. It can be seen as a parametric model reduction method (Benner et al., SIAM Rev 57(4):483–531, 2015), where greedy selection is combined with a projection space composed of solution snapshots. The approximation quality is certified by using rigorous error estimators. We apply the RBM to systems of Maxwell’s equations arising from electrical circuits. Using microstrip models, the input-output behaviour of the interconnect structures is approximated for a certain frequency range. Typically, an output is given by a linear functional, but in the case of impedance parameters (also called Z-parameters), the output is quadratic. An expanded formulation is used to rewrite the system to compliant form, i.e., a form, where the input and output are identical. This enables fast convergence in the approximation error and thus very low reduced model sizes. A numerical example from the microwave regime shows the advantage of this approach.