Trefftz Approximations in Complex Media: Accuracy and Applications

TL;DR

Trefftz function approximations offer high accuracy and exponential convergence in solving complex boundary value problems in physics, with broad applicability demonstrated through electromagnetics and photonics examples.

Contribution

This paper demonstrates the effectiveness of Trefftz approximations in complex media, extending their application beyond traditional homogeneous regions with theoretical and numerical support.

Findings

High-order convergence in electromagnetic simulations

Effective in both homogeneous and inhomogeneous media

Two mechanisms explain high accuracy: trigonometric interpolation and matrix well-posedness

Abstract

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential convergence with respect to the size of the basis set. We highlight two separate examples of that in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one -- somewhat surprisingly -- to well-posedness of random matrices.