author:
- Igor Tsukerman
- 'Shampy Mansha, Y. D. Chong'
- 'Vadim A. Markel'
bibliography:
- 'Igor\_reference\_dbase.bib'
- 'flame-slab-references.bib'
title: |
    Trefftz Approximations in Complex Media:\
    Accuracy and Applications
---

Trefftz approximations ,convergence ,Maxwell’s equations ,homogenization ,photonic devices ,wave scattering ,interpolation ,finite difference schemes ,random matrices 65M06 ,76M20 ,35B27 ,76M50 ,74Q15 ,35Q61 ,15B52

Introduction {#sec:Intro}
============

Many classical numerical methods for partial differential equations rely on polynomial or piecewise-polynomial approximations of the solution. Examples include traditional finite difference (FD) schemes, the finite element method (FEM), and the boundary element method (BEM). But a strong incentive to achieve qualitatively higher accuracy of the numerical solution has led, over several decades of research, to the development of Trefftz-based methods. By definition, Trefftz functions satisfy locally (in weak form) the underlying differential equations of the problem, which often results in high-order algebraic or even exponential convergence with respect to the dimension of the basis. This qualitative accuracy improvement has been demonstrated in a large variety of mathematical methods and engineering applications: Domain Decomposition [@Herrera00; @Farhat-DD-DG09], Generalized FEM [@Melenk96; @Babuska97; @Babuska-GFEM2004; @Plaks03; @Proekt02; @Strouboulis-GFEM-Helmholtz2006], Discontinuous Galerkin [@Farhat-DD-DG09; @Cockburn00; @Arnold02; @Buffa-Monk-ultraweak08; @Gittelson09; @Gabard-wave-based-DG-UW-LS11; @Hiptmair-PWDG-Helmholtz2011; @Kretzschmar-IMA16], and finite difference (“Flexible Local Approximation MEthods,” FLAME) [@Tsukerman-JCP10; @Tsukerman05; 