\underline{s} \in \mathrm{Null}(N^T), ~~~
\mathrm{where}~~  N^T_{\alpha \beta} = l_\beta (\psi_\alpha).$$ There are also least-squares versions of this idea [@Boag94; @Tsukerman-JCP10].

Many illustrative examples are given in [@Tsukerman-book07; @Tsukerman05; @Tsukerman06]. Here we mention just one of them, closely related to the construction of FLAME schemes in [Section \[sec:FLAME-slab\]]{}. For the 2D Helmholtz equation, one may consider a Trefftz basis set of eight plane waves traveling at the angles $\phi_0 + m \pi/4$ ($m = 0,1, \ldots, 7$), where $\phi_0$ is a given angle; practical choices are $\phi_0 = 0$ or $\phi_0 = \pi / 8$. Evaluating these plane waves over a standard $3 \times 3$ grid “molecule,” one obtains an $8 \times 9$ matrix $N^T$ whose null vector is the FLAME scheme. The result for $\phi_0 = 0$ is a nine-point ($3 \times 3$) order-six scheme [@Tsukerman06]. For $\phi_0 = \pi/8$, one arrives at a scheme derived by Babuška *et al*. in 1995 [@Babuska-Ihlenburg95] from very different considerations.

Trefftz Homogenization of Electromagnetic Structures {#sec:Trefftz-homogenization}
====================================================

We consider Trefftz-based homogenization of electromagnetic periodic structures (photonic crystals and metamaterials). The general description of the problem in this section follows [@Tsukerman-PLA17; @Tsukerman-Markel14] closely; but our focus here is on *Trefftz approximation*, the importance of 