@Tsukerman06; @Tsukerman-PBG08].

It is not our intention to review all, or even some, of these Trefftz-oriented methods; several good reviews are already available: [@Deckers-wave-method-overview2014; @Qin-Trefftz-FEM2005] and especially [@Hiptmair2016]. Rather, our focus is on one question central in these methods: *why are Trefftz approximations so effective*?

A simplified intuitive picture is shown in [Fig. \[fig:test-waves\]]{}, left panel. Several incident waves, schematically indicated with solid arrows, are impinging on an object (in general, physically inhomogeneous) and give rise to the respective total fields inside and to scattered fields outside that object. For visual clarity, only the incident waves are sketched in the figure, and their number is limited to three.

The total fields inside the scatterer, by definition, form a Trefftz set. One may view it as a “database” or “training set,” which can be precomputed and then used to approximate the field induced by another wave, indicated with a dashed arrow and a question mark in [Fig. \[fig:test-waves\]]{}. This approximation of one physically meaningful solution by other physically meaningful solutions (as opposed to, say, generic polynomials) certainly makes intuitive sense but is not trivial from the mathematical perspective.

\[fig:test-waves\] ![Several incident waves (schematically indicated with solid arrows) give rise to the 