  This work focuses on the study of partial differential equation (PDE) based
basis function for Discontinuous Galerkin methods to solve numerically
wave-related boundary value problems with variable coefficients. To tackle
problems with constant coefficients, wave-based methods have been widely
studied in the literature: they rely on the concept of Trefftz functions, i.e.
local solutions to the governing PDE, using oscillating basis functions rather
than polynomial functions to represent the numerical solution. Generalized
Plane Waves (GPWs) are an alternative developed to tackle problems with
variable coefficients, in which case Trefftz functions are not available. In a
similar way, they incorporate information on the PDE, however they are only
approximate Trefftz functions since they don't solve the governing PDE exactly,
but only an approximated PDE. Considering a new set of PDEs beyond the
Helmholtz equation, we propose to set a roadmap for the construction and study
of local interpolation properties of GPWs. Identifying carefully the various
steps of the process, we provide an algorithm to summarize the construction of
these functions, and establish necessary conditions to obtain high order
interpolation properties of the corresponding basis.
