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.

这项工作的重点是研究基于偏微分方程 (PDE) 的基函数，用于不连续伽辽金方法以数值求解具有可变系数的波浪相关边界值问题。为了解决常数系数问题，文献中广泛研究了基于波的方法：它们依赖于 Trefftz 函数的概念，即控制偏微分方程的局部解，使用振荡基函数而不是多项式函数来表示数值解。广义平面波 (GPW) 是为解决可变系数问题而开发的替代方法，在这种情况下，Trefftz 函数不可用。以类似的方式，它们包含关于 PDE 的信息，但是它们只是近似的 Trefftz 函数，因为它们不能准确地求解控制 PDE，但只是一个近似的 PDE。考虑到 Helmholtz 方程之外的一组新偏微分方程，我们建议为构建和研究 GPW 的局部插值特性制定路线图。仔细识别过程的各个步骤，我们提供了一种算法来总结这些函数的构造，并建立必要条件以获得相应基的高阶插值特性。