# A roadmap for Generalized Plane Waves and their interpolation properties

## TL;DR

This paper develops a systematic approach for constructing and analyzing Generalized Plane Waves (GPWs) as approximate solutions to variable coefficient PDEs, enhancing wave-based numerical methods like Discontinuous Galerkin.

## Contribution

It introduces a comprehensive roadmap and algorithm for building GPWs with high-order interpolation properties for PDEs beyond the Helmholtz equation.

## Key findings

- Established necessary conditions for high-order interpolation
- Provided an explicit construction algorithm for GPWs
- Extended the framework to new classes of PDEs

## Abstract

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.