广义平面波及其插值性质
A roadmap for Generalized Plane Waves and their interpolation properties
摘要：
本文主要研究了基于偏微分方程的间断Galerkin方法的基函数在变系数波动边值问题数值求解中的应用。为了解决常系数问题，基于波的方法在文献中得到了广泛的研究：它们依赖于Trefftz函数的概念，即控制PDE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）是为解决变系数问题而开发的一种替代方法，在这种情况下，Trefftz函数不可用。以类似的方式，它们包含PDE的信息，但是它们只是近似的Trefftz函数，因为它们没有精确地求解控制PDE，而只是近似的PDE。考虑到Helmholtz方程之外的一组新的偏微分方程，我们建议为GPW的局部插值性质的构造和研究设定一个路线图。仔细识别的过程中的各个步骤，我们提供了一个算法来总结这些函数的建设，并建立必要条件，以获得相应的基的高阶插值性质。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
