Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions, but rather local approximate solutions to the governing PDE. In order to develop quasi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation this work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. The polynomial basis shows significant promise as it does not suffer from the ill-conditioning issue inherent to wave-like bases.

中文翻译：

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三维对流亥姆霍兹方程的三种准 Trefftz 函数：构造和近似属性


Trefftz 方法是用于近似边界和/或初始值问题解的数值方法。它们是具有特定测试和试验函数的 Galerkin 方法，用于局部求解控制偏微分方程 （PDE）。此属性称为 Trefftz 属性。引入准 Trefftz 方法是为了利用 Trefftz 方法的优势来解决由可变系数偏微分方程控制的问题，通过将 Trefftz 属性放宽为所谓的准 Trefftz 属性：测试和试验函数不是精确解，而是控制偏微分方程的局部近似解。为了开发由对流亥姆霍兹方程控制的气动声学问题的准 Trefftz 方法，这项工作解决了准 Trefftz 函数的三类的定义、构造和近似性质的问题：两类基于平面波解的泛化，一类多项式。多项式基显示出巨大的前景，因为它没有波状基所固有的不良条件问题。