Generalized Plane Waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e. they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: they are only approximate solutions. They lead to high order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs, amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the pre-asymptotic regime, which will be tamed by avoiding high degree polynomials within an exponential. The new ansatz is introduces higher order terms in the amplitude rather than the phase of a plane wave as was initially proposed. The new functions' construction and the study of their interpolation properties are guided by the roadmap proposed in [16]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially-varying wavenumber. The extension to a range of operators allowing for anisotropy in the first and second order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions.

中文翻译：

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基于幅度的广义平面波：二维标量方程的新准Trefftz函数

引入广义平面波 (GPW) 以利用 Trefftz 方法解决由可变系数方程建模的问题。尽管 GPW 不满足 Trefftz 特性，即它们不是控制方程的精确解，但它们满足准 Trefftz 特性：它们只是近似解。它们导致了高阶数值方法，而这种准 Trefftz 特性对于它们的数值分析至关重要。目前的工作介绍了一个新的基于幅度的 GPW 系列。动机在于在渐近状态下基于相位的 GPW 近似的不良行为，这将通过避免指数内的高次多项式来解决。新的 ansatz 引入了振幅中的高阶项，而不是最初提出的平面波的相位。新函数的构建及其插值特性的研究由 [16] 中提出的路线图指导。为了清楚起见，首先关注具有空间变化波数的二维亥姆霍兹方程。允许一阶和二阶项中的各向异性的一系列算子的扩展如下。数值模拟说明了新的准 Trefftz 函数的理论研究。