Helmholtz Trefftz 间断伽辽金方法的自适应局部表示

中文摘要 AI 辅助

我们研究了Helmholtz方程Trefftz间断伽辽金离散中局部逼近空间的选择与稳定实现。一个缩放的Cauchy迹内积将平面波和Fourier-Bessel函数置于共同的几何框架中：Fourier-Bessel模态以显式权重正交，而相同的权重决定了等距平面波迹Gram矩阵的循环谱。这分离了振幅缩放与真实的迹秩损失，并为混合平面波-Fourier-Bessel空间导出了精确的最佳逼近恒等式。对于复平面波角，未解析的模态尾部是指数序列，因此传播和倏逝分量可以通过相同的ESPRIT/变量投影过程识别。我们证明了恢复角度的精确恢复和扰动估计，并在标准PWDG拟最优性界下，将这些扰动传递到DG误差。迹-Riesz正交归一化随后与组装算子的图-Riesz归一化分离。数值实验验证了这些恒等式，将稀疏射线场恢复到舍入精度，并在没有预设临界角的情况下解析了传播到倏逝的过渡。

英文摘要

We study the selection and stable realization of local approximation spaces in Trefftz discontinuous Galerkin discretizations of the Helmholtz equation. A scaled Cauchy-trace inner product places plane waves and Fourier-Bessel functions in a common geometry: Fourier-Bessel modes are orthogonal with explicit weights, while the same weights determine the circulant spectrum of an equispaced plane-wave trace Gram matrix. This separates amplitude scaling from genuine trace-rank loss and yields an exact best-approximation identity for mixed plane-wave--Fourier--Bessel spaces. With complex plane-wave angles, the unresolved modal tail is an exponential sequence, so propagating and evanescent components can be identified by the same ESPRIT/variable-projection procedure. We prove exact recovery and a perturbation estimate for the recovered angles and, under the standard PWDG quasi-optimality bound, transfer these perturbations to the DG error. Trace-Riesz orthonormalization is then separated from a graph--Riesz normalization of the assembled operator. Numerical experiments verify the identities, recover sparse ray fields to roundoff, and resolve a propagating-to-evanescent transition without a prescribed critical angle.

发表机构

• Texas A&M University（德克萨斯农工大学）

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