We study the acoustic Helmholtz equation with impedance boundary conditions formulated in terms of velocity, and analyze the stability and convergence properties of lowest-order Raviart-Thomas finite element discretizations. We focus on the high-wavenumber regime, where such discretizations suffer from the so-called “pollution effect”, and lack stability unless the mesh is sufficiently refined. We provide wavenumber-explicit mesh refinement conditions to ensure the well-posedness and stability of discrete scheme, as well as wavenumber-explicit error estimates. Our key result is that the condition “\(k^2 h\) is sufficiently small”, where k and h respectively denote the wavenumber and the mesh size, is sufficient to ensure the stability of the scheme. We also present numerical experiments that illustrate the theory and show that the derived stability condition is actually necessary.

中文翻译：

---

高波数声亥姆霍兹问题的混合有限元离散化

我们研究了以速度表示的具有阻抗边界条件的声学Helmholtz方程，并分析了最低阶Raviart-Thomas有限元离散化的稳定性和收敛性。我们将重点放在高波数体制上，在这种体制下，离散化会受到所谓的“污染效应”的影响，除非网格得到足够细化，否则其缺乏稳定性。我们提供了波数显式网格细化条件，以确保离散方案的适定性和稳定性，以及波数显式误差估计。我们的关键结果是条件“ \（k ^ 2 h \）足够小”，其中k和h分别表示波数和网格尺寸，足以确保方案的稳定性。我们还提供了数值实验来说明该理论，并表明导出的稳定条件实际上是必要的。