We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

翻译：本文研究$h$版本有限元方法（通过减小网格尺寸$h$同时保持多项式阶数$p$恒定来提高精度）应用于亥姆霍兹方程的问题。尽管"当波数$k$增加时，$h$必须以多快的速度减小才能维持精度"这一问题自20世纪90年代以来得到了深入研究，但现有严格的波数显式分析均未考虑几何逼近的影响。本文证明：对于使用直边单元求解的非俘获问题，几何误差的量级为$kh$，当$k$较大时，该误差小于污染误差$k(kh)^{2p}$；这一结论随后通过数值实验得到验证。更一般地，我们证明：即使对于存在强俘获效应的问题，采用四次（二维）或五次（三维）多项式与等参单元，也能确保在大多数大波数条件下几何误差小于污染误差。