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：我们考虑有限元法的 $h$ 版本（即通过减小网格宽度 $h$ 同时保持多项式阶数 $p$ 不变来提高精度）应用于亥姆霍兹方程。尽管自 20 世纪 90 年代以来，“为保持精度，$h$ 必须随波数 $k$ 的增加以多快的速度减小？”这一问题已被深入研究，但现有严格的波数显式分析均未考虑几何近似的影响。本文证明，对于使用直线单元求解的非捕获问题，几何误差为 $kh$ 阶，当 $k$ 较大时，该误差小于污染误差 $k(kh)^{2p}$；数值实验进一步验证了这一结论。更一般地，我们证明即使对于强捕获问题，在二维中使用四阶多项式、在三维中使用五阶多项式并结合等参单元，也能确保在大多数大波数下几何误差小于污染误差。