In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number $k$, the mesh size $h$ and the polynomial degree $p$ of the form ``$k(kh)^{p+1}$ sufficiently small'' and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in $h$ from the case $p=1$ in [H.~Wu, J.~Zou, \emph{SIAM J.~Numer.~Anal.} 56(3): 1338-1359, 2018] can be removed for $p\geq 2$. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.

翻译：本文研究了具有阻抗边界条件的非线性亥姆霍兹方程在任意固定多项式次数下的有限元方法。在波数$k$、网格尺寸$h$与多项式次数$p$满足形如"$k(kh)^{p+1}$充分小"的分辨率条件以及所谓数据小量假设下，我们证明了有限元解的适定性与误差估计。针对后者，我们证明对于$p\geq 2$的情形，可消除参考文献[H. Wu, J. Zou, \emph{SIAM J. Numer. Anal.} 56(3): 1338-1359, 2018]中$p=1$情形下对$h$的对数依赖性。我们展示了两种不同不动点迭代格式的收敛性。数值实验验证了理论结果，并比较了迭代格式对非线性强度及右端项数据的鲁棒性。