A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for the NLH problem, a priori stability and error estimates are established for the FEM on shape regular meshes including the case of locally refined meshes. Then a posteriori upper and lower bounds using a new residual-type error estimator, which is equivalent to the standard one, are derived for the FE solutions to the NLH problem. These a posteriori estimates have confirmed a significant fact that is also valid for the NLH problem, namely the residual-type estimator seriously underestimates the error of the FE solution in the preasymptotic regime, which was first observed by Babu\v{s}ka et al. [Int J Numer Methods Eng 40 (1997)] for a one-dimensional linear problem. Based on the new a posteriori error estimator, both the convergence and the quasi-optimality of the resulting adaptive finite element algorithm are proved the first time for the NLH problem, when the initial mesh size lying in the preasymptotic regime. Finally, numerical examples are presented to validate the theoretical findings and demonstrate that applying the continuous interior penalty (CIP) technique with appropriate penalty parameters can reduce the pollution errors efficiently. In particular, the nonlinear phenomenon of optical bistability with Gaussian incident waves is successfully simulated by the adaptive CIPFEM.

翻译：本文采用线性有限元方法对具有高频和角奇异的非线性亥姆霍兹方程进行离散化。在推导出NLH问题的波数显式稳定性估计及奇性分解后，首先在形状正则网格（包括局部加密网格情形）上建立了有限元解的先验稳定性与误差估计。随后，针对NLH问题的有限元解，利用一种与标准估计子等价的新型残差型误差估计子，推导了后验上界与下界估计。这些后验估计证实了一个对NLH问题同样成立的重要事实：残差型估计子在渐近前区域严重低估有限元解的误差，该现象最早由Babuška等人[Int J Numer Methods Eng 40 (1997)]在一维线性问题中发现。基于新型后验误差估计子，当初始网格尺寸处于渐近前区域时，首次证明了NLH问题自适应有限元算法的收敛性与拟最优性。最后，通过数值算例验证了理论结果，并证明采用适当罚参数的连续内罚技术能有效减少污染误差。特别地，自适应CIPFEM成功模拟了高斯入射波下的光学双稳态非线性现象。