A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM). Wave-number-explicit stability and regularity estimates and the exponential convergence are proved for the nonlinear truncated PML problem. Preasymptotic error estimates are obtained for the FEM, where the logarithmic factors in h required by the previous results for the NLH with impedance boundary condition are removed in the case of two dimensions. Moreover, local quadratic convergences of the Newton's methods are derived for both the NLH with PML and its FEM. Numerical examples are presented to verify the accuracy of the FEM, which demonstrate that the pollution errors may be greatly reduced by applying the interior penalty technique with proper penalty parameters to the FEM. The nonlinear phenomenon of optical bistability can be successfully simulated.
翻译:非线性Helmholtz方程式(NLH)具有高波数和Sommerfeld辐射条件,近似于完全匹配的层(PML)技术,然后由线性有限元素法(FEM)分离。波序数字显示的稳定性和规律性估计值以及指数趋同,对于非线性脱轨的PML问题,可以证明是非线性脱轨的PML问题。对FEM来说,测得的误差估计值为PLH先前结果中带有阻塞性边界条件的对数系数(h),在两个维度的情况中可以去除。此外,牛顿方法的局部二次方位趋同为NLH与PML及其FEM均取出。提供了数字示例,以核实FEM的准确性,这表明通过对FEM应用有适当惩罚参数的内限技术,污染误差可能大大降低。光性非线性光学微现象可以成功模拟。