Finite Element Method for a Nonlinear Perfectly Matched Layer Helmholtz Equation with High Wave Number
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DOI10.1137/21M1459381zbMATH Open1502.65204arXiv2207.04685MaRDI QIDQ5043047
Author name not available (Why is that?)
Publication date: 27 October 2022
Published in: (Search for Journal in Brave)
Abstract: 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.
Full work available at URL: https://arxiv.org/abs/2207.04685
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