Finite Element Method for a Nonlinear PML Helmholtz Equation with High Wave Number

TL;DR

This paper develops a finite element method for solving a high wave number nonlinear Helmholtz equation with PML, providing stability, convergence, and error estimates, and demonstrating its effectiveness through numerical examples.

Contribution

It introduces a wave-number-explicit analysis for a nonlinear PML Helmholtz problem, including error estimates and quadratic convergence of Newton's method, with improved accuracy for 2D cases.

Findings

Exponential convergence of the FEM for the nonlinear PML Helmholtz problem.

Preasymptotic error estimates without logarithmic factors in 2D.

Numerical verification of reduced pollution errors and optical bistability simulation.

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.