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

TL;DR

This paper develops an adaptive finite element method for solving a high-frequency nonlinear Helmholtz equation, providing stability, error estimates, and demonstrating effective error control and simulation of optical bistability.

Contribution

It introduces a new a posteriori error estimator and proves convergence and quasi-optimality of the adaptive FEM for the nonlinear Helmholtz problem with high wave numbers.

Findings

A new residual-type a posteriori error estimator is derived.

Adaptive FEM achieves convergence and quasi-optimality in the preasymptotic regime.

Numerical simulations validate the theoretical results and demonstrate effective pollution error reduction.

Abstract

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.