Higher-order finite element methods for the nonlinear Helmholtz equation

TL;DR

This paper analyzes higher-order finite element methods for the nonlinear Helmholtz equation, establishing well-posedness, error estimates, and convergence of iterative schemes under specific resolution conditions, supported by numerical experiments.

Contribution

It provides new error estimates and convergence results for finite element methods with arbitrary polynomial degree for the nonlinear Helmholtz equation, improving upon previous logarithmic dependencies.

Findings

Error estimates valid under a specific resolution condition

Convergence of two fixed-point iteration schemes

Numerical experiments confirming theoretical results

Abstract

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number $k$, the mesh size $h$ and the polynomial degree $p$ of the form ``$k(kh)^{p+1}$ sufficiently small'' and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in $h$ from the case $p=1$ in [H.~Wu, J.~Zou, \emph{SIAM J.~Numer.~Anal.} 56(3): 1338-1359, 2018] can be removed for $p\geq 2$. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.