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

Run Jiang, Haijun Wu, Yifeng Xu, Jun Zou

TL;DR

This work develops an adaptive finite element framework for a nonlinear Helmholtz equation with high wave number, incorporating corner singularities and Kerr-type nonlinearity. It extends an elliptic-projection-based analysis to the NLH, deriving wave-number–explicit stability, a singularity decomposition, and preasymptotic a priori estimates, and introduces a modified residual-type a posteriori estimator that remains equivalent to the standard one on the whole mesh. The authors prove convergence and quasi-optimality of the AFEM in the preasymptotic regime and demonstrate substantial pollution reduction using CIPFEM, including accurate simulation of optical bistability with Gaussian incident waves. Numerical results validate theoretical predictions and show that adaptive CIPFEM efficiently resolves both high-frequency oscillations and corner singularities, with practical implications for nonlinear wave scattering in optical media.

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 Babuška 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.

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

TL;DR

This work develops an adaptive finite element framework for a nonlinear Helmholtz equation with high wave number, incorporating corner singularities and Kerr-type nonlinearity. It extends an elliptic-projection-based analysis to the NLH, deriving wave-number–explicit stability, a singularity decomposition, and preasymptotic a priori estimates, and introduces a modified residual-type a posteriori estimator that remains equivalent to the standard one on the whole mesh. The authors prove convergence and quasi-optimality of the AFEM in the preasymptotic regime and demonstrate substantial pollution reduction using CIPFEM, including accurate simulation of optical bistability with Gaussian incident waves. Numerical results validate theoretical predictions and show that adaptive CIPFEM efficiently resolves both high-frequency oscillations and corner singularities, with practical implications for nonlinear wave scattering in optical media.

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 Babuška 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.
Paper Structure (17 sections, 23 theorems, 167 equations, 9 figures)

This paper contains 17 sections, 23 theorems, 167 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An auxiliary linearized problem
  4. Estimates of the NLH problem
  5. A priori estimates of the FEM
  6. A posteriori error estimates
  7. Error estimators
  8. Upper and lower bounds
  9. Equivalent relationships between two error estimators
  10. Convergence and quasi-optimality of the AFEM
  11. Adaptive algorithm
  12. Convergence of the AFEM
  13. Quasi-optimality of the AFEM
  14. Numerical results
  15. CIP Finite element method (CIPFEM)
  16. ...and 2 more sections

Key Result

Lemma 2.1

There exists a unique solution $w_0$ to eq:auxiliary cp with $\psi=0$ satisfying and $w_0$ has a decomposition: $w_0=w_{R}+\sum_{j=1}^NC_k^jS_j$ where $w_R\in H^1_{\Gamma_{\rm Dir}}(\Omega)\cap H^2(\Omega)$, $S_j(x)=\chi(r_j)r_j^{\alpha_j}\sin(\alpha_j\theta_j)$ with $\alpha_j=\frac{\pi}{\omega_j}$, and the constants $C_k^j\in \mathbb{C}$. Moreover, where $M(F,g)=\left\Vert F\right\Vert_{0}+\lef

Figures (9)

  • Figure 5.1: Top: The mesh after 20 adaptive iterations and $\Re(u)$ respectively; Bottom: The local meshes around $(\frac{1}{2},0)$ and $(0,0)$ respectively.
  • Figure 5.2: The relative $H^1$-error versus the number of elements for $k=80, q=20, h_{\mathcal{T}_0}=0.04.$
  • Figure 5.3: The relative $H^1$-errors of AFEM and FEM versus the number of elements for $k=5, q=1$, respectively.
  • Figure 5.4: Left: the scaled energy of the scattered field; Right: error estimators $\eta_\mathcal{T}$ at the point $C$.
  • Figure 5.5: Scattered field patterns (magnitude) of the three solutions marked as the points A, B, and C .
  • ...and 4 more figures

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 40 more