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<p>For the nonlinear Helmholtz equation (NLH), this paper extends several results from [<CitationRef CitationID="CR10">10</CitationRef>], which investigates wave-number-explicit regularity and pre-asymptotic error estimates for the linear finite element method (FEM). The new aspects lie on the transition to more general nonlinearities and a modified Dirichlet-to-Neumann (DtN) boundary condition used for truncating the perfectly matched layer (PML). Wave-number-explicit stability and regularity estimates and convergence are proved for the PML problem with modified DtN boundary condition. Preasymptotic error estimates are obtained for the FEM and local linear convergence of the modified Newton’s method is derived for both the NLH and its FEM.</p>

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Finite Element Method for Helmholtz Equations with Compactly Supported Nonlinearity

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Silin Wen,
  • Yu Du

摘要

For the nonlinear Helmholtz equation (NLH), this paper extends several results from [10], which investigates wave-number-explicit regularity and pre-asymptotic error estimates for the linear finite element method (FEM). The new aspects lie on the transition to more general nonlinearities and a modified Dirichlet-to-Neumann (DtN) boundary condition used for truncating the perfectly matched layer (PML). Wave-number-explicit stability and regularity estimates and convergence are proved for the PML problem with modified DtN boundary condition. Preasymptotic error estimates are obtained for the FEM and local linear convergence of the modified Newton’s method is derived for both the NLH and its FEM.