Finite element method for a nonlinear perfectly matched layer Helmholtz equation with high wave number. (English) Zbl 1502.65204
The authors set up a numerical algorithm in order to solve an exterior problem attached to a nonlinear high order wave number Helmholtz equation (NLH) along with Sommerfeld radiation condition at infinity. They use the perfectly matched layer (PML) boundary technique. The authors’ new idea is to introduce Newton’s sequences of approximate linearized problems to the continuous NLH problem with PML and its FEM, respectively. Then, they establish the well-posedness of both the NLH system and its linear finite element approximation; achieve the convergence of the two sequences and the pre-asymptotic error estimates between them. Some numerical experiments are carried out in order to verify the accuracy of the FEM and to show that the pollution error is greatly reduced by applying the continuous interior penalty FEM.
Reviewer: Calin Ioan Gheorghiu (Cluj-Napoca)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 78A45 | Diffraction, scattering |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
nonlinear Helmholtz equation; Kerr medium; high wave number; perfectly matched layer; Newton’s method; finite element method; preasymptotic error estimates; convergenceSoftware:
DLMFReferences:
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