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Journal of Scientific Computing

, Volume 76, Issue 3, pp 1813–1838 | Cite as

An Adaptive Finite Element Method for the Diffraction Grating Problem with PML and Few-Mode DtN Truncations

  • Weiqi Zhou
  • Haijun WuEmail author
Article
  • 174 Downloads

Abstract

The diffraction grating problem is modeled by a Helmholtz equation with PML boundary conditions. The PML is truncated by some few-mode Dirichlet to Neumann boundary conditions so that those Fourier modes that cannot be well absorbed by the PML pass through without reflections. Convergence of the truncated PML solution is proved, whose rate is exponential with respect to the PML parameters and uniform with respect to all modes. An a posteriori error estimate is derived for the finite element discretization. The a posteriori error estimate consists of two parts, the finite element discretization error and the PML truncation error which decays exponentially with respect to the PML parameters and uniformly with respect to all modes. Based on the a posteriori error control, a finite element adaptive strategy is established for the diffraction grating problem, such that the PML parameters are determined through the PML truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive algorithm.

Keywords

Adaptivity Perfectly matched layer Few-mode DtN operator A posteriori error estimates Diffraction gratings 

Mathematics Subject Classification

65N12 65N15 65N30 78A40 

Notes

Acknowledgements

The authors would like to thank Professor Zhiming Chen for suggesting this topic of research and thank the anonymous referees for their detailed comments and suggestions that improved the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina

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