Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers
Abstract
We study the uniaxial perfectly matched layer (UPML) problem for the Helmholtz equation in two and three dimensions and establish wavenumber-explicit stability estimates at high wavenumbers. For the full-space UPML problem, we prove the optimal stability estimate with the inf-sup constant satisfying μ∞eqsim k−1 for sufficiently large k if the absorption function σ∈ W1,∞ is a piecewise-power function of order at least one for which |σ'|L∞({t:0<σ(t)≤δ*})≤ ε*, where ε* and δ* depend only on the PML parameters. Using this stability estimate, we obtain the optimal inf-sup estimate μLeqsim k−1 for the truncated UPML problem. These estimates further yield preasymptotic error estimates for a linear continuous interior penalty finite element method (CIP-FEM). Numerical simulations confirmed the theoretical predictions and demonstrated the effectiveness of CIP-FEM under high-wavenumber conditions.
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BibTeX
@misc{wen2026optimal,
title = {{Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers}},
author = {Silin Wen and Yu Du},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06288},
eprint = {2611.06288},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06288},
}
RIS
TY - UNPB
TI - Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers
AU - Silin Wen
AU - Yu Du
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06288
UR - https://arxiv.org/abs/2611.06288
AB - We study the uniaxial perfectly matched layer (UPML) problem for the Helmholtz equation in two and three dimensions and establish wavenumber-explicit stability estimates at high wavenumbers. For the full-space UPML problem, we prove the optimal stability estimate with the inf-sup constant satisfying μ_∞eqsim k^−1 for sufficiently large k if the absorption function σ∈ W^1,∞ is a piecewise-power function of order at least one for which |σ'|_L^∞({t:0<σ(t)≤δ_*})≤ ε_*, where ε_* and δ_* depend only on the PML parameters. Using this stability estimate, we obtain the optimal inf-sup estimate μ_Leqsim k^−1 for the truncated UPML problem. These estimates further yield preasymptotic error estimates for a linear continuous interior penalty finite element method (CIP-FEM). Numerical simulations confirmed the theoretical predictions and demonstrated the effectiveness of CIP-FEM under high-wavenumber conditions.
ER -