Optimal stability estimates and preasymptotic error analysis of the CIP-FEM for the Helmholtz equation with uniaxial perfectly matched layers

arXiv:2611.06288v1 Announce Type: new 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 $\mu_\infty\eqsim k^{-1}$ for sufficiently large $k$ if the absorption function $\sigma\in W^{1,\infty}$ is a piecewise-power function of order at least one for which $\|\sigma'\|_{L^\infty(\{t:0<\sigma(t)\leq\delta_*\})}\leq \varepsilon_*$, where $\varepsilon_*$ and $\delta_*$ depend only on the PML parameters. Using this stability estimate, we obtain the optimal inf-sup estimate $\mu_L\eqsim k^{-1}$ 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.