AI 中文总结
研究二维亥姆霍兹散射问题有限元逼近，基于拉伸格林核估计，建立UPML问题显式稳定性及误差指数衰减估计，据此制定CIP-FEM方法，经数值实验验证其在高频区有效性，获波数显式预渐近误差估计。
AI 中文摘要
本文针对具有单轴完全匹配层（UPML）截断的二维亥姆霍兹散射问题的有限元逼近，开展了波数显式稳定性和预渐近误差分析。该分析基于与笛卡尔复坐标变换相关的拉伸格林核的直接估计。建立了截断UPML问题的显式稳定性，证明了在自然的k加权H^1范数下，截断UPML公式的下-上常数为μ_L = O(k^-1)，还得到了PML截断误差的指数衰减估计。基于此稳定性估计，在截断UPML域上制定了线性连续内部罚有限元方法（CIP-FEM）。分析的关键是建立分段H^{1+s}正则性结果（对于任何0<s<1），以考虑PML界面上的系数跳跃和笛卡尔角几何引起的奇异性。这种分数正则性足以推导波数显式预渐近误差估计。数值实验证实了理论预测，并说明了UPML CIP-FEM在高频区域的有效性。
英文摘要
This paper develops a wavenumber-explicit stability and preasymptotic error analysis for finite element approximation of the two-dimensional Helmholtz scattering problem with a uniaxial perfectly matched layer (UPML) truncation. The analysis is based on direct estimates of the stretched Green kernel associated with the Cartesian complex coordinate transformation. We establish explicit stability for the truncated UPML problem. In particular, we prove that the inf-sup constant of the truncated UPML formulation is $μ_L = O(k^{-1})$ in a natural $k$-weighted $H^1$ norm. As a consequence, we also obtain an exponential decay estimate for the PML truncation error. Based on this stability estimate, we formulate a linear continuous interior penalty finite element method (CIP-FEM) on the truncated UPML domain. A key ingredient of the analysis is to establish a piecewise $H^{1+s}$-regularity result (for any $0<s<1$), which accounts for coefficient jumps across PML interfaces and singularities induced by Cartesian corner geometries. This fractional regularity is sufficient to derive wavenumber-explicit preasymptotic error estimates. Numerical experiments confirm the theoretical predictions and illustrate the effectiveness of the UPML CIP-FEM in the high-frequency regime.
Comments30 pages, 5 figures
