发表机构
Universidad de Antioquia（安蒂奥基亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分析变传播速度Helmholtz方程的Cartesian PML近似，证明其适定性和指数收敛，并给出非陷获条件，数值实验验证了指数衰减。
AI 中文摘要
我们分析了二维Helmholtz方程$-c^2\Delta u-\omega^2 u=f$的Cartesian完美匹配层（PML）近似，其中传播速度$c$在感兴趣区域内可变，并在圆盘$B_R$外等于常数$c_\infty$。现有的Cartesian PML分析假设背景是均匀的或分段均匀的。我们仅假设$c$满足双侧界，因此涵盖了不连续速度——分层介质和紧致内含物。这是可能的，因为速度仅通过其零阶系数进入PML半双线性形式，主部仅由拉伸轮廓构建。物理问题在该正则性下的适定性由Lippmann-Schwinger方程得出，唯一性由Rellich引理和唯一延拓得出。Morawetz乘子产生关于$c$的尖锐非陷获条件，即对某个$x_0$有$\nabla c(x)\cdot(x-x_0)<c(x)$，在径向情形下简化为Herglotz条件$\frac{d}{d\rho}(\rho/c(\rho))>0$；其失效产生陷获双特征。可变速度对本征谱不可见：在分解出$c^2$后，两个算子相差一个紧支撑乘法，因此均匀Cartesian层的谱分析可直接应用。这得出截断问题对每个$\omega>0$的适定性，稳定性常数独立于层宽，并以$\omega\gamma_M\delta_M$的指数速率在$H^1$中收敛到物理解，其中$\gamma_M$和$\delta_M$分别是吸收强度和层厚。使用低速内含物的数值实验证实了在$\delta$和$\omega$上的指数衰减。
英文摘要
We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation $-c^2Δu-ω^2 u=f$, in which the propagation speed $c$ is variable inside the region of interest and equal to a constant $c_\infty$ outside a disc $B_R$. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of $c$ only a two-sided bound, so that discontinuous speeds -- layered media and compact inclusions -- are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich's lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on $c$, namely $\nabla c(x)\cdot(x-x_0)<c(x)$ for some $x_0$, which reduces in the radial case to the Herglotz condition $\frac{d}{dρ}(ρ/c(ρ))>0$; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out $c^2$, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every $ω>0$, with a stability constant independent of the layer width, and convergence to the physical solution in $H^1$ at a rate exponential in $ωγ_Mδ_M$, where $γ_M$ and $δ_M$ are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both $δ$ and $ω$.
